4.1 The beta sieve

Like most combinatorial sieves, the beta sieve provides a mechanism to estimate sifting functions of the form $S(\mathcal {A},\mathcal {P},z)$ given arithmetic information about the related quantities $|\mathcal {A}_d| := \sum _{d\mid n}a_n$ for squarefree integers d. More specifically, we require an approximation $|\mathcal {A}_d| = |\mathcal {A}_1|g(d) + r(d)$ , where $g(d)$ is a multiplicative function supported on squarefree integers and

$$\begin{align*}R(z) := \sum_{\substack{d\leqslant z\\ d \textrm{ squarefree}}}|r(d)| \end{align*}$$

is small. Define

$$ \begin{align*}V(z) = \sum_{d \mid P(z)}\mu(d)g(d) = \prod_{p \in \mathcal{P}_{\leqslant z}}(1-g(p)).\end{align*} $$

We shall assume that for some $\kappa , L \geqslant 0$ , we have

(4.4) $$ \begin{align} V(w) \leqslant \left(\frac{\log z}{\log w}\right)^{\kappa}\left(1+\frac{L}{\log w}\right)V(z) \end{align} $$

for all $2 \leqslant w \leqslant z$ .

For some choice of sieve weights $\Lambda ^{\pm } =(\lambda _d^{\pm })_{d \geqslant 1}$ , define

$$\begin{align*}V^{\pm}(z) = \sum_{d\mid P(z)}\lambda_d^{\pm}g(d). \end{align*}$$

We recall that if $\Lambda ^{\pm }$ are upper and lower bound sieves of level z (i.e., if $\lambda ^{\pm }_d$ are supported on squarefree integers $d<z$ and

$$\begin{align*}\sum_{d\mid m}\lambda^-_d \leqslant \sum_{d\mid m}\mu(d) \leqslant \sum_{d\mid m}\lambda^+_d \end{align*}$$

for all integers m), then we have

(4.5) $$ \begin{align} \begin{aligned} S(\mathcal{A}, \mathcal{P}, z) &\leqslant |\mathcal{A}_1|V^+(z) + R(z),\\ S(\mathcal{A}, \mathcal{P}, z) &\geqslant |\mathcal{A}_1|V^-(z) - R(z). \end{aligned} \end{align} $$

The main theorem of the beta sieve we apply is given in [Reference Friedlander and Iwaniec18, Theorem 11.12]. We record this theorem here for convenience, in the special case $s=1$ .

Theorem 4.1. Suppose $\kappa , L$ are such that the assumption (4.4) holds. Then there is a choice of upper and lower bound sieve weights $\Lambda ^{\pm }$ (taking values in $\{-1,0,1\}$ ), and explicit constants $A(\kappa ), B(\kappa ) \geqslant 0$ such that, as $z\rightarrow \infty $ , we have

(4.6) $$ \begin{align} \begin{aligned} V^+(z) &\leqslant \left(A(\kappa) + o(1)\right)V(z),\\ V^-(z) &\geqslant \left(B(\kappa) + o(1)\right)V(z). \end{aligned} \end{align} $$

In our applications of the beta sieve, the required bounds on $R(z)$ are provided by Corollary 3.2 and Lemma 3.6. For $i\in \{0,\ldots , k\}$ , we define multiplicative functions

$$ \begin{align*} \varrho _i(d_1,d_2) &=\#\{a,b \ (\mathrm{mod}\ d_1d_2): d_1\mid f_i(a,b), d_2\mid f(a,b)\},\\ \varrho _i(d) &=\#\{a,b \ (\mathrm{mod}\ d): f_i(a,b) \equiv 0 \ (\mathrm{mod}\ d)\},\\ \varrho (d) &= \#\{a,b \ (\mathrm{mod}\ d): f(a,b) \equiv 0 \ (\mathrm{mod}\ d)\}. \end{align*} $$

We note that the function $\varrho _i(d_1,d_2)$ is the same as the function $\varrho (d_1,d_2)$ from (3.2) with $g_1(x,y) = f_i(x,y)$ and $g_2(x,y) = f(x,y)$ , but in this section, we add a subscript to keep track of the dependence on i. When $\gcd (d_1,d_2) = 1$ , we have $\varrho _i(d_1,d_2) = \varrho _i(d_1)\varrho (d_2)$ . Moreover, for any $i\in \{1,\ldots , k\}$ and any prime $p\notin S$ , we have

(4.7) $$ \begin{align} \varrho _i(p) &= \nu_i(p)(p-1)+1, \end{align} $$

(4.8) $$ \begin{align} \varrho (p) &= \nu(p)(p-1)+1, \end{align} $$

where $\nu _i(p)$ and $\nu (p)$ are as defined in (2.2) and (2.3). We define multiplicative functions

$$\begin{align*}g(p) := \frac{\varrho (p)}{p^2}, \qquad g_i(p) := \frac{\varrho _i(p)}{p^2} \end{align*}$$

and define

(4.9) $$ \begin{align} V(x)= \prod_{\substack{p\in \mathcal{P}_{\leqslant z}}}\left(1-g(p)\right), \qquad V_i(x)= \prod_{\substack{p\in \mathcal{P}^{\prime}_{\leqslant z}}}\left(1-g_i(p)\right) \end{align} $$

for $i\in \{1,\ldots , k\}$ .

