2.1 General Galois extensions

As mentioned in the Introduction, we will translate fine distribution properties of Frobenius elements in terms of the representation theory of $G=\mathrm {Gal}(L/K)$ and the ramification data of $L/K$ . In this section, we state the precise results spelling out this idea. We refer the reader to [Reference Lagarias and OdlyzkoLO, (5.2)] for the definition of the Artin conductor $A(\chi )$ . Moreover, we let $\mathrm {rd}_L$ be the root discriminant of L, that is

(14) $$ \begin{align} \mathrm{rd}_L = d_L^{\frac 1{[L:{\mathbb{Q}}]}}\,,\\[-14pt]\nonumber \end{align} $$

where we recall that $d_L$ is the absolute value of the absolute discriminant of L. For convenience, we associate to any class function $t:G\rightarrow \mathbb C$ a formal object $L(s,L/K,t)$ for which we define the log derivative by extending the case of Artin L-functions:

(15) $$ \begin{align} \frac{L'(s,L/K,t)}{L(s,L/K,t)}:= \sum_{\chi \in \mathrm{Irr}(G)} {\widehat t(\chi)} \frac{L'(s,L/K,\chi)}{L(s,L/K,\chi)}\,. \\[-14pt]\nonumber\end{align} $$

Accordingly, we define the order of vanishing at some $s_0\in \mathbb C$ as follows:

(16) $$ \begin{align} \mathrm{ord}_{s=s_0} L(s,L/K,t) := \sum_{\chi\in\mathrm{Irr}(G)} {\widehat t(\chi)}\cdot \mathrm{ord}_{s=s_0} L(s,L/K,\chi)\,.\\[-14pt]\nonumber \end{align} $$

Our first main result is the following. We say that the function $E: \mathbb R_+\rightarrow \mathbb C$ admits the limiting distribution $\nu $ if $\nu $ is a probability measure on $\mathbb C$ , such that for any bounded continuous function $f:\mathbb C\rightarrow \mathbb R$ ,

$$ \begin{align*} \lim_{Y\rightarrow \infty} \frac 1Y\int_0^Y f(E(y)) \mathrm{d}y = \int_{\mathbb C} f \mathrm{d} \nu=\mathbb E[f]\,.\\[-14pt]\end{align*} $$

The mean and variance of the associated random variable $Z_\nu $ are defined by

$$ \begin{align*}\mathbb E[Z_\nu] =\mathbb E[\Re(Z_\nu)]+i \mathbb E[\Im(Z_\nu)]; \qquad \mathrm{Var}[Z_\nu] =\mathbb E[|Z_\nu-\mathbb E[Z_\nu]|^2].\\[-14pt]\end{align*} $$

Theorem 2.1. Let $L/K$ be a Galois extension of number fields and fix a class function $t:G\rightarrow \mathbb C$ . Recall that $\beta _L^t$ is defined by (10) and that the class function r is defined by (13). Then, $E(y;L/K,t)$ admits a limiting distribution whose mean is

$$ \begin{align*}\mu_{L/K,t}= -\langle t,r \rangle_G \delta_{\beta^t_L=\frac 12}-\frac 1{\beta_L^t}\mathrm{ord}_{s=\beta_L^t}L(s,L/K,t),\end{align*} $$

where $\delta $ is Kronecker’s delta and whose variance is

(17) $$ \begin{align} \sigma^2_{L/K,t}\ll {\lVert{t}\rVert}_1^2\log (d_L+2) \min(M_{L},\log (d_L+2)),\\[-14pt]\nonumber \end{align} $$

where $M_{L} := \max \big \{ \mathrm {ord}_{s=\rho } \zeta _L(s) : \Re (\rho )=\beta _L^t, 0<|\Im (\rho )|< \log (d_L+2)(\log \log (d_L+2))^2\big \}. $ Inother words, for generic values of y, we have the estimate

(18) $$ \begin{align} E(y;L/K,t)= \mu_{L/K,t}+O({\lVert{t}\rVert}_1 (\log (d_L+2))^{\frac 12} \min(M_{L},\log (d_L+2))^{\frac 12}).\\[-14pt]\nonumber \end{align} $$

Assuming that $L/{\mathbb {Q}}$ is Galois and that AC holds, we have the more preciseFootnote ⁸ bound

(19) $$ \begin{align} \sigma^2_{L/K,t}\ll (m_{L})^2 \log(\mathrm{rd}_L+2)\sum_{ \chi\in \mathrm{Irr}(G^+) } \chi(1)|\widehat{t^+}(\chi)|^2.\\[-15pt]\nonumber \end{align} $$

Here, denoting $T_{L}:= \log (\mathrm {rd}_L+2) \sum _{\substack {\chi \in \mathrm {Irr}(G^+) }} \chi (1) $ ,

$$ \begin{align*}m_{L} := \max \Big \{ \mathrm{ord}_{s=\rho} \Big(\prod_{\substack{\chi \in \mathrm{supp}(\widehat{t^+}) }} L(s,L/{\mathbb{Q}},\chi)\Big) : \Re(\rho)=\beta_L^t,0<|\Im(\rho)|< (T_L\log T_L)^2\Big\}.\end{align*} $$

Assuming, moreover, GRH andFootnote ⁹ LI, we have the lower bound

(20) $$ \begin{align} \sigma^2_{L/K,t}\gg \sum_{ \chi\in \mathrm{Irr}(G^+) } \chi(1)|\widehat{t^+}(\chi)|^2.\\[-15pt]\nonumber \end{align} $$

Finally, assuming, in addition, that each irreducible representation of $G^+=\mathrm {Gal}(L/{\mathbb {Q}})$ of dimension $\geq {\lVert {t^+}\rVert }_2 {\lVert {t^+}\rVert }_1^{-1} (2\#\mathrm { Irr}(G^+))^{-\frac 12}$ satisfies the bound $\max _{1\neq C \in (G^+)^\sharp } |\chi (C)| \leq (1-\eta )\chi (1)$ for some real number $0<\eta <1$ which depends on t and $G^+$ , then we have that

(21) $$ \begin{align} \sigma^2_{L/K,t}\gg \eta\log(\mathrm{rd}_L+2)\sum_{ \chi\in \mathrm{Irr}(G^+) } \chi(1)|\widehat{t^+}(\chi)|^2.\\[-15pt]\nonumber \end{align} $$

If $\widehat t\not \equiv 0$ , then the character sum in (21) satisfies the general bounds

$$ \begin{align*}\frac{{\lVert{t^+}\rVert}_2^3}{{\lVert{t^+}\rVert}_1(\#\mathrm{supp}(\widehat{t^+}))^{\frac 12}} \leq \sum_{ \chi\in \mathrm{Irr}(G^+) } \chi(1)|\widehat{t^+}(\chi)|^2 \leq |G^+|^{\frac 12} {\lVert{t^+}\rVert}_2.\\[-15pt]\end{align*} $$

Remark 2.2. The error term in (18) is significantly sharper than that in (2), as well as the further refinements of Murty–Murty–Saradha ([Reference Murty, Murty and SaradhaMMS, Section 3, in particular Corollary 3.7]) and Bellaïche ([Reference BellaïcheBe1, Théorème 1]). As a matter of comparison, taking $q\geq 3$ , $L={\mathbb {Q}}(\zeta _q)$ , $K={\mathbb {Q}}$ , $t=\phi (q) 1_a$ for some $a\in (\mathbb Z/q\mathbb Z)^\times $ and assuming BM, (18) translates to

$$ \begin{align*}E(y;{\mathbb{Q}}(\zeta_q)/{\mathbb{Q}},t) = \mu_{{\mathbb{Q}}(\zeta_q)/{\mathbb{Q}},t} + O((q\log q)^{\frac 12}),\\[-14pt]\end{align*} $$

which under GRH is of the strength of Montgomery’s conjecture (see [Reference Friedlander and GranvilleFG, Conjecture 1]) on primes in arithmetic progressions (since $E(y;{\mathbb {Q}}(\zeta _q)/{\mathbb {Q}},t) = y{\mathrm {e}}^{\frac y2}\phi (q)(\pi ({\mathrm {e}}^y;q,a)-\phi (q)^{-1}\mathrm { Li}({\mathrm {e}}^y))$ ). This answers a question of Murty–Murty–Saradha [Reference Murty, Murty and SaradhaMMS, Section 3.13] about the ‘true size’ of this error term, at least for generic and large enough values of y. A detailed generalisation of Montgomery’s conjecture with a range of validity and various applications will appear in a forthcoming paper joint with Morrison and Thorner. Comparing this with [Reference BellaïcheBe1, Théorème 1] (which holds for all y), we see that for an extension $L/{\mathbb {Q}}$ with $G=\mathrm {Gal}(L/{\mathbb {Q}})$ , the ratio between the bound on $E(y;L/{\mathbb {Q}},t)$ in loc. cit. and the bound given in (19) (i.e. our $L^2$ bound) is

$$ \begin{align*} \gg y \frac{ \log({\mathrm{e}}^y \mathrm{rd}_L)}{(\log(\mathrm{rd}_L+2))^{\frac 12}}\frac{\sum_{\chi \in \mathrm{Irr}(G)} \chi(1)|\widehat t(\chi)|}{\Big(\sum_{\chi \in \mathrm{Irr}(G)} \chi(1)|\widehat t(\chi)|^2\Big)^{\frac 12} } ,\end{align*} $$

which is clearly large. One can do a similar comparison with [Reference Murty, Murty and SaradhaMMS] for relative extensions.

As for our similar looking estimates (17) and (19), we see that for the extension ${\mathbb {Q}}(\zeta _q)/{\mathbb {Q}}$ , they are of the same quality since $\chi (1)=1$ . However, if, for example, we work with the class function $t_{C_1,C_2}$ in a family where $\mathrm {Gal}(L/{\mathbb {Q}})=S_n$ , then the ratio between (17) and (19) is $\gg \min (|C_1|,|C_2|)$ . As a more extreme example, we will see in Theorem 2.19 that there exist extensions for which the upper bound in (19) is identically zero. As this suggests, to fully understand the fluctuations of $E(y;L/K,t)$ , it is not sufficient to decompose it using the characters of the group G – zeros that are either multiple or common to different characters significantly affect the formula for the variance. To take this into account, we will formulate a transfer principle relating $E(y;L/K,t)$ to $E(y;L/{\mathbb {Q}},t^+)$ in Proposition 3.11 and Corollary 3.17 (see also Remark 3.19).

An interesting consequence of Theorem 2.1 (more precisely of Proposition 3.18, combined with (59) and Lemma 3.2) is that under AC and GRH, the limiting distributions of the functions $y{\mathrm {e}}^{-y/2} (|G^+| \pi ({\mathrm {e}}^y;L/{\mathbb {Q}},\{\mathrm {id}\}) - \mathrm {Li}({\mathrm {e}}^y))$ and $y{\mathrm {e}}^{-y/2} (|G| \pi ({\mathrm {e}}^y;L/K,\{\mathrm {id}\})-\mathrm { Li}({\mathrm {e}}^y))$ have the same variance, however, the mean of the first is always less than or equal to that of the second.

We now discuss applications of our ideas. We first focus on Linnik-type problems for Frobenius sets. Lagarias et al. [Reference Lagarias, Montgomery and OdlyzkoLMO] showed that under GRH, and for a given extension of number fields $L/K$ and for any conjugacy class $C\subset \mathrm {Gal}(L/K)$ , there exists an unramified prime ideal $\mathfrak p \triangleleft \mathcal O_K$ of norm $ \ll (\log (d_L+2))^2 $ for which $\mathrm {Frob}_{\mathfrak p}=C$ . Bellaïche [Reference BellaïcheBe2, Proposition 1] has shown that in the case $C=\{ \mathrm {id} \}$ , the exponent $2$ in this bound is best possible (see also [Reference FioriFio]). However, K. Murty conjectured [Reference MurtyMu2, Conjecture 2.2] that under GRH, we have the general bound $ \ll (\log (d_L+2))^2/|C| $ , which decreases when $|C|$ grows. This conjecture was motivated by [Reference MurtyMu2, Theorem 3.1], which shows that under the Riemann hypothesis and Artin’s conjecture for every $L(s,L/K,\chi )$ with $\chi \in \mathrm {Irr}(G)$ , we have the bound $ \ll [K:{\mathbb {Q}}]^2([L:K]\log [L:K] +\log (d_L+2))^2/|C| $ (the additional factors here come, in part, from the contribution of ramified prime ideals). In the case where $K={\mathbb {Q}}$ and the condition that $\mathfrak p \triangleleft \mathcal O_K$ is unramified is dropped, Bellaïche [Reference BellaïcheBe1, Théorème 3] showed that under AC and GRH, one can obtain sharper results in several important families. More precisely, one can obtain a bound in terms of the invariant

(22) $$ \begin{align} \varphi_G(C) := \inf\Big\{ \frac{\lambda(t)}{\widehat t(1)} \,\,\Big|\,\, t\colon G\rightarrow \mathbb R ;\,\widehat t(1)>0;\, \big(t(g) > 0\Rightarrow g\in C\big)\Big\} \ll \frac{|G|}{ |C|^{\frac 12}} \end{align} $$

(the bound follows from taking $t=1_C$ and applying Cauchy–Schwarz). We are now ready to state our bounds on the least unramified prime ideal in a given Frobenius set.

Theorem 2.3. Let $L/K$ be a Galois extension of number fields, and assume that the Riemann hypothesis and Artin’s conjecture hold for each $L(s,L/K,\chi )$ with $\chi \in \mathrm {Irr}(\mathrm { Gal}(L/K))$ . Then, Murty’s conjecture holds. In other words, for any conjugacy class $C\subset G$ , there exists an unramified prime ideal $\mathfrak p \triangleleft \mathcal O_K$ for which $\mathrm {Frob}_{\mathfrak p} = C$ and

(23) $$ \begin{align} \mathcal N\mathfrak p\ll \frac{(\log (d_L+2))^2}{|C|}.\\[-7pt]\nonumber \end{align} $$

More precisely, taking into account Bellaïche’s refinement,Footnote ¹⁰ for any class function $t:G \rightarrow \mathbb R$ such that $\widehat t(1)> 0$ , there exists an unramified prime ideal $\mathfrak p \triangleleft \mathcal O_K$ for which $t(\mathrm { Frob}_{\mathfrak p})>0$ and

(24) $$ \begin{align} \mathcal N\mathfrak p\ll \Big( \frac{\lambda(t)}{\widehat t(1)}\log(\mathrm{rd}_L+2) [K:{\mathbb{Q}}] \Big)^2.\\[-7pt]\nonumber \end{align} $$

If in addition $L/{\mathbb {Q}}$ is Galois and AC holds, then there exists an unramified prime ideal $\mathfrak p \triangleleft \mathcal O_K$ for which $\mathrm {Frob}_{\mathfrak p} = C$ and

(25) $$ \begin{align} \mathcal N\mathfrak p \ll \frac{ (\log (d_L+2) )^2}{|C^+|}+ \frac{ (\log (d_K+2) )^{\frac 43} |G|^{\frac 43}}{|C|^{\frac 23}}\\[-7pt]\nonumber \end{align} $$

(note that the second term in (25) is $\ll (\log (d_L+2))^{\frac 43} |C|^{-\frac 23} \ll (\log (d_L+2))^{2} (|G^+||C|)^{-\frac 23} $ ). Finally, under the same hypotheses and incorporating Bellaïche’s refinement, we obtain that for any class function $t:G \rightarrow \mathbb R$ , such that $\widehat t(1)>|G|^{-100}\sup |t|$ , there exists an unramified prime ideal $\mathfrak p \triangleleft \mathcal O_K$ for which $t(\mathrm {Frob}_{\mathfrak p})>0$ and

(26) $$ \begin{align} \mathcal N\mathfrak p &\ll \Big( \frac{\lambda(t^+)}{\widehat {t}(1)}\log(\mathrm{rd}_L+2) \Big)^2 + \frac{\lambda(t)}{\widehat t(1)}[K:{\mathbb{Q}}] \log(\mathrm{rd}_L+2) \\&\qquad\qquad +\sum_{\substack{2\leq \ell \ll \log\log d_L \nonumber\\ \mu^2(\ell)=1}} \Big(\Big( \frac{|\langle t,r_\ell \rangle_G|}{\widehat t(1)} \Big)^{\frac{\ell}{\ell-1}}+\Big(\frac{\lambda((t(\cdot^\ell) )^+)}{\widehat t(1)}\log(\mathrm{rd}_L+2) \Big)^{\frac{2\ell}{2\ell-1}}\Big).\\[-7pt]\nonumber \end{align} $$

Example 2.4. As an example in which (26) and (25) are significantly sharper than (23) and (24), consider any $S_n$ extension $L/{\mathbb {Q}}$ and $K=L^{\langle (12\cdots n)\rangle }$ . Clearly for $\sigma =(1\,2\cdots n)$ , one has $|\{\sigma \}^+| = (n-1)!, $ and likewise for any k coprime to n, taking $C=\{\sigma ^k\}$ , we have that $|C|=1$ and $|C^+|=(n-1)!$ . Thus, our bound on the unramified prime ideal $\mathfrak p$ of least norm for which $\mathrm {Frob}_{\mathfrak p}=C$ is $ \mathcal N\mathfrak p\ll (\log (d_L+2))^2/n!^{\frac 23} $ . In comparison, the bounds (23) and (24) are both $\asymp (\log (d_L+2))^2$ (see [Reference BellaïcheBe1, Proposition 17]).