4.2 Sieve dimensions

We prove in Lemma 4.3 that the functions $V,V_i$ defined above satisfy the hypothesis (4.4) with the sieve dimensions

where $\alpha , \theta _i$ and $\theta $ are as defined in (2.4), (2.5) and (2.6).

In the notation of Theorem 4.1, we write $A=A(\kappa ), B=B(\kappa )$ and $A_i = A(\kappa _i)$ . We assume throughout this section that $\kappa <1/2$ , and so $\kappa _i>1/2$ . Then A and B are defined in [Reference Friedlander and Iwaniec18, Equations (11.62, 11.63)] (see also Section 4.9), and $A_i$ is defined in [Reference Friedlander and Iwaniec18, Equations (11.42), (11.57)]. A table of numerical values of these constants can be found in [Reference Friedlander and Iwaniec18, Section 11.19].

Lemma 4.3. Let $x\geqslant 1$ . For $i\in \{1,\ldots ,k\}$ , and $V,V_i$ as in (4.9), there exist constants $c,c_i>0$ such that

(4.10) $$ \begin{align} V(x)&= \frac{c}{(\log x)^{\kappa}}\left(1+O((\log x)^{-1})\right), \end{align} $$

(4.11) $$ \begin{align} V_i(x)&= \frac{c_i}{(\log x)^{\kappa_i}}\left(1+O((\log x)^{-1})\right). \end{align} $$

The asymptotic in (4.11) also holds for $i=0$ when $f_0 \not \equiv 1$ .

Proof. We follow a similar approach to [Reference Irving27, Lemma 4.2]. Below, we denote by C a constant which is allowed to vary from line to line. We have

(4.12) $$ \begin{align} \log V(x)&=-\sum_{p \in \mathcal{P}_{\leqslant x}}\left(\sum_{m=1}^{\infty}\frac{\varrho (p)^m}{mp^ {2m}}\right) \end{align} $$

(4.13) $$ \begin{align} &=-\sum_{p\in \mathcal{P}_{\leqslant x}}\frac{\nu(p)(p-1)+1}{p^2} + C + O((\log x)^{-1}). \end{align} $$

(4.14) $$ \begin{align} &=-\sum_{p\in \mathcal{P}_{\leqslant x}}\frac{\nu(p)}{p} + C + O((\log x)^{-1}), \end{align} $$

where in (4.14) we have used that $\nu (p) \leqslant \deg f$ for all but finitely many primes p. To estimate the sum in (4.14), we apply partial summation, together with our assumption (2.6). For $t\geqslant 2$ , we define

(4.15) $$ \begin{align} A_t = \sum_{p \in \mathcal{P}_{\leqslant t}}\nu(p). \end{align} $$

Then

(4.16) $$ \begin{align} \sum_{p \in \mathcal{P}_{\leqslant x}}\frac{\nu(p)}{p}&=\frac{A_x}{x} + \int_{2}^x \frac{A_t}{t^2}\textrm{d}t\nonumber\\ &=\kappa\int_{2}^x \frac{\pi(t)\left(1+O\left((\log t)^{-1}\right)\right)}{t^2} \textrm{d}t + O((\log x)^{-1})\nonumber\\ &=\kappa\int_{2}^x \frac{\textrm{d}t}{t\log t} + C+O((\log x)^{-1})\nonumber\\ &=\kappa\log\log x + C + O((\log x)^{-1}). \end{align} $$

We deduce (4.10) by taking the exponential of (4.16).

We can prove (4.11) in a similar way. When $i=0$ and $f_0 \not \equiv 1$ , we have $\varrho _0(p) = p$ , and so the result is a consequence of Mertens’ theorem [Reference Iwaniec and Kowalski29, Equation (2.16)]. For any $i\in \{1,\ldots , k\}$ , we have

(4.17) $$ \begin{align} \log V_i(x)&=\sum_{p\in \mathcal{P'}_{\leqslant x}}\frac{\nu_i(p)}{p} + C + O((\log x)^{-1})\nonumber\\ &=\sum_{p\leqslant x \textrm{ prime}}\frac{\nu_i(p)}{p} - \sum_{p\in \mathcal{P}_{\leqslant x}}\frac{\nu_i(p)}{p} + C + O((\log x)^{-1}). \end{align} $$

Similarly to above, using partial summation and (2.5), we have

(4.18) $$ \begin{align}\sum_{p\in \mathcal{P}_{\leqslant x}}\frac{\nu_i(p)}{p} = \alpha\theta_i \log\log x + C+ O((\log x)^{-1}).\end{align} $$

To treat the first sum in (4.17), we define $L_i$ to be the number field generated by $f_i$ . For all but finitely many primes p, the quantity $\nu _i(p)$ is equal to the number of degree one prime ideals $\mathfrak {p}$ in $L_i$ above p. Let $\pi _{L_i}(x)$ denote the number of prime ideals $\mathfrak {p}$ in L of norm at most x. This count is dominated by degree one ideals. In fact, the number of prime ideals of degree at least $2$ enumerated by $\pi _{L_i}(x)$ is $O(x^{1/2})$ because such an ideal must lie over a rational prime $p \leqslant x^{1/2}$ and each rational prime p has at most $[L_i:\mathbb {Q}]$ prime ideals in $L_i$ lying above it. Therefore,

$$ \begin{align*}\sum_{p\leqslant x \textrm{ prime}}\nu_i(p) = \pi_{L_i}(x) + O(x^{1/2}).\end{align*} $$

Using partial summation, as above, together with the Prime ideal theorem [Reference Mitsui38], we deduce that

(4.19) $$ \begin{align} \sum_{p\leqslant x \textrm{ prime}}\frac{\nu_i(p)}{p} = \log\log x + C + O((\log x)^{-1}). \end{align} $$

Combining this with (4.18) and taking exponentials, we deduce the asymptotic in (4.11).

In the following lemma, we record three more useful estimates following similar arguments to Lemma 4.3.

Lemma 4.4. There exists constants $C,C_i>0$ such that

(4.20) $$ \begin{align} \sum_{p \in \mathcal{P}_{\leqslant x}} \frac{\varrho _i(p)}{p^2} &=(1-\kappa_i)\log\log x + C + O((\log x)^{-1}), \end{align} $$

(4.21) $$ \begin{align} \sum_{p \in \mathcal{P}^{\prime}_{\leqslant x}} \frac{\varrho _i(p)}{p^2} &=\kappa_i\log\log x + C_i + O((\log x)^{-1}), \qquad\end{align} $$

(4.22) $$ \begin{align} \sum_{p \in \mathcal{P}^{\prime}_{\leqslant x}} \frac{\varrho _i(p)}{p^2}\log p &=\kappa_i\log x +O(1). \qquad\qquad\qquad \qquad\qquad\end{align} $$

Proof. The estimates (4.20) and (4.21) are immediate consequences of (4.16), (4.18) and (4.19), together with fact that

$$ \begin{align*}\frac{\varrho _i(p)}{p^2} =\frac{(p-1)\nu_i(p)+1}{p^2}= \frac{\nu_i(p)}{p} + O(p^{-2}).\end{align*} $$

To prove (4.22), we proceed via partial summation in a very similar manner to (4.16). We recall from the Prime number theorem that

(4.23) $$ \begin{align} \pi(t) = \frac{t}{\log t}+\frac{t}{(\log t)^2} + O\left(\frac{t}{(\log t)^3}\right). \end{align} $$

For $A_t$ as defined in (4.15), we have

$$ \begin{align*} \sum_{p \in \mathcal{P}^{\prime}_{\leqslant x}}\frac{\varrho _i(p)}{p^2}\log p &= \sum_{p \in \mathcal{P}^{\prime}_{\leqslant x}}\frac{\nu_i(p)}{p}\log p + O(1)\\ &=\frac{A_x\log x}{x} - \int_{2}^x A_t\left(\frac{\log t}{t}\right)' \textrm{d}t + O(1)\\ &=\kappa_i\int_2^x \frac{(\log t -1)\pi(t)(1+O((\log t)^{-A})}{t^2} \textrm{d}t + O(1)\\ &=\kappa_i\int_{2}^x \frac{1}{t} + O\left(\frac{1}{t(\log t)^{2}}\right) \textrm{d}t + O(1) \quad (\textrm{from } (4.3))\\ &=\kappa_i\log x + O(1), \end{align*} $$

as required.