More generally, considering the extension $L/L^H$ , where $H \triangleleft S_n$ is a subgroup containing an element $h $ of cycle type $\lambda = (\lambda _1,\dots \lambda _k) \vdash n$ , the bound (25) is

$$ \begin{align*}\ll (\log (d_L+2))^2 \Big( \frac{\prod_{1\leq j \leq n} j^{a_j} a_j!}{n!}+ \frac 1{n!^{\frac 23}}\Big),\\[-9pt] \end{align*} $$

where $a_j = \# \{ i\leq k : \lambda _i=j\}$ .

Using similar arguments as in Theorem 2.3, we obtain a refinement of the Lagarias–Odlyzko–Serre, Murty–Murty–Saradha and Bellaïche bounds on the error term in Chebotarev’s density theorem.

Theorem 2.6. Let $L/K$ be a Galois extension of number fields for which $L/{\mathbb {Q}}$ is Galois, let $t:G\rightarrow \mathbb C$ be a class function and assume AC and GRH. Then for all $x\geq 2$ , we have the bound

(27) $$ \begin{align} \pi(x;L/K,t) - \widehat t(1)\mathrm{Li}(x) &\ll \lambda(t^+)x^{\frac 12} \log ( \mathrm{rd}_L x)\log x\\&\qquad + \sum_{\substack{2\leq \ell \leq 2\log x \nonumber\\ \mu^2(\ell)=1}}(x^{\frac 1\ell} |\langle t,r_\ell\rangle_G|+x^{\frac 1{2\ell}} \lambda((t(\cdot^\ell))^+) \log ( \mathrm{rd}_Lx )\log x).\\[-9pt]\nonumber \end{align} $$

Moreover, the quantity $\lambda ((t(\cdot ^\ell ))^+)$ can be replaced by $[K:{\mathbb {Q}}]\lambda (t(\cdot ^\ell ))$ . In the particular case $t=|G||C|^{-1}1_C$ , where $C\subset G$ is a conjugacy class, the right-hand side of (27) is $\ll ( |G^+||C^+|^{-\frac 12} x^{\frac 12} +|G^+||G|^{\frac 12}|C|^{-1} x^{\frac 14} )\log ( \mathrm {rd}_L x) \log x.$ Here, $G^+:=\mathrm {Gal}(L/{\mathbb {Q}})$ and $C^+$ is defined by (8).

Next, we turn to applications of our results to discrepancies in the distribution of Frobenius elements in conjugacy classes. We will combine Theorem 2.1 with estimates on Artin conductors (see Lemma 4.1) and probabilistic bounds on large deviations of random variables to detect when $\delta (L/K;t)$ (see (11)) is very close to $1$ , conditionally on AC, GRH and BM.

The following statement makes use of the Frobenius–Schur indicator $\varepsilon _2$ of an irreducible character (see (46) and Theorem 3.3): $\varepsilon _2$ takes the value $0$ at a nonreal irreducible character and the value $\pm 1$ at a real irreducible character. The value $+1$ (respectively, $-1$ ) corresponds to representations that are realizable (respectively, not realisable) over $\mathbb R$ .

Theorem 2.7. Let $L/K$ be an extension of number fields, such that $L/{\mathbb {Q}}$ is Galois, and fix $t:G\rightarrow \mathbb R$ a class function, such thatFootnote ¹¹ $\langle t ,r \rangle _G <0$ and $t^+\not \equiv 0$ . Assume that AC, GRH and BM hold. If for some small enough $\eta>0$ , the inequality

(28) $$ \begin{align} -\langle \widehat {t} , \varepsilon_2 \rangle_{\mathrm{Irr}(G)} -2\mathrm{ord}_{s=\frac 12} L(s,L/K,t)> \Big(\eta^{-1} \log (\mathrm{rd}_L+2) \sum_{\chi \in \mathrm{Irr}(G^+)} \chi(1) |\widehat{t^+}(\chi)|^2\Big)^{\frac 12} \end{align} $$

holds, then the fluctuations of $E(y;L/K,t)$ are dominated by a constant term, that is

$$ \begin{align*}\underline{\delta}(L/K;t)> 1-c_1\eta.\end{align*} $$

Under the additional assumption LI, we have the refined bound

$$ \begin{align*}\delta(L/K;t)> 1-\exp(-c_2\eta^{-1}).\end{align*} $$

Finally, if $K={\mathbb {Q}}$ and $|\widehat {t}(\chi )|\in \{0,1\}$ for all $\chi \in \mathrm {Irr}(G)$ , then we also have the upper bound

$$ \begin{align*}\delta(L/{\mathbb{Q}};t) < 1-\exp(-c_3\eta^{-1}).\end{align*} $$

Here, $c_1,c_2,c_3>0$ are absolute constants.

As a partial converse to Theorem 2.7, we show using an effective central limit theorem that up to the factor $\log (\mathrm { rd}_L+2)$ , the condition (28) is also sufficient. Here, the condition LI is required, since we need a lower bound on the variance and an estimate on higher moments.

Theorem 2.8. Let $L/K$ be an extension of number fields, such that $L/{\mathbb {Q}}$ is Galois, and for which AC, GRH $^-$ and LI hold. Fix a class function $t:G\rightarrow \mathbb R$ , such that $\widehat {t^+}\not \equiv 0$ , and let $\eta>0$ be small enough. If the condition

(29) $$ \begin{align} \Big(\langle \widehat {t} , \varepsilon_2 \rangle_{\mathrm{Irr}(G)} +2\mathrm{ord}_{s=\frac 12} L(s,L/K,t) \Big)^2 < \eta^2 \sum_{\chi \in \mathrm{Irr}(G^+)} \chi(1) |\widehat{t^+}(\chi)|^2\, \end{align} $$

is satisfied, then

(30) $$ \begin{align} \delta(L/K;t) - \frac 12 \ll \eta + \frac{{\lVert{t^+}\rVert}_1(\#\mathrm{Irr}(G^+))^{\frac 16}}{{\lVert{t^+}\rVert}_2} \,. \end{align} $$

Assuming further that $ |\langle \widehat t,\varepsilon _2\rangle _{\mathrm {Irr}(G)}+2\mathrm {ord}_{s=\frac 12} L(s,L/K,t)|\geq \eta ^{-\frac 12}, $ then the second error term on the right-hand side of (30) can be deleted.

Example 2.10. Take $K={\mathbb {Q}}$ and $L/{\mathbb {Q}}$ of even degree (so that there is at least one nontrivial real character) and $t=1-r$ , so that $\widehat {t}(\chi )=1_{\chi =1}-\varepsilon _2(\chi )$ . Assuming BM, we have the upper bound

$$ \begin{align*}|\langle \widehat {1-r} , \varepsilon_2 \rangle_{\mathrm{Irr}(G)} +2\mathrm{ord}_{s=\frac 12} L(s,L/{\mathbb{Q}},1-r)| \leq (M_0+1)\sum_{ \substack{1\neq\chi \in \mathrm{Irr}(G) \\ \chi \text{ real} }} 1, \end{align*} $$

and hence, (29) holds whenever

(31) $$ \begin{align} \sum_{ \substack{1\neq \chi \in \mathrm{Irr}(G)\\ \chi \text{ real} }} 1 < (M_0+1)^{-1}\eta \Big(\sum_{ \substack{1\neq \chi \in \mathrm{Irr}(G)\\ \chi \text{ real}}} \chi(1)\Big)^{\frac 12}. \end{align} $$

If this is the case, then thanks to (115), we conclude under AC, GRH $^{-}$ and LI that (see the proof of Theorem 5.10 in which $\mathbb E[X(L/{\mathbb {Q}};1-r)]\in \mathbb Z$ )

$$ \begin{align*}\delta(L/{\mathbb{Q}};1-r) -\frac 12 \ll \eta\,.\end{align*} $$

Moreover, we have the lower bound

(32) $$ \begin{align} \langle \widehat {1-r} , \varepsilon_2 \rangle_{\mathrm{Irr}(G)}&+ 2\sum_{\substack{ \chi \in \mathrm{Irr}(G) }} \widehat{1-r}(\chi)\mathrm{ord}_{s=\frac 12}L(s,L/{\mathbb{Q}},\chi) \nonumber\\ &\qquad \geq \#\{1\neq \chi\in \mathrm{Irr}(G): \chi \text{ real}\}-M_0\#\{ \chi\in \mathrm{Irr}(G): \varepsilon_2(\chi)=-1\}, \end{align} $$

and hence, ifFootnote ¹² $2M_0 \#\{ \chi \in \mathrm {Irr}(G): \varepsilon _2(\chi )=-1\} \leq \#\{ 1\neq \chi \in \mathrm {Irr}(G): \chi \,\mathrm {real}\},$ then the condition (28) holds whenever

(33) $$ \begin{align} \sum_{\substack{ 1\neq \chi \in \mathrm{Irr}(G)\\ \chi \,\mathrm{real} }} 1> \eta^{-\frac 12}\Big( \log (\mathrm{rd}_L+2) \sum_{\substack{1\neq \chi \in \mathrm{Irr}(G)\\ \chi \,\mathrm{real}}} \chi(1) \Big)^{\frac 12}. \end{align} $$

We expect the condition (31) to hold for many extensions, and hence, under AC, GRH $^-$ and LI, $\delta (L/{\mathbb {Q}};1-r)$ is often close to $\tfrac 12$ . Precisely, this holds if $G=G^+$ has a real irreducible representation of degree d and admits $o(\sqrt d)$ irreducible real representations. In the generic case $G=S_n$ , there exists exactly $p(n)\sim {\mathrm {e}}^{\pi \sqrt { \frac {2n}3}}/(4n\sqrt 3)$ (the number of partitions of n) real irreducible representations, one of which has degree $n!^{\frac 12-o(1)}$ (see Theorem 2.15).

Example 2.11. Take $L=\mathbb Q(\zeta _q)$ with $q\geq 3$ odd and squarefree in Example 2.10. Then the inequality (33) holds for all small enough $\eta>0$ satisfying

(34) $$ \begin{align} 2^{\omega(q)} \gg \eta^{-1} \log q\, \end{align} $$

(with an absolute implied constant), a condition which already appeared in [Reference FiorilliFi1]. For general finite abelian groups, the number of real characters is equal to the number of elements of order at most 2, hence, the inequality (33) translates to

$$ \begin{align*}|\{ 1\neq g \in G : g^2 = 1\}| \gg \eta^{-1} \log (\mathrm{rd}_L+2).\end{align*} $$

As a consequence, for $\delta (L/{\mathbb {Q}};1-r)$ to be close to $1$ , it is sufficient that $\mathrm {Gal}(L/{\mathbb {Q}})$ contains a substantial $2$ -torsion subgroup and that $d_L$ is of controlled size. A good example of such an extension is ${\mathbb {Q}}(\sqrt {p_1},\sqrt {p_2},...,\sqrt {p_k})/{\mathbb {Q}}$ (where the $p_i$ ’s are pairwise distinct primes). In this case, (33) holds provided

$$ \begin{align*}2^k \gg \eta^{-1} \sum_{i=1}^k \log p_i.\end{align*} $$

Interestingly, if we put $q:=\prod _{i=1}^kp_i$ , then this is exactly (34). We will see that the inequality (34) plays an explicit role in the statement of Theorems 2.21 and 2.24 (see also Remark 2.23 that discusses the density of integers q, such that (34) holds).

We now derive group theoretic criteria that ensure that (30) holds.

So far, we have shown that the limiting values $1$ or $\frac 12$ are expected for the density $\delta (L/K;t)$ in many natural examples. Taking $t=1-r$ and $K={\mathbb {Q}}$ , one could ask whether $\delta (L/{\mathbb {Q}};1-r)$ can plainly equal those limiting values. The following general result gives an effective negative answer to this question.

These bounds are essentially optimal. The first one is sharp (up to a log factor in the exponent) in the following cases:

• • the dihedral extensions considered in Theorem 2.17,

• • the extension $K_d/{\mathbb {Q}}$ , where $K_d$ is the Hilbert class field of a quadratic field ${\mathbb {Q}}(\sqrt {d})$ (see Theorems 1.1 and 2.18),

• • the abelian extension ${\mathbb {Q}}(\sqrt {p_1},\ldots ,\sqrt {p_m})/{\mathbb {Q}}$ (see Theorem 2.21).

As for (2) of Theorem 2.13, it is sharp in the case of p-cyclotomic extensions (where p is a prime number) as shown in [Reference Fiorilli and MartinFM, (3.20)]. There are also cases where the value of $\delta (L/{\mathbb {Q}};1-r)$ differs significantly from this bound, notably:

• • the case of the radical extensions considered in Theorem 2.19,

• • the case of $S_n$ -extensions (see Theorem 2.15).

The representation theoretic assumptions in Theorem 2.13(2) are essential since in the case where G is a generalised quaternion group (see [Reference BailleulBa1]) or $G=\mathrm {SL}_2(\mathbb F_3)$ , one can have $\delta (L/{\mathbb {Q}};1-r)=\frac 12$ (see Remark 2.25).

Remark 2.14. Note that even in the case where G admits no symplectic character, it would still be possible to have $\delta (L/{\mathbb {Q}};t)=\frac 12$ . However, if one, moreover, assumes that $\langle t,r \rangle _G \neq 0$ (recall (13)), then a lower bound on $|\delta (L/{\mathbb {Q}};t)-\tfrac 12|$ could be deduced from an estimate on

$$ \begin{align*}\log (\mathrm{rd}_L+2)\sum_{\chi\in\mathrm{Irr}(G)}\chi(1)|\widehat{t}(\chi)|^2\,. \end{align*} $$

This could be done by following the lines of the proof of Theorem 2.13.

In the following sections, we focus on the cases where the class function t considered is either $t=1-r$ (see (47)) or $t=t_{C_1,C_2}=|G||C_1|^{-1}1_{C_1} - |G||C_2|^{-1}1_{C_2}$ for distinct conjugacy classes $C_1, C_2$ of G.

2.2 Generic case: $S_n$ -extensions

The case where $\mathrm {Gal}(L/{\mathbb {Q}})=S_n$ is ‘generic’ in the sense that according to many orderings of number fields (see, e.g. [Reference GallagherGa, Reference MalleMal]), $S_n$ is the most common Galois group. In this case, our results rely on the rich and beautiful representation theory of the symmetric group that involves the combinatorics of partitions and tabloids. As an application, we answer positively and quantitatively a question of Ng [Reference NgNg, Section 5.3.5] about whether, for any conjugacy class $C\neq \{\mathrm {id} \} $ , we have $ r(\{ \mathrm {id}\})>r(C)$ , and as a result $\delta (L/{\mathbb {Q}};t_{C, \mathrm {id}} )> \frac 12$ (see (36) below). The exact bound we obtain in (131) is

$$ \begin{align*}r(\{\mathrm{id}\})-r(C) \geq n!^{\frac 12};\end{align*} $$

one can deduce sharper bounds for specific conjugacy classes using bounds on the characters of $S_n$ . Such bounds have been established in the important papers of Roichman [Reference RoichmanRo], Larsen–Shalev [Reference Larsen and ShalevLS] and Féray–Śniady [Reference Féray and ŚniadyFeS]. In our context, we are able to apply Roichman’s bound to obtain estimates for $\delta (L/{\mathbb {Q}};t_{C_1,C_2})$ that take into account the ramification data. This is specific to $S_n$ since the factor $(\log (\mathrm {rd}_L))^{-\frac 12}$ appearing in (35) is not present in Theorem 2.8. This leads to an estimate for Chebyshev’s bias that is superior to that following from Theorem 2.8. The resulting bound shows that the Chebyshev bias dissolves both in the horizontal (i.e. as the size of the root discriminant increases) and the vertical (i.e. as the size of the Galois group increases) limits.