4.3 The sum $S_1$

We apply the lower bound sieve $\Lambda ^{-}(\mathcal {A}, \mathcal {P}, g, N^{\gamma })$ and the level of distribution result from Corollary 3.2 with $g_1(x,y)=1, g_2(x,y)=f(x,y), D_1=1$ and $D_2= N^{\gamma }$ . The hypotheses of Corollary 3.2 require that $\gamma <1$ . We obtain

(4.24) $$ \begin{align} S_1 \geqslant (B+o(1))XV(N^{\gamma}). \end{align} $$

By Lemma 4.3, we have

$$ \begin{align*}V(N^{\gamma})\sim \frac{c}{(\log N^{\gamma})^{\kappa}}.\end{align*} $$

For any $\epsilon>0$ , taking $\gamma $ sufficiently close to $1$ , we obtain

(4.25) $$ \begin{align} S_1 \geqslant \frac{(cB-\epsilon+o(1))X}{(\log N)^{\kappa}}. \end{align} $$

4.4 The sums $S_2^{(i)}$

We write $f_i(a,b) = pr$ , for $p \in \mathcal {P}$ and $N^{\gamma }< p \ll N$ . We apply the switching principle, which transforms the sum over p defining $S_2^{(i)}$ into a much shorter sum over the variable r.

Let $R=N^{1-\gamma }$ . The sums $S_2^{(i)}$ only involve linear factors $f_i(x,y)$ since we assume $i\in \{0,\ldots , m\}$ . Therefore, for $(a,b) \in \mathcal {R}\cap \mathbb {Z}^2$ , we have $f_i(a,b) \ll N$ , and so $|r|\ll R$ . Let $z = N^{1/3}$ . We shall take $\gamma $ arbitrarily close to 1; for now, we assume that $\gamma>2/3$ . Then by definition of $S(\mathcal {A}_p^{(i)},\mathcal {P},p)$ , we know that $\gcd (r, P(R))=1$ and $\gcd (f(a,b),P(z))=1$ .

Let $r' = |r|/\gcd (r,\Delta )$ . We now explain why $r'$ only has prime factors in $\mathcal {P}'$ . Fix $l \in S$ . Recall the assumption that $\nu _l(f(a_0,b_0)) < \nu _l(\Delta )$ . Since $f_i(a_0,b_0) \mid f(a_0,b_0)$ and $f_i(a,b) \equiv f_i(a_0,b_0) \ (\mathrm {mod}\ \Delta )$ (by the condition $C(a,b)$ ), we have

(4.26) $$ \begin{align} \nu_l(r) = \nu_l(f_i(a, b)) \leqslant \nu_l(f(a,b)) = \nu_l(f(a_0,b_0)) < \nu_l(\Delta). \end{align} $$

Therefore, $\gcd (r',\Delta ) =1$ , and by the assumption that every prime in S divides $\Delta $ , this implies $r'$ has no prime factors in S. Moreover, p is the smallest prime in $\mathcal {P}$ dividing $f_i(a,b)$ . Since $r<p$ , this means that $r'$ has no prime factors in $\mathcal {P}$ , and hence only has prime factors in $\mathcal {P}'$ .

However, $f_i(a,b)/r'$ has no prime factors in $\mathcal {P}'$ . Therefore,

(4.27) $$ \begin{align} S_2^{(i)} \leqslant \sum_{\substack{r'\ll R\\ \gcd(r',P(R)\Delta)=1}}S_2^{(i)}(r'), \end{align} $$

where

(4.28) $$ \begin{align} S_2^{(i)}(r') = \#\left\{\!\!\!(a,b) \in \mathcal{R}\cap \mathbb{Z}^2: \begin{array}{l l l} &\displaystyle C(a,b), r'\mid f_i(a,b), \\ &\displaystyle \gcd(f_i(a,b)/r', P'(z))=1\\ &\displaystyle \gcd(f(a,b),P(z))=1 \end{array} \!\!\!\!\!\right\}. \end{align} $$

Below, for convenience, we change notation from $r'$ back to r.

Let $\Lambda _1^+, \Lambda _2^+$ be upper bound sieves of level z. Defining $A(dr,e), r(dr,e)$ as in Section 3 with $g_1(x,y) = f_i(x,y)$ and $g_2(x,y) = f(x,y)$ , and writing

$$\begin{align*}m_1 = \gcd\left(\frac{f_i(a,b)}{r}, P'(z)\right), \qquad m_2= \gcd(f(a,b), P(z)), \end{align*}$$

we obtain

(4.29) $$ \begin{align} S_2^{(i)}(r) &\leqslant \sum_{\substack{(a,b) \in \mathcal{R}\cap \mathbb{Z}^2\\ C(a,b)}}\sum_{d \mid m_1}\mu(d)\sum_{e\mid m_2}\mu(e)\nonumber\\ &\leqslant \sum_{\substack{(a,b) \in \mathcal{R}\cap \mathbb{Z}^2\\ C(a,b)}}\sum_{d \mid m_1}\lambda_1^+(d)\sum_{e\mid m_2}\lambda_2^+(e)\nonumber\\ &\leqslant \sum_{\substack{d \mid P'(z)\\ \gcd(d,r)=1}}\lambda_1^+(d)\sum_{e\mid P(z)}A(dr,e)\lambda_2^+(e)\nonumber\\ &=\sum_{\substack{d \mid P'(z)\\ \gcd(d,r)=1}}\lambda_1^+(d)\sum_{e\mid P(z)}\lambda_2^+(e)\left(\frac{X\varrho (dr,e)}{(dre)^2} + r(dr,e)\right)\nonumber\\ &\leqslant \left(\frac{X\varrho _i(r)}{r^2}\sum_{\substack{d \mid P'(z)\\ \gcd(d,r) = 1}}\lambda_1^+(d)g_i(d)\sum_{\substack{e \mid P(z)}}\lambda_2^+(e)g(d)\right) + \sum_{\substack{d,e \leqslant z\\ \gcd(dr,e) = 1\\ \gcd(dre, \Delta) = 1}}|r(dr,e)|. \end{align} $$