Theorem 2.15. Let $L/K$ be an extension of number fields for which L is Galois over ${\mathbb {Q}}$ . Assume that $G^+=\mathrm {Gal}(L/{\mathbb {Q}})=S_n$ with $n\geq 2$ , and that AC, GRH and LI hold. Fix $\varepsilon>0$ , and let $C_1, C_2$ be distinct elements of $G^\sharp \cup \{0\}$ for which $C_1^+\neq C_2^+$ and $\min (|C_1^+|,|C_2^+|) \leq n!^{1-\frac {4+\varepsilon }{{\mathrm {e}} \log n}}$ . Then, the functions $ E(y;L/K,t_{C_1,C_2}), E(y;L/{\mathbb {Q}},1-r)$ admit limiting distributions whose respective means areFootnote ¹⁴

$$ \begin{align*}\ll \Big( \frac{n! p(n)}{\min(|C_1^+|,|C_2^+|)} \Big) ^{\frac 12}; \hspace{1cm} =p(n)-1,\end{align*} $$

and whose respective variances are

$$ \begin{align*}\ll \frac{n!^{\frac 32}\log (\mathrm{rd}_L) }{ \min(|C_1^+|,|C_2^+|) } ;\hspace{1cm} \asymp \log (\mathrm{rd}_L) (n/{\mathrm{e}})^{n/2}{\mathrm{e}}^{\sqrt{n}}\,. \end{align*} $$

Moreover, the variance of the limiting distribution of $E(y;L/K,t_{C_1,C_2})$ is

$$ \begin{align*}\gg \Big(1-\frac{\log \min(|C_1^+|,|C_2^+|)}{\log n!} \Big) \frac{n!^{\frac 32}\log (\mathrm{rd}_L) }{ \min(|C_1^+|,|C_2^+|)^{\frac 32} p(n)^{\frac 12}},\end{align*} $$

and as a consequence, we have the upper bound

(35) $$ \begin{align} \delta(L/K;t_{C_1,C_2})-\frac 12 \ll \Big(1-\frac{\log \min(|C_1^+|,|C_2^+|)}{\log n!}\Big)^{-\frac 12}\cdot \frac{n!^{-\frac 1{4}} p(n)^{\frac 34} \min(|C_1^+|,|C_2^+|)^{\frac 1{4}}}{ (\log (\mathrm{rd}_L) )^{\frac 12}}\,. \end{align} $$

This estimate is essentially best possible in the sense that specialising to $K={\mathbb {Q}}$ and $C_2=\{\mathrm {id}\}$ , we have, for any conjugacy class $C_1$ , the lower bound

(36) $$ \begin{align} \delta(L/{\mathbb{Q}};t_{C_1,\{\mathrm{id}\}})-\frac 12 \geq c \frac{n!^{-\frac 1{4}} }{ (\log (\mathrm{rd}_L) )^{\frac 12}}, \end{align} $$

where $c>0$ is absolute. Finally,

(37) $$ \begin{align} \delta(L/{\mathbb{Q}};1-r) - \frac 12 \asymp \frac{ n!^{-\frac 14} p(n) {\mathrm{e}}^{-\frac{\sqrt{n}} 2}n^{\frac 18}}{(\log (\mathrm{rd}_L))^{\frac 12}}. \end{align} $$

As a consequence, we can quantify the idea that a ‘random’ Galois extension of the rationals rarely produces a high Chebyshev bias. This is the purpose of the following statement.

Corollary 2.16. For a polynomial $f\in \mathbb Z[T]$ , let $K_f\subset \mathbb C$ denote its splitting field over ${\mathbb {Q}}$ . For fixed integers $n,N\geq 2$ set:

$$ \begin{align*}E_n(N)=\{f\in \mathbb Z[T]\colon f\text{ monic of degree }n\text{ with all its coefficients in }[-N,N]\}\,. \end{align*} $$

The proportion $\eta _{n,N}$ of polynomials $f\in E_n(N)$ , such that

(38) $$ \begin{align} \delta(K_f/{\mathbb{Q}};1-r) - \frac 12 \asymp \frac{ n!^{-\frac 14} p(n) {\mathrm{e}}^{-\frac{\sqrt{n}} 2}n^{\frac 18}}{(\log (\mathrm{rd}_{K_f}))^{\frac 12}} \end{align} $$

satisfies, under AC, GRH and LI for every $K_f/{\mathbb {Q}}$ ,

$$ \begin{align*}\eta_{n,N}\geq 1-O\Big(n^3\frac{\log N}{\sqrt{N}}\Big)\,. \end{align*} $$

For the lower bound of the corollary to make sense, one should first pick a large value of n so that (38) implies that the density $\delta (L/{\mathbb {Q}};1-r)$ is close to $\frac 12$ . Then one selects a large value of N (explicitly, N of size $n^{6+\varepsilon }$ suffices) so that the upper bound of the corollary is small, that is, the proportion of admissible polynomials is close to $1$ .

The proof of the corollary follows easily from combining Theorem 2.15 with Gallagher’s theorem (see, e.g. [Reference KowalskiKow, Theorem 4.2]) that quantifies the fact that, generically, the splitting field over ${\mathbb {Q}}$ of a random monic integral polynomial of degree n has Galois group isomorphic to $S_n$ . Note that Gallagher’s bound has been improved (see, e.g. [Reference DietmannDi]), and therefore, the lower bound in Corollary 2.16 is not best possible (one conjectures that $1-\eta _{n,N}\asymp _n N^{-1}$ ); we will still apply Gallagher’s bound because of its uniformity with respect to n.

2.3 Explicit families

In this section, we discuss our results for some families of supersolvable extensions of number fields (we recall that AC is known for such extensions).

2.3.1 Dihedral extensions

Recall that for $n\geq 1$ , the dihedral group $D_n$ is defined by

(39) $$ \begin{align} D_n:=\langle \sigma,\tau\colon \sigma^n=\tau^2=1,\, \tau\sigma\tau=\sigma^{-1}\rangle\,. \end{align} $$

Dihedral groups have a substantial proportion of elements of order $2$ , and this translates into the existence of many real irreducible characters (set $h=1$ in (45), and note that $D_n$ only has irreducible representations of degree bounded by $2$ and admits no symplectic character). As a consequence, dihedral Galois extensions are natural candidates for extensions that may exhibit extreme biases in the distribution of Frobenius elements. The following result confirms this intuition by using a construction due to Klüners [Reference KlünersKl].

Theorem 2.17. There exists a sequence $(K_\ell /{\mathbb {Q}})_{\ell \geq 7}$ of dihedral extensions indexed by prime numbers $\ell \geq 7$ , such that $\mathrm {Gal}(K_\ell /{\mathbb {Q}})\simeq D_\ell $ and, such that, conditionally on GRH and BM for $K_\ell /{\mathbb {Q}}$ and for the choice $(C_1,C_2)=(\{\tau \sigma ^k\colon 0\leq k\leq \ell -1\},\{\mathrm {id}\})$ , the functions $ E(y;K_\ell /{\mathbb {Q}},t_{C_1,\{\mathrm {id}\}}), E(y;K_\ell /{\mathbb {Q}},1-r)$ admit limiting distributions whose means are both $\gg \ell $ and whose variances are both $\ll \ell \log \ell $ . As a result, the fluctuations of these functions are dominated by a constant term, and one has that

$$ \begin{align*}\min\big(\underline{\delta}(K_\ell/{\mathbb{Q}};t_{C_1,C_2}), \underline{\delta}(K_\ell/{\mathbb{Q}};1-r)\big)\geq 1-O\Big(\frac{\log\ell}\ell\Big)\,. \end{align*} $$

If one additionally assumes LI for $K_\ell /{\mathbb {Q}}$ , then both $\delta (K_\ell /{\mathbb {Q}};1-r)$ and $\delta (K_\ell /{\mathbb {Q}};t_{C_1,C_2})$ exist and one has the refined bounds

$$ \begin{align*}&\exp(-c_1\ell)\leq 1-\delta(K_\ell/{\mathbb{Q}};1-r)\leq \exp\Big(-c_2\frac{\ell}{\log\ell}\Big); \\ &\delta(K_\ell/{\mathbb{Q}};t_{C_1,C_2})\geq 1-\exp\Big(-c_3\frac{\ell}{\log\ell}\Big), \end{align*} $$

where the constants $c_1, c_2, c_3>0$ are absolute.

2.3.3 Radical extensions

In contrast with the two previous families, we now consider extensions of number fields exhibiting this time a moderate Chebyshev bias. We will consider splitting fields $K_{a,p}/{\mathbb {Q}}$ of polynomials $f(X)=X^p-a$ , where p and a are distinct odd prime numbers. To simplify the analysis, we make the extra assumption that $a^{p-1}\not \equiv 1 (\bmod \ p^2)$ , in other words, ‘p is not a Wieferich prime to base a’ (see, e.g. [Reference KatzKat, Sections 1–3] for a nice account on the theory of such prime numbers). Let $G=\mathrm {Gal}(K_{a,p}/{\mathbb {Q}})$ . We have the following group isomorphism:

(40) $$ \begin{align} G\simeq\left\{ \left(\begin{array}{cc} c & d \\ 0 & 1 \end{array}\right) : c\in \mathbb F_p^*, d\in \mathbb F_p \right\}\,.\\[-7pt]\nonumber \end{align} $$

In particular, G is supersolvable (consider the cyclic maximal unipotent subgroup H of G), so that Artin’s conjecture holds for $K_{a,p}/{\mathbb {Q}}$ . In this case, we apply the work of [Reference VivianiVi] and explicitly compute the filtration of inertia and, in particular, obtain an exact formula for the Artin conductor of each irreducible character of G.

Theorem 2.19. Let $a,p$ be primes, such that $a^{p-1}\not \equiv 1 (\bmod \ p^2)$ , and assume GRH and LI for the extension $K_{a,p}/\mathbb Q$ . Let $C_1$ , $C_2$ be distinct conjugacy classes of G. Then the functions $ E(y;K_{a,p}/{\mathbb {Q}},t_{C_1,C_2})$ , $E(y;K_{a,p}/{\mathbb {Q}},1-r)$ admit limiting distributions and if $C_1,C_2\neq \{\mathrm {id}\}$ , then the means are both $\ll 1$ , and the variances are both $\ll p\log (ap)$ and $\gg p\log p$ . If one of $C_1$ or $C_2$ (say $C_2$ ) is the trivial conjugacy class, then the mean of $ E(y;K_{a,p}/{\mathbb {Q}},t_{C_1,\{\mathrm {id}\}})$ is $\asymp p$ and the variance $\asymp p^3 \log (ap)$ . As a result, we have the following estimates.

1. (1) For the class function $t=1-r$ ,

   (41) $$ \begin{align} \delta(K_{a,p}/{\mathbb{Q}};1-r) - \frac 12 \asymp \frac 1{\sqrt{p\log (a p)}}\,. \end{align} $$

2. (2) For $c\in \mathbb F_p^\times \setminus \{1\}$ , let $c^+$ be the conjugacy class of G (recall (40)) consisting of all matrices with first row $(c,d)$ , where d runs over $\mathbb F_p$ . If $C_1$ and $C_2$ are not both of type $c^+$ , thenFootnote ¹⁵

   $$ \begin{align*}\left| \delta(K_{a,p}/{\mathbb{Q}};t_{C_1,C_2})-\frac 12 \right| \asymp \frac 1{\sqrt{p\log(ap)}}\,.\\[-7pt] \end{align*} $$

3. (3) If $C_1=x^+$ and $C_2=y^+$ for distinct $x,y\in \mathbb F_p^\times \setminus \{1\}$ , then $\delta (K_{a,p}/{\mathbb {Q}};t_{x^+,y^+})= \delta (p;x,y)$ , which denotes the density of the classical Chebyshev biasFootnote ¹⁶ for the couple of residue classes $(x,y)$ modulo p.

Finally, in the relative case $K={\mathbb {Q}}(\zeta _p)$ (where $\mathrm {Gal}(K_{a,p}/K)\simeq \mathbb Z/p\mathbb Z$ is the maximal unipotent subgroup of G), for any distinct $d_1,d_2\in \mathbb Z/p\mathbb Z$ , the function $E(y;K_{a,p}/K,t_{\{d_1\},\{d_2\}})$ admits a limiting distribution. The mean is always $0$ , and the variance is $\asymp p^3 \log (ap)$ in the case $d_1d_2=0$ , and $0$ otherwise. If $d_1d_2=0$ , then we have $\delta (K_{a,p}/K;t_{\{d_1\},\{d_2\}})=\tfrac 12$ (if $d_1d_2\neq 0$ , then we have no result on $\delta (K_{a,p}/K;t_{\{d_1\},\{d_2\}})$ ).

By estimating the density of the couples of primes $(a,p)$ , such that p is not Wieferich to base a, we deduce the following statement.

Corollary 2.20. Assume GRH and LI for every $K_{a,p}$ with $a,p$ running over all primes. The proportion of couples of primes $(a,p)$ with $a\leq A$ and $p\leq P$ , such that (41) holds is

$$ \begin{align*}1- O\Big(\frac{\log P(\log A\log\log A)^{\frac 12}}{P} + \frac{P\log A}{A}\Big)\\[-7pt] \end{align*} $$

in the range $A,P\geq 3$ , $ P\log P \leq A \leq {\mathrm {e}}^{P^2/(\log P)^3}$ .

We proceed by considering abelian extensions of number fields.

2.3.4 Iterated quadratic extensions

We first describe the case of a Galois group with a ‘big’ $2$ -torsion subgroup. We see that if the product of ramified primes belongs to a certain subset of $\mathbb {N}$ of density $0$ (see Remark 2.23), then we obtain an extreme Chebyshev bias.

Theorem 2.21. Let $L={\mathbb {Q}}(\sqrt {p_1},...,\sqrt {p_m})$ , where $p_1<p_2<...<p_m$ are distinct odd primes. Let $G=\mathrm {Gal}(L/{\mathbb {Q}})\simeq \{\pm 1\}^m$ and $q:=\prod _i p_i$ . AssumeFootnote ¹⁷ GRH and LI, and let $\varepsilon>0$ be small enough. Then for any $a,b\in G$ , the functions $ E(y;L/{\mathbb {Q}},t_{a,b}) $ and $E(y;L/{\mathbb {Q}},1-r)$ admit limiting distributions of respective means $|G|(\delta _{b=\mathbf {1}}-\delta _{a=\mathbf {1}})$ (where $\mathbf {1}=(1,\ldots ,1)$ ) and $|G|$ , and both of variance $\asymp |G|\log q$ . As a result, for any $a\in G\setminus \{ \mathbf {1}\}$ , we have that

$$ \begin{align*}\delta(L/{\mathbb{Q}};t_{a,\mathbf{1}})= \begin{cases} 1-O(\exp(-c 2^{\omega(q)}/\log q )) &\text{ if } 2^{-\omega(q)}\log q \leq \varepsilon \\ \frac 12+O\Big(\sqrt{2^{\omega(q)}/\log q}\Big) &\text{ if } 2^{-\omega(q)}\log q \geq \varepsilon^{-1} \end{cases} ,\end{align*} $$

where c is some positive absolute constant. The same estimate holds for $\delta (L/{\mathbb {Q}};1-r)$ .

Remark 2.22. The reason we chose $b=\mathbf {1}$ in the second part of the statement is because one can show that $\delta (L/{\mathbb {Q}};t_{a,b})=\frac 12$ as soon as $a\neq \mathbf {1}$ and $b\neq \mathbf {1}$ .

Remark 2.23. It follows from Theorem 2.21 and [Reference TenenbaumTe, Chapter 2, Theorem 6.4] that there exists a subset $S\subseteq \mathbb {N}$ , the natural density of which is given by

$$ \begin{align*}\#S\cap [1,X]=\frac{X}{(\log X)^{1-\frac{1+\log\log 2}{\log 2}+o(1)}} \qquad (X\to \infty)\,, \end{align*} $$

such that for any nontrivial $a\in \mathrm {Gal}({\mathbb {Q}}(\sqrt {p_1},\ldots ,\sqrt {p_m})/{\mathbb {Q}})$ , one has

$$\begin{align*}\delta({\mathbb{Q}}(\sqrt{p_1},\ldots,\sqrt{p_m})/{\mathbb{Q}};t_{a,\mathbf{1}}) =1-o_{q\rightarrow\infty}(1)\qquad (q = p_1 \cdots p_m\in S)\,. \end{align*}$$

2.3.5 Hilbert class fields of quadratic extensions: the relative case

In this section, the setting is as in Section 2.3.2: d is a fundamental discriminant satisfying $|d|>1$ , and $K_d$ denotes the Hilbert class field of the quadratic field ${\mathbb {Q}}(\sqrt {d})$ , therefore, $K_d/{\mathbb {Q}}(\sqrt {d})$ is Galois with group $G\simeq \mathrm {Cl}_d$ . In [Reference NgNg, Section 6.2], Ng studies discrepancies in the distribution of prime ideals according to their class in $\mathrm {Cl}_d$ being either trivial or any fixed nontrivial class. In the next result, we consider two possible choices for $(C_1,C_2)$ ; in the case $(C_1,C_2)=(\{\overline {\mathfrak a}\},\{\overline {1}\})$ , we recover precisely Ng’s Theorem [Reference NgNg, Theorem 6.2.1].

Theorem 2.24. Let d be a fundamental discriminant, and assume that $h(d)>1$ . Let $K_d$ be the Hilbert class field of ${\mathbb {Q}}(\sqrt {d})$ , and assume that GRH holds for the extension $K_d/{\mathbb {Q}}$ . We identify $\mathrm {Gal}(K_d/{\mathbb {Q}}(\sqrt {d}))$ with the class group $\mathrm {Cl}_d$ , and we let $\overline {\mathfrak a}$ be a nontrivial ideal class. Choosing $(C_1,C_2)$ to be either $(\{\overline {\mathfrak a}\},\{\overline {1}\})$ or $(\{\overline {1}\},0)$ , the Frobenius counting function $E(y;K_d/{\mathbb {Q}}(\sqrt {d}),t_{C_1,C_2})$ defined in (9) admits a limiting distribution of mean and variance respectively denoted $\mu _{K_d/{\mathbb {Q}}(\sqrt d)}(C_1,C_2)$ , and $\sigma ^2_{K_d/{\mathbb {Q}}(\sqrt d)}(C_1,C_2)$ , satisfying

$$ \begin{align*} |\mu_{K_d/{\mathbb{Q}}(\sqrt d)}(C_1,C_2)|&\leq 2^{\omega(d)}+4\sum_{1\neq\chi\in\mathrm{Irr}(\mathrm{Cl}_d)}\mathrm{ord}_{s=\frac 12}L(s,K_d/{\mathbb{Q}}(\sqrt{d}),\chi)\,, \\ \sigma^2_{K_d/{\mathbb{Q}}(\sqrt d)}(C_1,C_2)& \gg h(d)\,\,\,(\text{only for the choice } (C_1,C_2)=(\{\overline{1}\},0))\,. \end{align*} $$

Under the extra assumption LI, one has that $|\mu _{K_d/{\mathbb {Q}}(\sqrt d)}(C_1,C_2)|\leq 2^{\omega (d)}$ and $\sigma ^2_{K_d/{\mathbb {Q}}(\sqrt d)}\gg h(d)$ for both choices of $(C_1,C_2)$ , and as a consequence, one deduces

$$ \begin{align*}\delta(K_d/{\mathbb{Q}}(\sqrt{d}),t_{C_1,C_2}\})-\frac 12\ll \frac{2^{\omega(d)}}{\sqrt{h(d)}}\,. \end{align*} $$

For the set of all negative fundamental discriminants, or the family of positive discriminants in Lemma 9.5, the last upper bound of the above result implies that the density $\delta (K_d/{\mathbb {Q}}(\sqrt {d}),t_{C_1,C_2}\})$ approaches $\frac 12$ as d grows.

Remark 2.25. Number field extensions with Galois group $G=\mathrm {SL}_2(\mathbb F_p)$ have the peculiarity that many representations of G are symplectic, and hence, there is potentially a large supply of real zeros. For some Galois extensions $L/{\mathbb {Q}}$ , the existence of real zeros of Artin L-functions has the dramatic effect that $\delta (L/{\mathbb {Q}};1-r) =\tfrac 12$ (see [Reference BailleulBa1] for the impact on the bias of central zeros in the case of generalised quaternion extensions). This is in total contradiction with the usual Chebyshev bias philosophy, which says that ‘primes congruent to quadratic nonresidues are more abundant than primes congruent to quadratic residues’. For $\mathrm {SL}_2(\mathbb F_p)$ -extensions $L/{\mathbb {Q}}$ , one can showFootnote ¹⁸ that $\delta (L/{\mathbb {Q}};1-r)$ takes values between $\tfrac 12$ and $\eta $ for some absolute $\eta <1$ , and one expects that $\tfrac 12 \leq \delta (L/{\mathbb {Q}};1-r) \leq \tfrac 12+c(\log (\mathrm {rd}_L))^{-\frac 12}$ for some absolute $c>0$ . The case $\delta (L/{\mathbb {Q}};1-r)=\tfrac 12$ is achievedFootnote ¹⁹ with extensions for which the Artin root number of every symplectic character is $-1$ (see also [Reference BailleulBa1]).

3.1 Representation theory of finite groups

For completeness and because of its crucial importance in our work, let us first recall some basic representation theory of finite groups. We let $\mathrm {Irr} (G)$ and $G^\sharp $ denote, respectively, the set of irreducible characters and the set of conjugacy classes of the group G. Keeping the notation of Bellaïche [Reference BellaïcheBe1], for a class function $t:G \rightarrow \mathbb C$ and a character $\chi \in \mathrm {Irr}(G)$ , we define the Fourier transform

$$ \begin{align*}\widehat t (\chi):=\langle t,\chi\rangle_G=\frac 1{|G|}\sum_{g\in G} t(g) \overline {\chi(g)} = \sum_{C \in G^\sharp} \frac{|C|}{|G|} t(C) \overline {\chi(C)} \,.\end{align*} $$

Note that if $1_D$ is the characteristic function of a given conjugacy-invariant set $D\subset G$ , then

$$ \begin{align*}\widehat{1_D} (\chi) =\frac 1{|G|}\sum_{\substack{C\in G^\sharp: \\ C \subset D}}|C|\overline{\chi(C)}\,.\end{align*} $$

Lemma 3.1 (Orthogonality relations).

Let G be a finite group. If $g_1,g_2 \in G$ , then

(42) $$ \begin{align} \sum_{\chi \in \mathrm{Irr}(G)} \chi(g_1)\overline{\chi}(g_2) = \begin{cases} \frac{|G|}{|C|} & \text{ if } g_1 \text{ and } g_2 \text{ are both in the same conjugacy class }C,\\ 0 & \text{ otherwise.} \end{cases} \end{align} $$

Moreover, if $\chi ,\psi \in \mathrm {Irr}(G)$ , then

(43) $$ \begin{align} \frac 1{|G|} \sum_{g\in G} \chi(g)\overline{\psi}(g) = \begin{cases} 1 & \text{ if } \chi = \psi,\\ 0 & \text{ otherwise}. \end{cases} \end{align} $$

As a consequence of (43), we have the formula

(44) $$ \begin{align} t = \sum_{\chi \in \mathrm{Irr}(G)} \widehat t (\chi) \chi. \end{align} $$

We will often count elements of order $2$ with characters, using the following Lemma.

Lemma 3.2. Let G be a finite group and $ k\in \mathbb N$ . Then for any $h\in G$ , we have the identity

(45) $$ \begin{align} \#\{g\in G : g^k = h\} = \sum_{\chi \in \mathrm{Irr}(G)} \overline{\varepsilon_k(\chi)} \chi(h), \end{align} $$

(46) $$ \begin{align} \text{where}\qquad \varepsilon_k(\chi):= \frac 1{|G|} \sum_{g\in G}\chi (g^k). \end{align} $$

Proof. Since $r_k(h):=\#\{g\in G : g^k = h\}$ defines a class function on G, we have

$$ \begin{align*}\#\{g\in G : g^k = h\} = \sum_{\chi \in \mathrm{Irr}(G)} \widehat{r_k}(\chi)\chi(h)\,.\end{align*} $$

The proof follows by definition of Fourier coefficients.

The number

$$ \begin{align*}\varepsilon_2(\chi)=\frac{1}{|G|}\sum_{g\in G} \chi(g^2)\end{align*} $$

is called the Frobenius–Schur indicator of $\chi $ and is central in our analysis. If $\chi $ is irreducible, then for $r=r_2$ , one has $\varepsilon _2(\chi )=\widehat {r}(\overline {\chi })=\widehat {r}(\chi ) \in \{ -1,0,1\}$ (see [Reference HuppertHu, Theorem 8.7]), and moreover, each of these three possible values has a precise meaning in terms of the $\mathbb R$ -rationality of $\chi $ and of the underlying representation $\rho $ .

A direct consequence of Lemma 3.2 and Theorem 3.3 is the following formula for the class function r introduced in (13):

(47) $$ \begin{align} r=\sum_{\substack{\chi \in \mathrm{Irr}(G)\\ \chi \,\mathrm{real}}} \varepsilon_2(\chi) \chi. \end{align} $$

Lemma 3.4. Let G be a finite group, and let $t:G \rightarrow \mathbb C$ be a class function. We have the identity

(48) $$ \begin{align} {\lVert{\widehat t}\rVert} _2^2 :=\sum_{\chi \in \mathrm{Irr}(G)} |\widehat t (\chi)|^2 = \sum_{C\in G^\sharp}\frac{|C|}{|G|} |t(C)|^2 = {\lVert{t}\rVert}^2_2. \end{align} $$

In particular, if $C_1,...,C_k \in G^\sharp $ are distinct and $\alpha _1,...,\alpha _k \in \mathbb C$ , then

$$ \begin{align*}\sum_{\chi \in \mathrm{Irr}(G)} |\alpha_1\chi(C_1)+\dots +\alpha_k \chi(C_k) |^2 = |\alpha_1|^2 \frac{|C_1|}{|G|} +\cdots +|\alpha_k|^2 \frac{|C_k|}{|G|}.\end{align*} $$

We will also need a pointwise bound on the Fourier coefficients $\widehat t(\chi )$ .

Lemma 3.5. Let G be a finite group, $t:G \rightarrow \mathbb C$ be a class function and $\chi \in \mathrm {Irr}(G)$ . Then we have the bound

$$ \begin{align*}|\widehat t(\chi)| \leq \chi(1) {\lVert{t}\rVert}_1.\end{align*} $$

From the definition $t^+:=\mathrm {Ind}_G^{G^+} (t)$ , one easily sees that

$$\begin{align*}t^+= \sum_{\chi \in \mathrm{Irr}(G)} \widehat t(\chi) \mathrm{Ind}_G^{G^+} (\chi)\,. \end{align*}$$

Applying Frobenius reciprocity, we can compute the Fourier transform of $t^+$ in terms of that of t.

Lemma 3.6. Let $G^+$ be a finite group, let G be a subgroup and let $t\colon G\rightarrow \mathbb C$ be a class function on G. Then, for any $\chi \in \mathrm {Irr}(G^+)$ , we have the formula

(49) $$ \begin{align} \widehat {t^+}(\chi)=\langle t,\chi|_G\rangle_G\,. \end{align} $$

Proof. Frobenius reciprocity gives that

$$ \begin{align*}\widehat {t^+}(\chi) = \sum_{\eta \in \mathrm{Irr}(G)} \widehat t(\eta) \langle \mathrm{Ind}_G^{G^+} (\eta),\chi \rangle_{G^+} = \sum_{\eta \in \mathrm{Irr}(G)} \widehat t(\eta) \langle \eta,\chi|_{G}\rangle_{G} = \langle t,\chi|_G\rangle_G\,.\\[-42pt] \end{align*} $$

Under certain assumptions, we can also compare the $2$ -norm of $t^+$ in terms of that of t.

Proof of Lemma 3.7

We first expand the norm of $t^+$ :

(50) $$ \begin{align} {\lVert{t^+}\rVert}^2_2 = \frac 1{|G^+|} \sum_{g\in G^+} \Big| \sum_{\substack{aG \in G^+/G \\ a^{-1}ga\in G}} t(a^{-1}ga)\Big|^2 = \frac 1{|G^+|} \sum_{bG \in G^+/G} \sum_{g\in G} \Big| \sum_{\substack{aG \in G^+/G \\ a^{-1}bga\in G}} t(a^{-1}bga)\Big|^2\,. \end{align} $$

Now we prove (1). Using Cauchy–Schwarz, we deduce from (50) that

$$ \begin{align*}{\lVert{t^+}\rVert}^2_2 \leq \frac 1{|G^+|} \sum_{bG \in G^+/G} \sum_{g\in G}\Big( \sum_{\substack{aG \in G^+/G \\ a^{-1}bga\in G}} 1 \Big)\Big(\sum_{\substack{aG \in G^+/G \\ a^{-1}bga\in G}} |t(a^{-1}bga)|^2\Big)\,. \end{align*} $$

We bound the first sum in parentheses trivially, and we exploit the fact that $bg\in aGa^{-1}$ is equivalent to $bg\in G$ for any $a\in G^+$ , since G is a normal subgroup. Therefore, the condition on the left coset $bG$ in the second sum in parentheses imposes $bG$ to be the trivial left coset G. We conclude that

$$ \begin{align*}{\lVert{t^+}\rVert}_2 \leq \frac 1{|G|} \sum_{g\in G}\sum_{\substack{aG \in G^+/G }} |t(a^{-1}ga)|^2 = \frac 1{|G|}|G^+/G| \sum_{g\in G} |t(g)|^2, \end{align*} $$

using the fact that conjugation by any $a\in G^+$ induces a bijection of G. The claimed upper bound follows.

For the lower bound (2), we apply positivity in (50) and deduce that

$$ \begin{align*} {\lVert{t^+}\rVert}^2_2& \geq \frac 1{|G^+|} \sum_{bG \in G^+/G}\sum_{g\in G} \sum_{\substack{aG \in G^+/G \\ a^{-1}bga\in G}} |t(a^{-1}bga)|^2 \\ &=\frac 1{|G^+|} \sum_{g\in G} \sum_{\substack{aG \in G^+/G }} \sum_{\substack{bG = aGa^{-1}} } |t(a^{-1}bga)|^2= {\lVert{t}\rVert}^2_2. \end{align*} $$

We finish this section by computing $r^+$ (recall (47)) and by deducing a consequence which will be useful in showing that, up to ramified primes, $\pi (x;L/K,t)$ is determined by $t^+$ .

Lemma 3.9. Let $G^+$ be a finite group, and let G be a normal subgroup. For any $k\in \mathbb N$ and for any class function $t\colon G^+\rightarrow \mathbb C$ , we define the class function $r_{k,t}\colon G\rightarrow \mathbb C$ by setting

$$ \begin{align*}r_{k,t}(g) := \sum_{\substack{h\in G \\ h^k=g }} t(h).\end{align*} $$

Then we have the following equality of class functions:

$$ \begin{align*}r_{k,t}^+\big|_G = [G^+:G] \cdot r_{k,t}.\end{align*} $$

Proof. First note that if for some $g,h\in G$ and $a\in G^+$ we have that $h^k =a^{-1} g a$ , then $(aha^{-1})^k=g$ . In other words, since for each fixed value of a, we have $aG=Ga$ , there is a bijection between the sets $\{ h\in G: h^k=a^{-1} g a\}$ and $\{ h\in G : h^k= g\}$ . Hence, for any $g\in G$ ,

$$ \begin{align*}r_{k,t}^+\big|_G(g) = \sum_{\substack{aG \in G^+/G \\ a^{-1} g a \in G}}r_{k,t}(a^{-1} g a) = \sum_{h\in G} t(h) \sum_{\substack{aG \in G^+/G \\ a^{-1} g a =h^k}} 1 =\sum_{\substack{aG \in G^+/G }} \sum_{\substack{h\in G \\ h^k=g}} t(a^{-1}ha) .\end{align*} $$

The claim follows since t is a class function on $G^+$ .

Proof. By Frobenius reciprocity and Lemma 3.9, we have that

$$ \begin{align*}\langle t, r_k \rangle_G\cdot [G^+:G]=\langle t, r_k^+\big|_G \rangle_G = \langle t^+, r_k^+ \rangle_{G^+}=0.\\[-32pt]\end{align*} $$

3.2 Explicit formulas and limiting distributions

We fix a Galois extension of number fields $L/K$ and let $G=\mathrm {Gal}(L/K)$ . For a class function $t\colon G \rightarrow \mathbb C$ , we define the following prime ideal counting function:

$$ \begin{align*}\psi(x;L/K,t):=\sum_{\substack{\mathfrak p\triangleleft \mathcal O_K \\ m\geq 1\\ \mathcal N\mathfrak p^m \leq x}} t(\varphi_{\mathfrak p}^m) \log (\mathcal N\mathfrak p),\end{align*} $$

where $\varphi _{\mathfrak p}$ is shorthand for $\mathrm {Frob}_{\mathfrak p}$ , the conjugacy class of a lift (defined up to inertia) of the Frobenius automorphism on the residue field $\mathcal O_L/\mathfrak P$ for some (any) $\mathfrak P\triangleleft \mathcal O_L$ above $\mathfrak p$ , and

(51) $$ \begin{align} t(\varphi_{\mathfrak p}^m) := \frac 1{|I_{\mathfrak p}|} \sum_{ i \in I_{\mathfrak p}} t(\varphi_{\mathfrak p}^m i), \end{align} $$

where $I_{\mathfrak p}$ is the inertia group attached to $\mathfrak p$ and any $\mathfrak P\triangleleft \mathcal O_L$ above $\mathfrak p$ . If $D\subset G$ is conjugacy invariant, then we define $\psi (x;L/K,D):=\psi (x;L/K,1_D)$ , where $1_{ D}$ is the indicator function of D. We also recall the definition (1) of the prime ideal counting function attached to a conjugacy class C of G which we extend in the obvious way to conjugacy invariant sets $D\subset G$ .