Choose $\lambda _1^+,\lambda _2^+$ to be the beta sieves $\Lambda ^+(\mathcal {P}'\backslash \{p\mid r\}, g_i, N^{1/3}), \Lambda ^+(\mathcal {P}, g, N^{1/3})$ , respectively. Then the remainder term in (4.29) can be bounded by

$$ \begin{align*}\sum_{\substack{d,e \leqslant N^{1/3}\\ \gcd(dr,e) = 1\\ \gcd(dre, \Delta) = 1}}|r(dr,e)| \leqslant \sum_{\substack{d \ll N^{1/3}R\\e \ll N^{1/3}\\ \gcd(dr,e) = 1\\ \gcd(dre, \Delta) = 1}}|r(d,e)|,\end{align*} $$

which is negligible by Lemma 3.6 because $N^{2/3}R = N^{5/3-\gamma } \ll N^{1-\epsilon }$ .

Define a multiplicative function $h_i(r)$ supported on squarefree integers r, which is zero unless all prime factors of r are in $\mathcal {P}'$ and

(4.30) $$ \begin{align} h_i(r)=\frac{\varrho _i(r)}{r^2}\prod_{p\mid r}\left(1-\frac{\varrho _i(p)}{p^2}\right)^{-1} \end{align} $$

otherwise. Applying Theorem 4.1, we conclude that

$$ \begin{align*}S_2^{(i)}(r) \leqslant Xh_i(r)V(N^{1/3})V_i(N^{1/3})(AA_i + o(1)).\end{align*} $$

From Lemma 4.3, we obtain

$$\begin{align*}S_2^{(i)}\leqslant \frac{(cc_iAA_i+o(1))X}{(\log N^{1/3})^{\kappa + \kappa_i}}\sum_{\substack{r \ll R}}h_i(r). \end{align*}$$

To deal with the sum over $h_i(r)$ , we note that

(4.31) $$ \begin{align} \sum_{\substack{r \ll R}}h_i(r) \leqslant \prod_{\substack{p \in \mathcal{P}'\\p\ll R}}\left(1+\sum_{m=1}^{\infty} h_i(p^m)\right)= \prod_{\substack{p \in \mathcal{P}'\\p \ll R}}\left(1-g_i(p)\right)^{-1}(1+O(p^{-2})). \end{align} $$

Applying Lemma 4.3 to the product, we obtain

$$ \begin{align*}\sum_{\substack{r \ll R}}h_i(r) \ll (\log R)^{\kappa_i}.\end{align*} $$

Since $R=N^{1-\gamma }$ , we deduce that

$$ \begin{align*}S_2^{(i)} \ll \frac{cc_iAA_iX}{(\log N)^{\kappa}}\left(\frac{(1-\gamma)^{\kappa_i}}{(1/3)^{\kappa_i+\kappa}}\right).\end{align*} $$

Therefore, $S_2^{(i)}$ can be made negligible compared to $S_1$ by taking $\gamma $ arbitrarily close to 1.

4.5 The sums $S_3^{(i)}$

For a fixed prime p and an upper bound sieve $\lambda ^+$ of level z, we have

$$ \begin{align*}S(\mathcal{A}_p^{(i)},\mathcal{P}, z) \leqslant \sum_{d\mid P(z)}\lambda^+(d)A(p,d),\end{align*} $$

where $A(p,d)$ is as in Section 3 with $g_1(x,y) = f_i(x,y)$ and $g_2(x,y) = f(x,y)$ .

We first deal with the primes p in the interval $I:=(N^{\gamma }, N^{2-\gamma -\epsilon }]$ . For any $p\in I$ , let $\lambda ^+$ be the beta sieve $\Lambda ^+(\mathcal {A}_p^{(i)}, \mathcal {P}, g_i(p)g, N^{\gamma })$ . We obtain

(4.32) $$ \begin{align} S(\mathcal{A}_p^{(i)}, \mathcal{P},p) &\leqslant S(\mathcal{A}_p^{(i)}, \mathcal{P},N^{\gamma})\nonumber\\ &\leqslant (A+o(1))XV(N^{\gamma})g_i(p) + \sum_{\substack{d \leqslant N^{\gamma}\\ \gcd(d,p\Delta)=1}}|r(p,N^{\gamma})|. \end{align} $$

We apply Corollary 3.2 with $D_1 = N^{2-\gamma - \epsilon }$ and $D_2 = N^{\gamma }$ . These choices of $D_1,D_2$ satisfy the hypotheses of Corollary 3.2. Taking a sum over $p\in I$ , the contribution from the remainder term in (4.32) is negligible. We obtain

(4.33) $$ \begin{align} \sum_{\substack{p \in I\cap \mathcal{P}}}S(\mathcal{A}_p^{(i)}, \mathcal{P},p)\leqslant (A+o(1)) XV(N^{\gamma})\sum_{p \in I\cap \mathcal{P}}g_i(p). \end{align} $$

It follows from Lemma 4.4 that

$$ \begin{align*} \sum_{\substack{N^{\gamma}<p\leqslant N^{2-\gamma-\epsilon}}}g_i(p) &= \log\log N^{2-\gamma-\epsilon} - \log\log N^{\gamma} + o(1)\\ &= \log(2-\gamma - \epsilon) - \log \gamma +o(1). \end{align*} $$

Therefore, the contribution to $S_3^{(i)}$ from this range is negligible if we take $\gamma $ arbitrarily close to 1.

In the remaining range $N^{2-\gamma -\epsilon }< p <N^{\beta _i}$ , we split into dyadic intervals $(R,2R]$ . Note that for $p\in (R,2R]$ , the assumption $\gamma <1$ implies that $N^{2-\epsilon }/R <p$ , so $S(\mathcal {A}_p^{(i)}, \mathcal {P}, p) \leqslant S(\mathcal {A}_p^{(i)}, \mathcal {P}, N^{2-\epsilon }/R)$ . For each dyadic interval, we apply the beta sieve $\Lambda ^+(\mathcal {A}_p^{(i)}, \mathcal {P}, g_i(p)g, N^{2-\epsilon }/R)$ and the level of distribution result from Corollary 3.2 with $D_1=2R$ and $D_2 = N^{2-\epsilon }/R$ . At this point, we need to assume that $\beta _i<2$ for all i, so that $D_2\geqslant 1$ . We obtain

(4.34) $$ \begin{align} &\sum_{\substack{N^{2-\gamma-\epsilon}<p< N^{\beta_i}\\ p \in \mathcal{P}}}S(\mathcal{A}_p^{(i)}, \mathcal{P},p) \end{align} $$

(4.35) $$ \begin{align} &\leqslant \sum_{\substack{R \textrm{ dyadic}\\N^{2-\gamma -\epsilon}<R < N^{\beta_i}}}\sum_{\substack{p \in (R,2R]\\ p \in \mathcal{P}}}S(\mathcal{A}_p^{(i)}, \mathcal{P},N^{2-\epsilon}/R)\nonumber\\ &\leqslant (A+o(1))X\sum_{\substack{R \textrm{ dyadic}\\ N^{2-\gamma -\epsilon}<R < N^{\beta_i}}}V(N^{2-\epsilon}/R)\sum_{\substack{p \in (R,2R]\\p \in \mathcal{P}}}g_i(p)\nonumber\\ &\leqslant \frac{(cA+o(1))X}{(\log N)^{\kappa}}\sum_{\substack{ N^{2-\gamma-\epsilon} \leqslant p< N^{\beta_i}\\p \in \mathcal{P}}}\frac{g_i(p)}{(2-\epsilon - \frac{\log p}{\log N})^{\kappa}}, \end{align} $$

where the last line follows from Lemma 4.3 and the fact that $V(N^{2-\epsilon }/R)<V(N^{2-\epsilon -\log p/\log N})$ for all $p\in (R,2R]$ .

We denote the sum in (4.35) by $T(\beta _i, \kappa )$ . Since $\gamma <1$ , we have $2-\gamma -\epsilon>1$ for sufficiently small $\epsilon $ , and so we may upper bound $T(\beta _i,\kappa )$ by enlarging its range of summation to $N<p< N^{\beta _i}$ . Define

$$ \begin{align*}A(t) = \sum_{p \in \mathcal{P}_{\leqslant t}}g_i(p), \qquad h(t) = \left(2-\epsilon - \frac{\log t}{\log N}\right)^{-\kappa}.\end{align*} $$

From Lemma 4.4, we have $A(t)= (1-\kappa _i)\log \log t + C+o(1)$ for some constant C. In particular, for any $r>0$ , we have $A(N^r)-A(N) = (1-\kappa _i)\log r + o(1)$ . Applying summation by parts, followed by the substitution $t=N^s$ , we obtain