Our goal is to express $\pi (x;L/K,t)$ in terms of the zeros of primitive Artin L-functions; this will prevent arithmetic multiplicities from occurring in our formulas. To do so, we will first relate the prime ideal counting functions $\psi (x;L/K,t)$ and $\psi (x;L/{\mathbb {Q}},t^+)$ using the induction property for Artin L-functions.

Proposition 3.11. Let $L/K/M$ be a tower of number fields for which $L/M$ is Galois, let $G=\mathrm {Gal}(L/K)$ and $G^+=\mathrm {Gal}(L/M)$ . For any class function $t\colon G\rightarrow \mathbb C$ , we have the identity

(52) $$ \begin{align} \psi(x;L/K,t) = \psi(x;L/M,t^+).\\[-9pt]\nonumber \end{align} $$

As a consequence, if $D\subset G$ is conjugacy invariant, then

(53) $$ \begin{align} \psi(x;L/K,D)=\frac{|G^+|}{|G|} \sum_{\substack{C\in G^\sharp: \\ C \subset D}} \frac{|C|}{|C^+|}\psi(x;L/M,C^+),\\[-9pt]\nonumber \end{align} $$

where $C^+$ is defined by (8).

Proof of Proposition 3.11

For any $\chi \in \mathrm {Irr}(G)$ , we have the identity (see, e.g. [Reference MartinetMar, Section 4])

(54) $$ \begin{align} L(s,L/K,\chi)= L(s,L/M, \mathrm{Ind}_G^{G+}(\chi)),\\[-9pt]\nonumber \end{align} $$

and hence

(55) $$ \begin{align} \psi(x;L/K,\chi) = -\frac 1{2\pi i}\int_{\Re(s)=2} \frac{L'(s,L/K,\chi)}{L(s,L/K,\chi)} \frac{x^s}{s} \mathrm{d}s = \psi(x;L/M,\mathrm{Ind}_G^{G+}(\chi)).\\[-9pt]\nonumber \end{align} $$

As a consequence,

$$ \begin{align*}\psi(x;L/K,t) = \!\sum_{\chi \in \mathrm{Irr}(G)} \!\widehat {t}(\chi) \psi(x;L/K,\chi)=\!\sum_{\chi \in \mathrm{Irr}(G)} \!\widehat {t}(\chi) \psi(x;L/M,\mathrm{ Ind}_G^{G^+}\chi)=\psi(x;L/M,t^+)\,.\\[-9pt]\end{align*} $$

Now, if $D\subset G$ is conjugacy invariant and $\chi \in \mathrm {Irr}(G^+)$ , then by Lemma 3.6,

$$ \begin{align*}\widehat {1^+_{D}}(\chi) = \widehat{1_{D}}(\chi|_G) = \frac 1{|G|} \sum_{\substack{C\in G^\sharp: \\ C \subset D}} \overline{\chi(C^+)}|C|,\end{align*} $$

since $\chi |_G(C)=\chi (C^+)$ . It follows that

(56) $$ \begin{align} 1^+_{D} = \sum_{\substack{C\in G^\sharp: \\ C \subset D}} \frac{|C|}{|G|}\sum_{\chi \in \mathrm{Irr}(G^+)} \overline{\chi}(C^+)\chi = \sum_{\substack{C\in G^\sharp: \\ C \subset D}} \frac{|C|}{|G|}\frac{|G^+|}{|C^+|} 1_{C^+}\,. \end{align} $$

The proof of (53) follows from combining this with (52) in the form

$$ \begin{align*}\psi(x;L/K,1_D) = \psi(x;L/M,1^+_D).\\[-36pt]\end{align*} $$

In the next lemma, we show that up to ramified primes, the counting function $\pi (x;L/K,t)$ is determined by $t^+$ , rather than by t. Note, however, that $\pi (x;L/K,t)$ and $\pi (x;L/{\mathbb {Q}},t^+)$ are not equal in general.

Lemma 3.13. Let $L/K$ be an extension of number fields, such that $L/{\mathbb {Q}}$ and $K/{\mathbb {Q}}$ are Galois. If $t_1,t_2:G\rightarrow \mathbb C$ are class functions, such that $t_1^+=t_2^+$ , then

$$ \begin{align*}|\pi(x;L/K,t_1)-\pi(x;L/K,t_2)|\leq \sup(|t_1-t_2|) \cdot \#\{ \mathfrak p\triangleleft \mathcal O_K \text{ ramified in }L/K\}.\end{align*} $$

Proof. We let $t:=t_1-t_2$ , so that $t^+\equiv 0$ . For any $\chi \in \mathrm {Irr}(G^+)$ and $\ell \in \mathbb N$ , Lemma 3.9 implies that

$$ \begin{align*}r_{\ell,\chi}^+|_G = [G^+:G] \cdot r_{\ell,\chi}\,.\end{align*} $$

Moreover, using Frobenius reciprocity, one has

$$ \begin{align*}\widehat{(t(\cdot^\ell))^+}(\chi)= \langle (t(\cdot^\ell))^+,\chi \rangle_{G^+}= \langle t(\cdot^\ell),\chi|_G \rangle_{G} = \langle t,r_{\ell,\chi|_G} \rangle_G=\langle t,r_{\ell,\chi} \rangle_G \,,\end{align*} $$

which by the class function equality above equals

$$ \begin{align*} [G^+:G]^{-1}\langle t, r_{\ell,\chi}^+|_G\rangle_G& = [G^+:G]^{-1} \langle t^+,r_{\ell,\chi}^+\rangle_{G^+}=0\,. \end{align*} $$

We deduce that $(t(\cdot ^\ell ))^+ \equiv 0$ , and as such, applying inclusion-exclusion,

$$ \begin{align*} \theta(x;L/K,t)&:= \sum_{ \substack{ \mathfrak p \triangleleft \mathcal O_K \\ \mathcal N\mathfrak p \leq x }} t(\varphi_{\mathfrak p}) \log (\mathcal N\mathfrak p) = \sum_{\ell\geq 1} \mu(\ell) \psi(x^{\frac 1\ell},L/K,t(\cdot^\ell)) \\&=\sum_{\ell\geq 1} \mu(\ell) \psi(x^{\frac 1\ell},L/{\mathbb{Q}},(t(\cdot^\ell))^+)=0, \end{align*} $$

by Proposition 3.11. Denoting by $D_{L/K}$ the relative discriminant of $L/K$ and applying summation by parts, we deduce that

$$ \begin{align*}\pi(x;L/K,t)+\sum_{\substack{\mathfrak p \mid D_{L/K} \\ \mathcal N\mathfrak p\leq x}} t(\varphi_{\mathfrak p}) = \int_1^{x} \frac{\mathrm{d} \theta(u;L/K,t) }{\log u}=0.\\[-46pt]\end{align*} $$

In Proposition 3.11, we reduced our counting problem to one which will involve zeros of primitive L-functions, at the cost of working in a larger Galois group. In some of our results, we will circumvent AC by doing the exact opposite (as is done classically): we will work in an abelian Galois group, and allow imprimitive L-functions. This will be done using a result of Serre which goes back to Deuring and is in the spirit of Chebotarev’s original reduction.

Lemma 3.14 [Reference SerreSe3, Section 2.7].

Let $L/K$ be a Galois extension with Galois group G, and let $C\in G^\sharp $ . For any $g\in C$ , let $\mathrm {ord}(g)$ denote the order of the subgroup $\langle g\rangle $ generated by g. We have the equality

$$ \begin{align*}\psi(x;L/K , 1_C) = \frac{|C| \mathrm{ord}(g)}{|G|}\psi(x;L/L^{\langle g\rangle},1_{\{g\}}).\end{align*} $$

This identity follows again from Artin induction in the form

(57) $$ \begin{align} L(s,L/K,\frac{|G|}{|C|}1_C) = L(s,L/L^{\langle g \rangle},\mathrm{ord}(g_C) 1_{\{g_C\}}), \end{align} $$

which holds since $ \mathrm {Ind}_{\langle g \rangle }^G (\mathrm {ord}(g)1_{\{g\}})= {|G|}{|C|^{-1}}1_C$ (recall (56)).

We will also need the following consequence of Proposition 3.11 and Lemma 3.14.

Lemma 3.15. Let $L/K$ be a Galois extension of number fields, and let $C\in G^\sharp $ . For any $s_0\in \mathbb C$ and $g\in C$ , we have that

(58) $$ \begin{align} \sum_{ \chi\in \mathrm{Irr}(G)} \chi(C^{-1}) \mathrm{ord}_{s=s_0} L(s,L/K,\chi) = \sum_{ \chi\in \mathrm{Irr}(\langle g \rangle)} \chi(g^{-1}) \mathrm{ord}_{s=s_0} L(s,L/L^{\langle g \rangle},\chi). \end{align} $$

Assuming, moreover, that $L/{\mathbb {Q}}$ is Galois, for any class function $t:G\rightarrow \mathbb C$ and $s_0 \in \mathbb C$ , we have that

(59) $$ \begin{align} \sum_{ \chi\in \mathrm{Irr}(G)} \widehat {t} (\chi) \mathrm{ord}_{s=s_0} L(s,L/K,\chi) = \sum_{ \chi\in \mathrm{Irr}(G^+)} \widehat {t^+} (\chi) \mathrm{ord}_{s=s_0} L(s,L/{\mathbb{Q}},\chi). \end{align} $$

Proof. The first claimed identity clearly follows from the following:

$$ \begin{align*}-\sum_{ \chi\in \mathrm{Irr}(G)} \chi(C^{-1}) \frac{L'(s,L/K,\chi)}{L(s,L/K,\chi)} = -\sum_{ \chi\in \mathrm{Irr}(\langle g \rangle)} \chi(g^{-1})\frac{L'(s,L/L^{\langle g \rangle},\chi)}{L(s,L/L^{\langle g \rangle},\chi)},\end{align*} $$

which we will establish for $s>1$ using the induction property for Artin L-functions. By uniqueness of analytic continuation, this is sufficient.

The summatory function of the coefficients of the Dirichlet series on the left-hand side is given by

(60) $$ \begin{align} \sum_{ \chi\in \mathrm{Irr}(G)} \chi(C^{-1}) \sum_{\substack{\mathfrak p\triangleleft \mathcal O_{K}\\ \mathcal N \mathfrak p^m \leq x \\ m\geq 1}}\chi(\varphi_{\mathfrak p}^m)\log (\mathcal N \mathfrak p)=\frac{|G|}{|C|}\sum_{\substack{\mathfrak p\triangleleft \mathcal O_{K} \\ \mathcal N \mathfrak p^m \leq x \\ m\geq 1}} \mathbf 1_{ C}(\varphi_{\mathfrak p}^m) \log (\mathcal N \mathfrak p); \end{align} $$

the same holds for the coefficients of the Dirichlet series on the right-hand side of the equality to be established, by virtue of Lemma 3.14. This concludes the proof of (58). The proof of (59) is similar using Proposition 3.11.

We are now ready to relate $\psi (x;L/K,t)$ and $\pi (x;L/K,t)$ (respectively, $\psi (x;L/K,C)$ and $\pi (x;L/K,C)$ ) with the zeros of Artin L-functions associated to the extension $L/{\mathbb {Q}}$ (respectively, $L/L^{\langle g \rangle }$ , for any $g\in C$ ), which ultimately will allow us to use the language of random variables. Under AC, the calculation of the mean and variance in Proposition 3.18 below can be deduced from combining [Reference DevinDe, Theorem 2.1] (see also [Reference FiorilliFi2]) with induction properties of Artin L-functions. For the sake of completeness and in order to provide a full decomposition into sums of independent random variables, we decided to give further details while trying to stay brief. This closely follows [Reference DevinDe, Theorem 2.1], [Reference FiorilliFi2] and [Reference NgNg, Section 5.1].

Lemma 3.16. Let $L/K$ be a Galois extension of number fields for which AC holds. If $\chi $ is an irreducible character of $G=\mathrm {Gal}(L/K)$ and $C\subset G$ is a conjugacy class, then for any $x,X\geq 2$ ,

(61) $$ \begin{align} \psi(x;L/K,\chi) &=\delta_{\chi=1}x-\sum_{\substack{\rho_{\chi} \\ |\Im(\rho_{\chi})| \leq X}}\frac{x^{\rho_{\chi}}}{\rho_{\chi}} +O_{L,K}\Big(\log x +\frac xX (\log (xX))^2\Big); \end{align} $$

(62) $$ \begin{align} \psi(x;L/K,C) &= \frac{|C|}{|G|} x-\frac{|C|}{|G|} \sum_{ \chi \in \mathrm{Irr}(G)} \chi(C^{-1}) \sum_{\substack{\rho_{\chi} \\ |\Im(\rho_{\chi})| \leq X}}\frac{x^{\rho_{\chi}}}{\rho_{\chi}} +O_{L,K}\Big(\log x +\frac xX (\log (xX))^2\Big), \end{align} $$

where $\delta _{\chi =1}$ is $1$ when $\chi $ is the trivial character, and $0$ otherwise. In both formulas, the sum is over the zeros $\rho _\chi $ (counted with multiplicity) of the Artin L-function $L(s,L/K,\chi )$ in the critical strip $\Re (s) \in (0,1)$ .

Corollary 3.17. Let $L/K$ be a Galois extension of number fields, let $C\subset \mathrm {Gal}(L/K)$ be a conjugacy class and let $g_C$ be any representative of C. For $ x,X\geq 2$ , we have the estimate

(63) $$ \begin{align} \frac{\log x}{x^{\beta^t_{L}}}\Big( \frac{|G|}{|C|}\pi(x;L/K,C)-\mathrm{Li}(x) \Big)= -\delta_{\beta^t_{L}=\frac 12}r(C)-\frac 1{\beta^t_{L}}\sum_{\chi\in \mathrm{Irr}(G)} \overline{\chi}(C) \mathrm{ord}_{s=\beta^t_{L}} L(s,L/K,\chi)\nonumber\\- \sum_{ \chi \in \mathrm{ Irr}(\langle g_C \rangle)} \overline{\chi}(g_C) \sum_{\substack{\rho_{\chi} \\ 0< |\gamma_{\chi}|\leq X }} \frac{x^{\rho_{\chi}-\beta^t_L}}{\rho_{\chi}} +O_{L,K}\Big(\frac 1{\log x}+ \frac{x^{1-\beta^t_L}}X (\log(xX))^2\Big)\,, \end{align} $$

where $\rho _{\chi }$ runs through the nontrivial zeros of $L(s,L/K,\chi )$ . If, in addition, we assume that $L/{\mathbb {Q}}$ is Galois and that AC holds, then for any class function $t:G\rightarrow \mathbb C$ ,

(64) $$ \begin{align} \frac{\log x}{x^{\beta^t_{L}}}&\Big( \pi(x;L/K,t)- \widehat{t}(1)\mathrm{Li}(x) \Big)= -\delta_{\beta^t_{L}=\frac 12} \langle t,r \rangle_G-\frac 1{\beta^t_{L}}\sum_{\chi\in \mathrm{Irr}(G)} \widehat {t^+} (\chi) \mathrm{ord}_{s=\beta^t_{L}} L(s,L/K,\chi)\\&\qquad\quad - \sum_{ \chi \in \mathrm{Irr}(G^+)} \widehat{{t^+}} (\chi) \sum_{\substack{\rho_{\chi} \nonumber\\ 0<|\gamma_{\chi}|\leq X }} \frac{x^{\rho_{\chi}-\beta^t_L}}{\rho_{\chi}} +O_{L,K}\Big(\frac 1{\log x}+ \frac{x^{1-\beta^t_L}}X (\log(xX))^2\Big)\,. \end{align} $$

Proof. We first establish (64). Arguing as in [Reference NgNg, Section 5.1] and [Reference DevinDe, Section 4.3], we see that

(65) $$ \begin{align} &\frac{\log x}{x^{\beta^t_{L}}}\Big( \pi(x;L/K,t)-\widehat{t}(1)\mathrm{Li}(x) \Big) \nonumber\\&\qquad\qquad\qquad\qquad = x^{-\beta^t_{L}} \Big(\psi(x;L/K,t)-\widehat{t}(1)x-\widehat{t(\cdot^2)}(1)x^{\frac 12}\Big) +O_{L,K}\Big( \frac {1}{\log x}\Big). \end{align} $$

By Proposition 3.11, this is

$$ \begin{align*} &= x^{-\beta^t_{L}} \psi(x; L/{\mathbb{Q}}, t^+ ) -\widehat{t}(1)x^{1-\beta^t_{L}}-\delta_{\beta^t_{L}=\frac 12}\langle t,r\rangle_G+O_{L,K}\Big( \frac 1{\log x}\Big)\\ &=x^{-\beta^t_{L}} \sum_{\chi \in \mathrm{Irr}(G^+)} \widehat {t^+} (\chi)\psi(x; L/{\mathbb{Q}}, \chi)-\widehat{t}(1)x^{1-\beta^t_{L}}-\delta_{\beta^t_{L}=\frac 12}\langle t,r\rangle_G+O_{L,K}\Big( \frac 1{\log x}\Big). \end{align*} $$

The estimate (64) then follows from applying Lemma 3.16. The proof of (63) is similar (note that $\mathrm {Ind}_{\langle g_C \rangle }^G (\mathrm {ord}(g_C)1_{g_C})=\frac {|G|}{|C|} 1_C$ ).

To state the next proposition, we first define the following multisets of zeros of Artin L-functions, where $\beta _{L}$ and $\beta ^{\,\mathrm {real}}_{L/K}$ are defined in Theorem 2.1:

(66) $$ \begin{align} Z_{L}:=\{ \gamma \in \mathbb R_{>0} : \zeta_L(\beta_{L}+i\gamma)=0\}; \end{align} $$

(67) $$ \begin{align} Z^{t}_{L}:=\bigcup_{\substack{\chi \in \mathrm{Irr}(\mathrm{Gal}(L/{\mathbb{Q}}))\\ \widehat{t^+}(\chi)\neq 0 }}\{ \gamma \in \mathbb R_{>0} : L(\beta^{t}_{L}+i\gamma,\chi)=0\}. \end{align} $$

Recall also the definition (16).

Proposition 3.18. Let $L/K$ be a Galois extension of number fields, let $G=\mathrm {Gal}(L/K)$ and fix $t:G\rightarrow \mathbb C$ a class function. The function $E(y;L/K,t)$ admits a limiting distribution. Moreover, the associated random variable $X(L/K;t)$ is such that

(68) $$ \begin{align} \notag \mathbb E[X(L/K;t)] &=-\delta_{\beta^t_{L}=\frac 12}\langle t,r \rangle_G-\frac 1{\beta^t_{L}}\mathrm{ord}_{s=\beta^t_{L}}L(s,L/K,t) \notag \\ &=-\delta_{\beta^t_{L}=\frac 12}\langle \widehat t, \varepsilon_2 \rangle_{\mathrm{Irr}(G)}-\frac 1{\beta^t_{L}}\mathrm{ord}_{s=\beta^t_{L}}L(s,L/K,t). \end{align} $$

Furthermore, we have that

(69) $$ \begin{align} \mathrm{Var}[X(L/K;t)]=2\mathop{\sum \Bigl.^{*}}\limits_{\gamma \in Z_{L} } \frac{|\mathrm{ord}_{s=\beta^t_L+i\gamma}L(s,L/K,t)|^2}{(\beta^t_{L})^2+\gamma^2}, \end{align} $$

where the starred sum means a sum without multiplicities. If $L/{\mathbb {Q}}$ is Galois, then we have the alternativeFootnote ²⁰ formula

(70) $$ \begin{align} \mathrm{Var}[X(L/K;t)]=2\mathop{\sum \Bigl.^{*}}\limits_{\gamma \in Z^t_{L} } \frac{|\mathrm{ord}_{s=\beta^t_L+i\gamma}L(s,L/{\mathbb{Q}},t^+)|^2}{(\beta^t_{L})^2+\gamma^2} .\end{align} $$

Assuming, in addition, that LI $^{-}$ holds, we have the simplified formula

$$ \begin{align*}\mathrm{Var}[X(L/K;t)] &= 2\sum_{\chi\in \mathrm{Irr}(G^+)} |\widehat {t^+}(\chi)|^2 \sum_{\gamma_{\chi}\neq 0} \frac {1}{(\beta^t_{L})^2+\gamma_{\chi}^2}. \end{align*} $$

Proof of Proposition 3.18

We will combine the arguments in [Reference NgNg, Section 5.1], [Reference FiorilliFi2, Lemma 2.6], [Reference DevinDe, Theorem 2.1] and [Reference Akbary, Ng and ShahabiANS, Theorem 1.2] (one cannot apply those results directly, since we are not assuming GRH and, moreover, t is complex-valued). For any $T\geq 1$ , we define

(71) $$ \begin{align} \beta^t_{L}(T)= \begin{cases} \sup\{ \Re(\rho) < \beta^t_{L} : |\Im(\rho)|\leq T,L(\rho,L/{\mathbb{Q}},\chi)=0;&\\ \qquad \chi \in \mathrm{supp}(\widehat{t^+}) \} & \text{if AC holds for } L/{\mathbb{Q}} ;\\ \sup\{ \Re(\rho)<\beta^t_{L} : |\Im(\rho)|\leq T,\zeta_L(\rho)=0 \} &\text{otherwise}, \end{cases} \end{align} $$

where we recall the definition (10). Note that $\beta ^t_{L}(T)<\beta ^t_{L}$ . Using the decomposition $ t= \sum _{C\in G^\sharp } t(C) 1_C $ and letting $g_C$ be any element of C, we deduce from Corollary 3.17 that for $x\geq 2$ and $X\geq T\geq 2$ ,

(72) $$ \begin{align} &\frac{\log x}{x^{\beta^t_{L}}} (\pi(x;L/K,t)-\widehat t(1)\mathrm{Li}(x)) = -\delta_{\beta^t_{L}=\frac 12}\langle t,r \rangle_G-\frac{1}{\beta^t_{L}} \mathrm{ord}_{s=\beta^t_{L}} L(s,L/K,t)\nonumber\\ &\qquad\qquad -\sum_{C\in G^\sharp} t(C) \sum_{ \chi \in \mathrm{Irr}(\langle g_C\rangle)} \overline{\chi}(g_C) \Big(\sum_{\substack{\rho_\chi = \beta^t_L+i\gamma_\chi\nonumber\\ 0<|\gamma_\chi|\leq T}} +\sum_{\substack{\rho_\chi = \beta^t_L+i\gamma_\chi\\ T<|\gamma_\chi|\leq X}} \Big) \frac{x^{i\gamma_{\chi}}}{\rho_{\chi}}\nonumber\\ &\qquad\qquad +\,O_{L,K}\Big(\frac 1{\log x}+ \frac{x^{1-\beta^t_L}}X (\log(xX))^2+x^{\beta^t_L(T)-\beta^t_L}(\log X)^2\Big) \,. \end{align} $$

Taking $X=x=e^y$ , we see that

$$ \begin{align*}\int_{2}^Y \Big|\sum_{C\in G^\sharp} t(C)\sum_{1\neq \chi \in \mathrm{Irr}(\langle g_C\rangle)} \overline{\chi}(g_C) \sum_{\substack{\rho_\chi\\ T<|\gamma_\chi|\leq {\mathrm{e}}^y}} \frac{{\mathrm{e}}^{y \rho_{\chi}}}{\rho_{\chi}}\Big|^2\mathrm{d}y \ll_{L,K} \frac {(Y+\log T) (\log(d_L T ))^2}{T},\end{align*} $$

and we deduce as in [Reference NgNg, Section 5.1], [Reference FiorilliFi2, Lemma 2.6], [Reference DevinDe, Theorem 2.1] and [Reference Akbary, Ng and ShahabiANS, Theorem 1.2] that the function

$$ \begin{align*}\overrightarrow E(y) = \frac{y}{{\mathrm{e}}^{\beta^t_{L}}} (\Re(\pi({\mathrm{e}}^y;L/K,t)-\widehat t(1)\mathrm{Li}(x)),\Im(\pi({\mathrm{e}}^y;L/K,t)-\widehat t(1)\mathrm{ Li}(x)))\end{align*} $$

is $B^2$ almost-periodic. In particular, this function admits a limiting distribution.

To compute the first two moments of this distribution, we deduce using the arguments in the proofs of [Reference FiorilliFi2, Lemmas 2.5, 2.6] (see also [Reference Akbary, Ng and ShahabiANS, Theorem 1.14] and [Reference DevinDe, Theorem 2.1]) that

$$ \begin{align*} \mathbb E[X_\nu] &= \lim_{Y\rightarrow \infty} \frac 1Y \int_2^Y \frac{y}{{\mathrm{e}}^{\beta^t_{L}}}(\pi({\mathrm{e}}^y;L/K,t)-\widehat t(1)\mathrm{Li}({\mathrm{e}}^y)) \mathrm{ d}y\\ &=-\delta_{\beta^t_{L}=\frac 12}\langle t, r \rangle_{G}-\frac 1{\beta^t_{L}}\mathrm{ord}_{s=\beta^t_{L}}L(s,L/K,t). \end{align*} $$

Similarly,

(73) $$ \begin{align} \mathrm{Var}[X_\nu] &=2\mathop{\sum \Bigl.^{*}}\limits_{\gamma \in Z_{L} } \frac{1}{(\beta^t_{L})^2+\gamma^2} \Big|\sum_{C\in G^\sharp} t(C)\sum_{ \chi \in \mathrm{Irr}(\langle g_C\rangle)}\overline{\chi}(g_C)\mathrm{ord}_{s=\beta^t_{L}+i\gamma}L(s,L/L^{\langle g_C\rangle},\chi )\Big|^2 \end{align} $$

$$ \begin{align*} &=2\mathop{\sum \Bigl.^{*}}\limits_{\gamma \in Z_{L} } \frac{|\mathrm{ord}_{s=\beta^t_L+i\gamma}L(s,L/K,t)|^2}{(\beta^t_{L})^2+\gamma^2}, \end{align*} $$

by (57). Moreover, if $L/{\mathbb {Q}}$ is Galois, then by (54), this is

$$ \begin{align*}= 2\mathop{\sum \Bigl.^{*}}\limits_{\gamma \in Z_{L} } \frac{1}{(\beta^t_{L})^2+\gamma^2} |\mathrm{ord}_{s=\beta^t_{L}+i\gamma}L(s,L/{\mathbb{Q}},t^+ )|^2.\\[-39pt] \end{align*} $$

Under AC, GRH and LI $^-$ and for real-valued class functions t, we give an explicit expression for the random variables in Proposition 3.18. We stress that in order for the random variables appearing in this expression to be independent, it is crucial to express $\pi (x;L/K,C)$ in terms of zeros of $L(s,L/{\mathbb {Q}},\chi )$ (rather than $L(s,L/K,\chi )$ ) associated to irreducible characters of $\mathrm {Gal}(L/{\mathbb {Q}})$ ; indeed, these L-functions are believed to be primitive.

Lemma 3.20. Let $L/K$ be an extension of number fields, such that $L/{\mathbb {Q}}$ is Galois, and for which AC, GRH and LI $^-$ hold. Let $G=\mathrm {Gal}(L/K)$ , $G^+=\mathrm { Gal}(L/{\mathbb {Q}})$ , and fix a class function $t:G\rightarrow \mathbb R$ . Then, we have the following equality (in distribution) of random variables:

(74) $$ \begin{align} X(L/K;t)\overset{\mathrm{d}}= -\langle t,r\rangle_G - 2\mathrm{ord}_{s=\frac 12}L(s,L/K,t) + \sum_{\substack{1\neq \chi\in \mathrm{Irr}(G^+)}}|\widehat{t^+}(\chi)|\sum_{\gamma_{\chi}>0} \frac{2 X_{\gamma_{\chi}}}{\big(\frac 14+\gamma_{\chi}^2\big)^{\frac 12}}. \end{align} $$

Here, the random variables $X_{\gamma }$ are defined by $X_{\gamma }=\Re (Z_{\gamma })$ , where the $Z_{\gamma }$ are i.i.d. random variables uniformly distributed on the unit circle in $\mathbb C$ .

4.1 Link with ramification and representation theory

In this section, we analyse the ramification data of a given Galois extension $L/K$ . These data are related with the expressions obtained for the variance of the random variable $X(L/K;t)$ in Proposition 3.18.

Let us first review the definition of the Artin conductor $A(\chi )$ , following [Reference FröhlichFr] (this is a quite standard invariant to consider; see, e.g. [Reference Martin, Scarfy, Bahrini, Bajpai, Downey, Parvardi, Simpson and WhiteMM, Chapter 2, Section 2] or [Reference Lagarias and OdlyzkoLO, (5.2)]). Consider a finite Galois extension of number fields $L/K$ with Galois group G. For $\mathfrak p$ a prime ideal of $\mathcal O_K$ and $\mathfrak P$ a prime ideal of $\mathcal O_L$ lying above $\mathfrak p$ , the higher ramification groups form a sequence $(G_i(\mathfrak P/\mathfrak p))_{i\geq 0}$ of subgroups of G (called filtration of the inertia group $\mathrm {I}(\mathfrak P/\mathfrak p)$ ) defined as follows:

$$ \begin{align*}G_i(\mathfrak P/\mathfrak p) :=\{ \sigma\in G : \forall z \in \mathcal O_L,\, (\sigma z - z) \in \mathfrak P^{i+1}\}.\end{align*} $$

Each $G_i(\mathfrak P/\mathfrak p)$ only depends on $\mathfrak p$ up to conjugation and $G_0(\mathfrak P/\mathfrak p)=\mathrm {I}(\mathfrak P/\mathfrak p)$ . For clarity, let us fix prime ideals $\mathfrak p$ and $\mathfrak P$ as above and write $G_i$ for $G_i(\mathfrak P/\mathfrak p)$ . Given a representation $\rho \colon G \rightarrow GL(V)$ on a complex vector space V, the subgroups $G_i$ act on V through $\rho $ , and we will denote by $V^{G_i}\subset V$ the subspace of $G_i$ -invariant vectors. Let $\chi $ be the character of $\rho $ and

(75) $$ \begin{align} n(\chi,\mathfrak p):=\sum_{i=0}^{\infty} \frac{|G_i|}{|G_0|} \mathrm{codim} V^{G_i}, \end{align} $$

which was shown by Artin to be an integer (see, e.g. [Reference SerreSe1, Chapter $6$ , Theorem $1$ ’]). The Artin conductor of $\chi $ is the ideal

$$ \begin{align*}\mathfrak f(L/K,\chi):=\prod_{\mathfrak p} \mathfrak p^{n(\chi,\mathfrak p)}\,. \end{align*} $$

Note that the set indexing the above product is finite since only finitely many prime ideals $\mathfrak p$ of $\mathcal O_K$ ramify in $L/K$ . We set

(76) $$ \begin{align} A(\chi):=d_K^{\chi(1)}\mathcal N_{K/{\mathbb{Q}}}(\mathfrak f(L/K,\chi))\,, \end{align} $$

where $d_K$ is the absolute value of the absolute discriminant of the number field K and $\mathcal N_{K/{\mathbb {Q}}}$ is the relative ideal norm with respect to $K/{\mathbb {Q}}$ (we will use the slight abuse of notation that identifies the value taken by this relative norm map with the positive generator of the corresponding ideal). One can show (see, e.g. [Reference SerreSe1, Chapter $6$ , consequences of Proposition $6$ ]) the following equalities of ideals in $\mathcal O_K$ (known as the conductor–discriminant formula):

(77) $$ \begin{align} \mathfrak f(L/K,\chi_{\mathrm{reg}})=\prod_{\chi \in \mathrm{Irr}(G)} \mathfrak f(L/K,\chi)^{\chi(1)} =D_{L/K}\,,\\[-9pt]\nonumber \end{align} $$

where $D_{L/K}$ is the relative discriminant of $L/K$ and $\chi _{\mathrm {reg}}$ is the character of the regular representation of G. In particular, by [Reference Zariski and SamuelZS, Chapter 5, Theorem 31], we have the identity

(78) $$ \begin{align} d_K^{|G|}\mathcal N_{K/{\mathbb{Q}}}(D_{L/K})= A(\chi_{\mathrm{reg}})=d_L.\\[-9pt]\nonumber \end{align} $$

We now estimate $A(\chi )$ for irreducible characters $\chi \in \mathrm {Irr}(G)$ .