(4.36) $$ \begin{align} T(\beta_i,\kappa) &\leqslant (A(N^{\beta_i})-A(N))h(N^{\beta_i}) - \int_{N}^{N^{\beta_i}}(A(t)-A(N))h'(t) \textrm{d}t \nonumber \\ & =(A(N^{\beta_i})-A(N))h(N^{\beta_i}) - \int_{1}^{\beta_i}(A(N^s)-A(N))\frac{\partial h(N^s)}{\partial s}\textrm{d}s \nonumber \\ &=(1-\kappa_i)\log \beta_i h(N^{\beta_i}) -(1-\kappa_i)\int_{1}^{\beta_i}(\log s) \frac{\partial h(N^s)}{\partial s}\textrm{d}s + o(1) \nonumber \\ &=(1-\kappa_i)\int_{1}^{\beta_i}\frac{h(N^s)}{s}\textrm{d}s + o(1). \end{align} $$

Taking $\epsilon $ sufficiently small and redefining it appropriately, we may replace $h(N^s)$ by $(2-s)^{-\kappa }$ at the cost of adding $\epsilon $ in (4.36). Combining with (4.35), we conclude that for any $\epsilon>0$ ,

(4.37) $$ \begin{align} S_3^{(i)}\leqslant \frac{(cA+\epsilon+o(1))X}{(\log N)^{\kappa}}\cdot (1-\kappa_i) \int_{1}^{\beta_i}(2-s)^{-\kappa}\frac{\textrm{d}s}{s}. \end{align} $$

Due to the factor $1-\kappa _i = \alpha \theta _i$ appearing in the above estimate, $S_3^{(i)}$ becomes negligible compared to $S_1$ as $\alpha \rightarrow 0$ . We perform a more precise quantitative comparison in Section 4.7.

4.6 The sums $S_4^{(i)}$

We begin in a similar manner to the sums $S_2^{(i)}$ , by writing $f_i(a,b) = pr$ , for $p \in \mathcal {P}$ , where now $N^{\beta _i}\leqslant p < x$ and $R=x/N^{\beta _i}$ . Let $D_1 = N^{\eta _1}$ and $D_2 = N^{\eta _2}$ for parameters $\eta _1, \eta _2>0$ (which may depend on r and i) to be chosen later. We assume $\eta _2 < \beta _i$ so that the condition $\gcd (f(a,b), P(p)) =1$ can be replaced by the weaker condition $\gcd (f(a,b), P(N^{\eta _2}))=1$ . Proceeding as in Section 4.4, we have

(4.38) $$ \begin{align} S_4^{(i)} \leqslant \sum_{\substack{r\ll R\\ \gcd(r,P(N^{\beta_i})\Delta)=1}}S_4^{(i)}(r), \end{align} $$

where

(4.39) $$ \begin{align} S_4^{(i)}(r) = \#\left\{\!(a,b) \in \mathcal{R}\cap \mathbb{Z}^2: \begin{array}{l l l} &\displaystyle C(a,b), r\mid f_i(a,b), \\ &\displaystyle \gcd(f_i(a,b)/r, P'(N^{\eta_1}))=1\\ &\displaystyle \gcd(f(a,b),P(N^{\eta_2}))=1 \end{array} \!\!\!\!\right\}. \end{align} $$

Similarly to (4.29), for any upper bound sieves $\lambda _1^+, \lambda _2^+$ of levels $N^{\eta _1}, N^{\eta _2}$ , the quantity $S_4^{(i)}(r)$ is bounded above by

$$\begin{align*}\left(\frac{X\varrho _i(r)}{r^2}\sum_{\substack{d \mid P'(N^{\eta_1})\\ \gcd(d,r) = 1}}\lambda_1^+(d)g_i(d)\sum_{\substack{e \mid P(N^{\eta_2})}}\lambda_2^+(e)g(d)\right) + \sum_{\substack{d\leqslant N^{\eta_1}, e \leqslant N^{\eta_2}\\ \gcd(dr,e) = 1\\ \gcd(dre, \Delta) = 1}}|r(dr,e)|. \end{align*}$$

In order to ensure the error terms from applying the level of distribution result from Corollary 3.2 are negligible after summing over r, we need $\eta _1,\eta _2$ to satisfy

(4.40) $$ \begin{align} \eta_1, \eta_2>0, \quad \eta_2 \leqslant 1-\delta \qquad (\textrm{for all }r\leqslant R), \end{align} $$

(4.41) $$ \begin{align} \sum_{r\leqslant R}N^{\eta_1+\eta_2}\leqslant N^{2-\delta}. \end{align} $$

Choose $\lambda ^+_1, \lambda ^+_2$ to be the beta sieves $\Lambda ^+(\mathcal {P}'\backslash \{p\mid r\}, g_i, N^{\eta _1}), \Lambda ^+(\mathcal {P}, g, N^{\eta _2})$ , respectively. Recalling the definition of $h_i(r)$ from (4.30), we obtain