Lemma 4.1. Let $L/K$ be a finite Galois extension. Let $\chi $ be an irreducible character of $G=\mathrm {Gal}(L/K)$ , and assume that either $K\neq {\mathbb {Q}}$ , or that $\chi $ is nontrivial. Then, one has the bounds

$$ \begin{align*}\max(1,[K:{\mathbb{Q}}]/2)\chi(1) \leq \log A(\chi) \leq 2\chi(1) [K:{\mathbb{Q}}]\log (\mathrm{rd}_L)\,,\\[-9pt]\end{align*} $$

where the root discriminant $\mathrm {rd}_L$ is defined by (14). The upper bound, due to [Reference Martin, Scarfy, Bahrini, Bajpai, Downey, Parvardi, Simpson and WhiteMM], is unconditional. The lower bound is unconditionalFootnote ²¹ if $K\neq {\mathbb {Q}}$ , and holds assuming $L(s,L/{\mathbb {Q}},\chi )$ can be extended to an entire function otherwise.

Proof. In Lemma 4.2, we will reproduce the proof of the upper bound found in [Reference Martin, Scarfy, Bahrini, Bajpai, Downey, Parvardi, Simpson and WhiteMM]. For the lower bound, we consider two cases. Suppose first that $[K:{\mathbb {Q}}]\geq 2$ . Then we use the lower bound for the absolute discriminant of a number field obtained, that is, in [Reference Brueggeman and DoudBD, Theorem 2.4(1)] (noting that the sum over $\mathfrak P$ on the right-hand side of their formula is positive) and which holds for any $y>0$ :

(79) $$ \begin{align} \log d_K\geq r_1(1-I_1(y))+[K:{\mathbb{Q}}](\gamma+\log(4\pi)-I_2(y))-\frac{12\pi}{5\sqrt{y}} ,\\[-9pt]\nonumber\end{align} $$

where $\gamma $ is the Euler constant, $r_1$ is the number of real embeddings of $K/{\mathbb {Q}}$ and

$$ \begin{align*}f(x)=\left(\frac{3}{x^3}(\sin(x)-x\cos(x))\right)^2\,,\end{align*} $$

$$ \begin{align*}I_1(y)= \int_0^\infty\frac{1-f(x\sqrt{y})}{2\cosh^2(x/2)}\mathrm{d}x \,,\hspace{1cm} I_2(y)= \int_0^\infty\frac{1-f(x\sqrt{y})}{\sinh(x)}\mathrm{d}x\,. \end{align*} $$

Setting $y=20$ and using the numerical integration method implemented in SageMath ([Sage]), we obtain that $I_1(20)<0.08$ and $I_2(20)<1.73$ . In particular, the quantity $\gamma -1/2+\log (4\pi )-I_2(20)$ is positive, and we deduce from (79) that

$$ \begin{align*} \log d_K-\frac{[K:{\mathbb{Q}}]}{2}&\geq [K:{\mathbb{Q}}]\bigl(\gamma-\frac{1}{2}+\log(4\pi)-I_2(20)\bigr)-\frac{6\pi}{5\sqrt{5}}\\ &\geq 2\bigl(\gamma-\frac{1}{2}+\log(4\pi)-I_2(20)\bigr)-\frac{6\pi}{5\sqrt{5}}\\ &\geq 0.07\,. \end{align*} $$

Moreover, $\mathcal N_{K/{\mathbb {Q}}}(\mathfrak f(\chi ))\geq 1$ , so that one trivially deduces $\log A(\chi )\geq \chi (1)\log d_K\geq \chi (1)[K:{\mathbb {Q}}]/2$ . If $K={\mathbb {Q}}$ , the lower bound is a consequence of Odlyzko type lower bounds on Artin conductors (see, e.g. [Reference Pizarro-MadariagaPi1, Theorem 3.2], where the author proves $\mathfrak f(\chi )\geq 2.91^{\chi (1)}$ ), that are conditional on Artin’s conjecture.

It is known (see [Reference Pizarro-MadariagaPi2]) that the lower bound in Lemma 4.1 is optimal. Nevertheless, will show in the next lemma that if one has good estimates on character values, then it is possible to improve both bounds in Lemma 4.1, and in some cases to deduce the exact order of magnitude of $\log A(\chi )$ (for instance, in the case of $G=S_n$ ; see Lemma 7.4).

Lemma 4.2. Let $L/K$ be a finite Galois extension. For any character $\chi $ of $G=\mathrm {Gal}(L/K)$ , we defineFootnote ²²

$$ \begin{align*}M_{\chi}:=\max_{1\neq \sigma \in G}\frac{|\chi(\sigma)|}{\chi(1)}\leq 1\,.\end{align*} $$

Then we have the bounds

$$ \begin{align*}(1-M_{\chi}) \chi(1) [K:{\mathbb{Q}}]\log (\mathrm{rd}_L) \leq \log A(\chi) \leq (1+M_{\chi}) \chi(1) [K:{\mathbb{Q}}]\log (\mathrm{rd}_L)\,.\end{align*} $$

Proof. The proof is inspired by [Reference Martin, Scarfy, Bahrini, Bajpai, Downey, Parvardi, Simpson and WhiteMM, Proof of Proposition 7.4]. Let $\rho \colon G\rightarrow GL(V)$ be a complex representation with character $\tau $ , let $\mathfrak p$ be a prime ideal of $\mathcal O_K$ and let $(G_i)$ be the attached filtration of inertia (defined up to conjugation in G). We start with the following identity:

(80) $$ \begin{align} \mathrm{codim}\ V^{G_i}=\tau(1)-\dim V^{G_i} =\tau(1)-\langle \tau,\mathbf{1}\rangle_{G_i}=\frac 1{|G_i|} \sum_{a\in G_i} \bigl(\tau(1)-\tau(a)\bigr)\,, \end{align} $$

where $\langle \,,\,\rangle _{G_i}$ is defined as in Section 3.1 and $\mathbf {1}$ is the trivial representation. If $\tau =\chi _{\mathrm {reg}}$ is the character of the regular representation of G, then for any $a\in G\setminus \{\mathrm {id}\}$ ,

$$ \begin{align*}\chi_{\mathrm{reg}}(a)=\sum_{\varphi\in\mathrm{Irr}(G)}\varphi(1)\varphi(a)=0 \end{align*} $$

by the orthogonality relation (42). Hence, combining (75) with (80), we obtain that

(81) $$ \begin{align} n(\chi_{\mathrm{reg}},\mathfrak p) = \frac 1{|G_0|}\sum_{i\geq 0} \sum_{1\neq a\in G_i}\chi_{\mathrm{reg}}(1)=\frac{|G|}{|G_0|}\sum_{i\geq 0}\bigl(|G_i|-1\bigr). \end{align} $$

Similarly, setting $\tau =\chi $ in (80), we have that

(82) $$ \begin{align} n(\chi,\mathfrak p)= \frac{\chi(1)}{|G_0|} \sum_{i\geq 0} \sum_{1\neq a\in G_i} \left(1-\frac{\chi(a)}{\chi(1)}\right)\,. \end{align} $$

Combining our expressions for $n(\chi _{\mathrm {reg}},\mathfrak p)$ and $n(\chi ,\mathfrak p)$ yields the bound

(83) $$ \begin{align} \left|n(\chi,\mathfrak p)-\frac{\chi(1)}{|G|}n(\chi_{\mathrm{reg}},\mathfrak p)\right| \leq M_\chi\frac{\chi(1)}{|G|}n(\chi_{\mathrm{reg}},\mathfrak p)\,. \end{align} $$

We now establish the claimed bound on $\log A(\chi )$ . Let $\nu _{\mathfrak p}$ denote the $\mathfrak p$ -adic valuation on $\mathcal O_K$ , and observe that (83) implies the bound

$$ \begin{align*}n(\chi,\mathfrak p)=\nu_{\mathfrak p}(\mathfrak f(L/K,\chi))\leq \frac{\chi(1)}{|G|}(1+M_\chi)n(\chi_{\mathrm{reg}},\mathfrak p)= \frac{\chi(1)}{|G|}(1+M_\chi)\nu_{\mathfrak p}(D_{L/K})\,. \end{align*} $$

We deduce that

$$ \begin{align*}\log \mathcal N_{K/{\mathbb{Q}}}\bigl(\mathfrak f(L/K,\chi)\bigr)\leq \frac{\chi(1)}{|G|}(1+M_\chi) \log \mathcal N_{K/{\mathbb{Q}}}(D_{L/K})\,. \end{align*} $$

By adding $\chi (1)\log (d_K)$ on both sides, we obtain the bound

$$ \begin{align*} \log A(\chi)&\leq \chi(1)(1+M_\chi)\biggl(\log (d_K)+\log\bigl(\mathcal N_{K/{\mathbb{Q}}} (D_{L/K})^{\frac{1}{|G|}}\bigr)\biggr)\\ &\leq \chi(1)(1+M_\chi)\log\bigl(d_L^{\frac{1}{|G|}}\bigr)\\ &\leq \chi(1)(1+M_\chi)[K:{\mathbb{Q}}]\log ( \mathrm{rd}_L)\,. \end{align*} $$

The lower bound of the lemma is deduced from (83) in an analogous fashion.

4.2 Variance associated to the limiting distribution

We now consider a Galois extension of number fields $L/K$ of group G and estimate various sums indexed by zeros of the associated Artin L-functions. For class functions $t\colon G\rightarrow \mathbb C$ , these sums are related to the variance and fourth moment of the random variable $X(L/K;t)$ defined in Proposition 3.18. For $\chi \in \mathrm {Irr}(\mathrm {Gal}(L/K))$ , we define

(84) $$ \begin{align} B(\chi):=\sum_{\gamma_\chi} \frac{1}{\frac 14+\gamma_{\chi}^2}; \hspace{1cm} B_0(\chi):= \sum_{\gamma_\chi\neq 0} \frac{1}{\frac 14+\gamma_{\chi}^2}; \hspace{1cm} B_2(\chi) :=\sum_{\gamma_\chi\neq 0} \frac{1}{(\frac 14+\gamma_{\chi}^2)^2}\,, \end{align} $$

where the sums are indexed by the imaginary parts of the ordinates of the nontrivial zeros of $L(s,L/K,\chi )$ , counted with multiplicities. In the next lemma, we will determine the order of magnitude of $B(\chi ),B_0(\chi )$ and $B_2(\chi )$ . Note that under GRH, every nontrivial zero of $L(s,L/K,\chi )$ is of the form $\rho _\chi =\frac 12+i \gamma _\chi $ with $\gamma _\chi \in \mathbb R$ , and thus, $\frac 14+\gamma _\chi ^2 = |\rho _\chi |^2$ . However, the constant $\frac 14$ in (84) could be replaced by any fixed real number $\beta ^2 \in [\frac 14,1]$ , and that would not change the orders of magnitude of $B(\chi ),B_0(\chi )$ and $B_2(\chi )$ (with constants independent of $\beta )$ . Indeed, for $\gamma \in \mathbb R$ , we have that $ (\frac 14 +\gamma ^2) \leq (\beta ^2 +\gamma ^2) \leq 4 (\frac 14 +\gamma ^2)$ .

Lemma 4.3. Let $L/K$ be a finite Galois extension for which AC holds. For any character $\chi $ of $G=\mathrm {Gal}(L/K)$ , we have the estimates

$$ \begin{align*}B(\chi)\asymp B_0(\chi)\asymp B_2(\chi) \asymp \log (A(\chi)+2)\,.\end{align*} $$

Proof. We begin with the Riemann–von Mangoldt formula [Reference Iwaniec and KowalskiIK, Theorem 5.8], which we combine with the bound on the analytic conductor given in [Reference Iwaniec and KowalskiIK, Section 5.13]. In the notation of loc. cit., the degree d of the L-function $L(s,L/K,\chi )$ is relative to ${\mathbb {Q}}$ and thus equals $[K:{\mathbb {Q}}]\chi (1)$ . For $T\geq 1$ , we obtain the estimate

(85) $$ \begin{align} N(T,\chi):= |\{\gamma_{\chi}\colon |\gamma_{\chi}|\leq T \}|&=\frac T{\pi} \log \Big(\frac{A(\chi)T^{[K:{\mathbb{Q}}]\chi(1)}}{(2\pi e)^{[K:{\mathbb{Q}}]\chi(1)}}\Big)\nonumber\\ &\quad + O(\log ((A(\chi)+2)(T+4)^{[K:{\mathbb{Q}}]\chi(1)}))\,. \end{align} $$

It follows that

$$ \begin{align*}N(2T,\chi)-N(T,\chi) = \frac T{\pi} \log \Big(\frac{A(\chi)(2T)^{[K:{\mathbb{Q}}]\chi(1)}}{(\pi e)^{[K:{\mathbb{Q}}]\chi(1)}}\Big) +O(\log((A(\chi)+2)(2T+8)^{[K:{\mathbb{Q}}]\chi(1)}))\,.\end{align*} $$

It is easy to see that for T larger than an absolute constant, the main term is at least twice as big as the error term (e.g. if we let $C_0$ be the implied constant in the error term above, it suffices to take T larger than $2\pi C_0$ and, such that $T\log (2T/\pi e)\geq 2\pi C_0\log (2T+8)$ ).

Therefore, there exists an absolute constant $T_0\geq 4\pi e$ , such that

$$ \begin{align*}N(2T_0,\chi)-N(T_0,\chi) \geq \frac {T_0}{2\pi} \log\Big( A(\chi) \Big(\frac{2T_0}{\pi e}\Big)^{[K:{\mathbb{Q}}]\chi(1)}\Big)\geq \frac{T_0}{2\pi} \log (A(\chi)+2),\end{align*} $$

and hence

$$ \begin{align*}B_0(\chi) \geq \frac 1{\frac 14 + (2T_0)^2}\frac{T_0}{2\pi} \log (A(\chi)+2) \gg \log (A(\chi)+2).\end{align*} $$

For the upper bound, one easily deduces from (85) that $ N(T,\chi ) \ll T \log \frac {A(\chi )T^{[K:{\mathbb {Q}}]\chi (1)}}{(2\pi e)^{[K:{\mathbb {Q}}]\chi (1)}}$ for $T\geq 4\pi e$ , and hence, summation by parts yields that

$$ \begin{align*} B_0(\chi)\leq B(\chi) &= \sum_{\gamma_\chi} \frac{1}{\frac 14+\gamma_{\chi}^2} \leq 4N(2,\chi)+\int_2^{\infty} \frac{\mathrm{d}N(t,\chi)}{\frac 14+t^2} \\ & \ll \log (A(\chi)+2)+[K:{\mathbb{Q}}]\chi(1) + \int_2^{\infty} \frac{t^2 (\log A(\chi)+[K:{\mathbb{Q}}]\chi(1)\log t)}{(1/4+t^2)^2}\mathrm{d}t \\ &\ll \log (A(\chi)+2)+[K:{\mathbb{Q}}]\chi(1)\,. \end{align*} $$

Since we are assuming AC, we can apply Lemma 4.1 to deduce that $B_0(\chi ),B(\chi )\asymp \log (A(\chi )+2)$ . The proof is similar for $B_2(\chi )$ .

Remark 4.4. If $\chi $ is a Dirichlet character of conductor $q^*\geq 3$ , then under GRH, we can give an exact formula for $B(\chi )$ and $B_2(\chi )$ , and deduce the more precise estimate

$$ \begin{align*}B(\chi) =\log q^* + O(\log\log q^*)\end{align*} $$

(this is achieved, e.g. by applying Littlewood’s conditional bound on $\frac {L'(1,\chi )}{L(1,\chi )}$ to [Reference Montgomery and VaughanMV, (10.39)]). Such an estimate is harder to establish for a general extension $L/K$ . We have by [Reference Lagarias and OdlyzkoLO, (5.11)] that

$$ \begin{align*}2B(\chi)= \log A(\chi) +2\frac{\gamma_{\chi}'(1)}{\gamma_{\chi}(1)}+2\Re \frac{L'(1,\chi)}{L(1,\chi)},\end{align*} $$

where the gamma factor is given by

$$ \begin{align*}\gamma_{\chi}(s):= \Big( \pi^{-\frac{s+1}2} \Gamma (\tfrac{s+1}2) \Big)^{b(\chi)} \Big( \pi^{-\frac{s}2} \Gamma (\tfrac{s}2) \Big)^{a(\chi)}\end{align*} $$

for some nonnegative integers $a(\chi ),b(\chi )$ , such that $a(\chi )+b(\chi )=\chi (1)$ . It follows that

$$ \begin{align*}\frac{\gamma_{\chi}'(1)}{\gamma_{\chi}(1)} \asymp \chi(1).\end{align*} $$

As for the ‘analytic term,’ we could either use the following bound (see [Reference NgNg, Proposition 2.4.2.3])

$$ \begin{align*}\frac{L'(1,\chi)}{L(1,\chi)} \ll \chi(1) \log\log (A(\chi)+2),\end{align*} $$

or an estimate for its average, as in [Reference Fiorilli and MartinFM, Theorem 1.7]. The problem with this individual bound for a given $\chi $ is that it seems hard, in general, to improve the bound $\chi (1)\ll \log A(\chi )$ (one can, however, do this in the specific case $G=S_n$ , and we put this to use in Proposition 7.6). As for the bound, on average, it works quite well for some abelian extensions (see [Reference Fiorilli and MartinFM]), however, there are examples such as Theorem 2.19 in which there is a unique nonabelian character of degree comparable to $|G|$ , hence, the averaging will not succeed in this case.

4.3 Proofs of Theorems 2.1 and 2.3

We first state and prove Proposition 4.6, which implies Theorem 2.1. This will require the following lemma.

Lemma 4.5. Let $L/K$ be a finite Galois extension for which AC holds, and let $\chi $ be an irreducible character of $G=\mathrm {Gal}(L/K)$ . For $T\geq 1$ , $\beta \in [\tfrac 14,1]$ and $j\in \mathbb Z_{\geq 0}$ , we have the estimate

$$ \begin{align*}\sum_{|\gamma_{\chi}|>T} \frac {(\log(|\gamma_{\chi}|+4))^j}{\beta+\gamma_{\chi}^2} \ll_j \frac{(\log (T+4))^j\log(A(\chi)(T+4)^{[K:{\mathbb{Q}}]\chi(1)})}T,\end{align*} $$

where the sum on the left-hand side is over imaginary parts of zeros of $L(s,L/K,\chi )$ and where the implied constant is independent of $\beta $ .

Proof. By (85), we have that

$$ \begin{align*}N(T,\chi)= |\{\gamma_{\chi}\colon |\gamma_{\chi}|\leq T \}|\ll T\log (A(\chi)(T+4)^{[K:{\mathbb{Q}}]\chi(1)}). \end{align*} $$

With a summation by parts, we obtain that for $j\geq 1$ ,

$$ \begin{align*} & \sum_{\substack{|\gamma_{\chi}|> T }} \frac {(\log(|\gamma_{\chi}|+4))^j}{\beta+\gamma_{\chi}^2} \leq \sum_{\substack{|\gamma_{\chi}| > T }} \frac {(\log(|\gamma_{\chi}|+4))^j}{\gamma_{\chi}^2} = \int_T^{\infty} \frac{(\log(t+4))^j\mathrm{d}N(t,\chi)}{t^2} \\&=- \frac{(\log(T+4))^jN(T,\chi)}{T^2} + \int_T^{\infty}\Big( j\frac{(\log(t+4))^{j-1}}{t^2(t+4)} -2\frac{(\log(t+4))^j}{t^3}\Big) N(t,\chi)\mathrm{d}t \\ &\ll j \int_T^{\infty} \frac{ (\log(t+4))^j\log (A(\chi)(t+4)^{[K:{\mathbb{Q}}]\chi(1)}) }{ t^2} \mathrm{d}t \\ &\hspace{2cm}+ \frac{(\log(T+4))^j\log (A(\chi)(T+4)^{[K:{\mathbb{Q}}]\chi(1)})}T.\\[-9pt] \end{align*} $$

The proof follows and is similar in the case $j=0$ .

In Proposition 4.6, we will use the bound $ |\widehat t(\chi )| \leq \chi (1) {\lVert {t}\rVert }_1$ , which follows from the triangle inequality.

Proposition 4.6. Let $L/K$ be a Galois extension of number fields, and let $G=\mathrm {Gal}(L/K)$ . Then for any class function $t\colon G\rightarrow \mathbb C$ , we have the upper bound

(86) $$ \begin{align} \mathrm{Var}[X(L/K;t)] \ll {\lVert{t}\rVert}_1^2\log (d_L+2) \min(M_L, \log (d_L+2)),\\[-9pt]\nonumber \end{align} $$

where

$$ \begin{align*}M_{L} := \max \Big \{ \mathrm{ord}_{s=\rho} \zeta_L(s) :\Re(\rho)=\beta_L^t, 0<|\Im(\rho)|< \log (d_L+2)(\log\log (d_L+2))^2\Big\}.\\[-9pt]\end{align*} $$

If $L/{\mathbb {Q}}$ is Galois and AC holds, then we have the bound

(87) $$ \begin{align} \mathrm{Var}[X(L/K;t)] \ll (m^t_{L})^2\sum_{\chi\in \mathrm{Irr}(G^+) }|\widehat{t^+}(\chi)|^2 \log (A(\chi)+2)\,, \\[-9pt]\nonumber\end{align} $$

where

$$ \begin{align*}m^t_{L} := \max \Big \{ \mathrm{ord}_{s=\rho} \Big( \prod_{\substack{\chi \in \mathrm{supp}(\widehat {t^+}) }} L(s,L/{\mathbb{Q}},\chi)\Big) : \Re(\rho)=\beta_L^t, 0<|\Im(\rho)|< (T_{L}\log (T_{L}))^2\},\end{align*} $$

with $T_{L}:=\log (\mathrm {rd}_L+2)\max _{\substack { \chi \in \mathrm {Irr}(G^+) }}\chi (1)$ . Assuming, moreover, that $\beta _{L}^{t}=\frac 12$ , andFootnote ²³ $\widehat {t^+}(\chi ) \not \equiv 0$ , we have, under LI $^-$ , the lower bound

(88) $$ \begin{align} \mathrm{Var}[X(L/K;t)] \gg \sum_{ \chi\in \mathrm{Irr}(G^+) }|\widehat{t^+}(\chi)|^2 \log (A(\chi)+2)\,. \end{align} $$

In the particular case $t=|G|1_{\{\mathrm {id}\}}$ , the lower bound $ \mathrm {Var}[X(L/K;|G|1_{\{\mathrm {id}\}} )]\gg \log (d_L+2)$ holds assuming only the Riemann hypothesis for $\zeta _L(s)$ (without requiring $L/{\mathbb {Q}}$ to be Galois).

We recall that if $L/{\mathbb {Q}}$ is Galois and under AC and LI $^-$ , $m^t_{L}\leq 1$ and $M_L\leq \max _{\substack {\chi \in \mathrm {Irr}(G^+) }}\chi (1) $ .

Proof of Proposition 4.6

We start by establishing (86). Note that for any $s_0\in \mathbb C$ , $C\in G^\sharp $ and $g_C\in C$ ,

$$ \begin{align*}\Big|\sum_{\chi \in \mathrm{Irr}(\langle g_C\rangle)} \chi(g_C) \mathrm{ord}_{s=s_0}L(s,L/L^{\langle g_C\rangle},\chi )\Big| \leq \mathrm{ord}_{s=s_0}\zeta_L(s).\\[-9pt]\end{align*} $$

We have used the crucial fact that $\mathrm {ord}_{s=s_0}L(s,L/L^{\langle g_C\rangle },\chi )\geq 0$ (since AC holds for the abelian extension $L/L^{\langle g_C \rangle }$ ). Hence, (73) implies that

$$ \begin{align*} \mathrm{Var}[X(L/K;t)]&\leq 2{\lVert{t}\rVert}^2_1\mathop{\sum \Bigl.^{*}}\limits_{\gamma \in Z_{L} } \frac{(\mathrm{ord}_{s=\beta^t_L+i\gamma}\zeta_L(s))^2}{(\beta^t_{L})^2+\gamma^2}\,,\\[-9pt] \end{align*} $$

where $Z_L$ is defined in (66). Now, we have the classical unconditional upper bound

(89) $$ \begin{align} \mathrm{ord}_{s=\rho}\zeta_L(s) \ll \log (d_L(|\Im(\rho)|+4)^{[L:{\mathbb{Q}}]})\\[-9pt]\nonumber \end{align} $$

(see [Reference Iwaniec and KowalskiIK, (5.27)]); we deduce that

$$ \begin{align*}\mathrm{Var}[X(L/K;t)] \ll {\lVert{t}\rVert}^2_1 (\log (d_L+2))^2+ {\lVert{t}\rVert}^2_1|G^+|\log (d_L+2) \ll {\lVert{t}\rVert}^2_1(\log (d_L+2))^2,\end{align*} $$

by Lemma 4.1. To prove (86), we apply Lemma 4.5 to the trivial extension $L/L$ ; this takes the form

$$ \begin{align*}\mathop{\sum \Bigl.^{*}}\limits_{\substack{\gamma \in Z_{L} \\ |\gamma|>T}} \frac{\mathrm{ord}_{s=\beta^t_L+i\gamma}\zeta_L(s)}{(\beta_L^t)^2+\gamma^2} \ll \frac{\log(d_L(T+4)^{[L:{\mathbb{Q}}]})}{T}.\\[-9pt] \end{align*} $$

We deduce that for any $T\geq 1,$

$$ \begin{align*} \mathop{\sum \Bigl.^{*}}\limits_{\substack{\gamma \in Z_{L} \\ |\gamma|>T}} \frac{(\mathrm{ord}_{s=\beta^t_L+i\gamma}\zeta_L(s))^2}{(\beta^t_L)^2+\gamma^2} &\ll \mathop{\sum \Bigl.^{*}}\limits_{\substack{\gamma \in Z_{L} \\ |\gamma|>T}} \frac{\log (d_L(|\gamma|+4)^{[L:{\mathbb{Q}}]})\mathrm{ord}_{s=\beta^t_L+i\gamma}\zeta_L(s)}{\beta_{L}^2+\gamma^2} \\ &\ll \frac{(\log(d_L(T+4)^{[L:{\mathbb{Q}}]}))^2}{T}. \end{align*} $$

Moreover, by Lemma 4.3 (see the comments before this lemma about replacing $\frac 14$ by $(\beta _L^t)^2\in [\frac 14,1 ]$ ), taking $T= \log (d_L+2) (\log \log (d_L+2))^2$ and applying Lemma 4.1,

$$ \begin{align*}\mathop{\sum \Bigl.^{*}}\limits_{\substack{\gamma \in Z_{L} \\ |\gamma|\leq T}} \frac{(\mathrm{ord}_{s=\beta^t_L+i\gamma}\zeta_L(s))^2}{(\beta^t_L)^2+\gamma^2} \ll M_L \log (d_L+2). \end{align*} $$

The upper bound (86) follows. Also, under GRH, Proposition 3.18 reads

$$ \begin{align*}\mathrm{Var}[X(L/K;|G|1_{\{\mathrm{id}\}})]=2\mathop{\sum \Bigl.^{*}}\limits_{\gamma \in Z_{L} } \frac{(\mathrm{ord}_{s=\beta^t_L+i\gamma}\zeta_L(s))^2}{\tfrac 14+\gamma^2}\geq \sum_{\gamma \in Z_{L} } \frac{1}{\tfrac 14+\gamma^2} \gg \log (d_L+2). \end{align*} $$

We now move to (87). We enumerate the characters $\chi \in \mathrm {Irr} (G^+) = \{ \chi _1,\chi _2, \cdots , \chi _k\}$ in such a way that for each $1\leq j\leq k-1$ , $ |\widehat {t^+}(\chi _j)| \geq |\widehat {t^+}(\chi _{j+1})|. $ Then, by Proposition 3.18, we have that

$$ \begin{align*} \mathrm{Var}[X(L/K;t)] &= 2\mathop{\sum \Bigl.^{*}}\limits_{\gamma \in Z_{L} } \frac{1}{\beta_{L}^2+\gamma^2} |\mathrm{ord}_{s=\beta^t_L+i\gamma}L(s,L/{\mathbb{Q}},t^+ )|^2 \\ &\leq 2 \sum_{j=1}^k \mathop{\sum \Bigl.^{*}}\limits_{ \substack{\gamma \in Z_{L} \\ L(\beta_{L}+i\gamma,\chi_j)=0 \\ L(\beta_{L}+i\gamma,\chi_{\ell})\neq 0 \text{ for } \ell <j}} \\&\quad \times \frac{1}{\beta_{L}^2+\gamma^2} \Big(\sum_{ \chi \in \mathrm{Irr}(G^+)} |\widehat{t^+}(\chi)|\mathrm{ord}_{s=\beta^t_L+i\gamma}L(s,L/{\mathbb{Q}},\chi )\Big)^2 \\ &= V_{\leq T}+V_{>T} , \end{align*} $$

where $V_{\leq T}$ denotes the sum over $\gamma \leq T$ and $V_{> T}$ that over $\gamma>T$ . Now, if $L(\beta _L^t+i\gamma ,L/{\mathbb {Q}},\chi _j)=0$ and $L(\beta _L^t+i\gamma ,L/{\mathbb {Q}},\chi _{\ell })\neq 0 \text { for } \ell <j$ , then

$$ \begin{align*}\sum_{\chi \in \mathrm{Irr}(G^+)}|\widehat{t^+}(\chi)|\mathrm{ord}_{s=\beta^t_L+i\gamma}L(s,L/{\mathbb{Q}},\chi ) \leq |\widehat{t^+}(\chi_j)|m^t_{L}(\beta_{L}+i\gamma) ,\end{align*} $$

where $m^t_{L}(\rho )$ is the order of vanishing of $\prod _{\chi \in \mathrm {supp}(\widehat {t^+})}L(s,L/{\mathbb {Q}},\chi )$ at $s=\rho $ . Hence, for any $T>1$ and denoting

$$ \begin{align*}m^t_{L}(T) := \max \Big \{ \mathrm{ord}_{s=\rho} \Big( \prod_{\substack{\chi \in \mathrm{supp}(\widehat{t^+}) }} L(s,L/{\mathbb{Q}},\chi)\Big) : \Re(\rho)=\beta_L^t, 0<|\Im(\rho)| \leq T\},\end{align*} $$

we have that

$$ \begin{align*} V_{\leq T}&\ll (m^t_{L}(T))^2 \sum_{j=1}^k |\widehat{t^+}(\chi_j)|^2 \mathop{\sum \Bigl.^{*}}\limits_{ \substack{\gamma \in Z_{L} \\ L(\beta^t_{L}+i\gamma,\chi_j)=0 \\ L(\beta^t_{L}+i\gamma,\chi_{\ell})\neq 0 \text{ for } \ell <j \\ \gamma \leq T}} \frac{1}{(\beta_L^t)^2+\gamma^2} \\ & \ll (m^t_{L}(T))^2 \sum_{\chi \in \mathrm{Irr}(G^+) } |\widehat{t^+}(\chi)|^2 \mathop{\sum \Bigl.^{*}}\limits_{\gamma_{\chi}} \frac{1}{(\beta_L^t)^2+\gamma_{\chi}^2} \\ &\ll (m^t_{L}(T))^2 \sum_{\chi \in \mathrm{Irr}(G^+) } |\widehat{t^+}(\chi)|^2\log (A(\chi)+2). \end{align*} $$

In a similar fashion, and by applying (89) and Lemma 4.5 (with $j=0,2$ ), we see that

$$ \begin{align*} V_{> T}\ll\frac 1T \sum_{\chi \in \mathrm{Irr}(G^+)}\chi(1) \log(\mathrm{rd}_L+2) \sum_{\chi \in \mathrm{Irr}(G^+) } |\widehat{t^+}(\chi)|^2\log (A(\chi)+2),\\[-9pt] \end{align*} $$

and (87) follows from taking

$$ \begin{align*}T=\Big( \Big(\log (\mathrm{rd}_L+2)\max_{\substack{ \chi \in \mathrm{Irr}(G^+) }} \chi(1)\Big) \cdot \log\Big( \log (\mathrm{rd}_L+2)\max_{\substack{ \chi \in \mathrm{ Irr}(G^+) }} \chi(1)\Big)\Big)^2 .\\[-9pt]\end{align*} $$

Coming back to the general case and assuming $\beta ^t_L=\tfrac 12$ and LI $^-$ , we see that for $\chi _1\neq \chi _2\in \mathrm {Irr}(G^+)$ , the sets of nonreal zeros of $L(s,\chi _1) $ and $L(s,\chi _2)$ are disjoint. Hence, Proposition 3.18 takes the form

$$ \begin{align*}\mathrm{Var}[X(L/K;t)] &= \sum_{\chi\in \mathrm{Irr}(G^+)} |\widehat {t^+}(\chi)|^2 \sum_{\gamma_{\chi}\neq 0} \frac {1}{\frac 14+\gamma_{\chi}^2}.\\[-9pt] \end{align*} $$

The lower bound (88) follows once more from applying Lemma 4.3.