(4.42) $$ \begin{align} S_4^{(i)}(r) \leqslant (AA_i+o(1))Xh_i(r)V_i(N^{\eta_1})V(N^{\eta_2}).\end{align} $$

Using Lemma 4.3 to estimate the products in (4.42), and taking a sum over r, we obtain

(4.43) $$ \begin{align} S_4^{(i)}\leqslant \frac{(cc_iAA_i+o(1))X}{(\log N)^{\kappa_i+\kappa}}\sum_{r\leqslant R}\frac{h_i(r)}{\eta_1^{\kappa_i}\eta_2^{\kappa}}. \end{align} $$

We divide the sum over r into dyadic intervals $r \in (R_1,2R_1]$ , and take $\eta _1,\eta _2$ depending only on $R_1$ and i. To obtain a good estimate for (4.43), we maximise $\eta _1^{\kappa _i}\eta _2^{\kappa }$ subject to the constraints

$$ \begin{align*}\eta_1,\eta_2>0, \qquad \eta_2\leqslant 1-\delta, \qquad \eta_1+\eta_2 \leqslant 2-\delta - \frac{\log R_1}{\log N}.\end{align*} $$

By a similar computation to [Reference Irving27, Section 6.5], the optimal solution is

$$ \begin{align*} \eta_1 &= \frac{\kappa_i}{\kappa+\kappa_i}\left(2-\delta-\frac{\log R_1}{\log N}\right),\\ \eta_2 &= \frac{\kappa}{\kappa+\kappa_i}\left(2-\delta-\frac{\log R_1}{\log N}\right). \end{align*} $$

We note that for $\delta>0$ sufficiently small, this solution satisfies $\eta _2 \leqslant 1-\delta $ due to the assumption $\kappa <1/2$ . Substituting this choice of $\eta _1,\eta _2$ into (4.43), we obtain

(4.44) $$ \begin{align} \sum_{r\leqslant R}\frac{h_i(r)}{\eta_1^{\kappa_i}\eta_2^{\kappa}}\leqslant \sum_{r\leqslant R}w(r,\delta)h_i(r), \end{align} $$

where

(4.45) $$ \begin{align} w(r,\delta) = \left(\frac{\kappa}{\kappa+\kappa_i}\right)^{-\kappa}\left(\frac{\kappa_i}{\kappa+\kappa_i}\right)^{-\kappa_i}\left(2-\delta -\frac{\log r}{\log N}\right)^{-(\kappa+\kappa_i)}. \end{align} $$

We treat the sum in (4.44) using partial summation, for which we require estimates for $\sum _{r\leqslant t}h_i(r)$ . For the case $d=2$ , the estimate already found in (4.31) is sufficient. Below, we find a more refined estimate which we use in the case $d=3$ .

We first consider the contribution to (4.44) from squarefree values of r. We would like to apply [Reference Friedlander and Iwaniec18, Theorem A.5], which states that under certain hypothesis on the function $h_i(r)$ , we have

(4.46) $$ \begin{align} \sum_{\substack{m \leqslant x\\ \mu^2(m)=1}}h_i(m)= c_{h_i}(\log x)^{\kappa_i} + O((\log x)^{\kappa_i -1}), \end{align} $$

where

$$ \begin{align*}c_{h_i} = \frac{1}{\Gamma(\kappa_i+1)}\prod_p \left(1-\frac{1}{p}\right)^{\kappa_i}(1+h_i(p)).\end{align*} $$

In the following lemma, we verify that the function $h_i(r)$ satisfies the required hypotheses for [Reference Friedlander and Iwaniec18, Theorem A.5].

Lemma 4.6. For any $x\geqslant 1$ and any $2\leqslant w< z$ , the function $h_i(r)$ satisfies the following estimates:

(4.47) $$ \begin{align} \prod_{w\leqslant p <z}(1+h_i(p)) &\ll \left(\frac{\log z}{\log w}\right)^{\kappa_i}, \end{align} $$

(4.48) $$ \begin{align} \sum_p h_i(p)^2\log p &< \infty, \end{align} $$

(4.49) $$ \begin{align} \sum_{p \leqslant x}h_i(p)\log p &= \kappa_i\log x + O(1). \end{align} $$

Proof. To prove (4.47), we note that $1+h_i(p) = \left (1-\frac {\varrho _i(p)}{p^2}\right )^{-1}$ for all $p\in \mathcal {P}'$ . The result is then immediate from Lemma 4.3. To prove (4.45), we recall that $\varrho _i(p) \ll p$ , and so $h_i(p) \ll p^{-1}$ . Therefore,

$$ \begin{align*}\sum_{p}h_i(p)^2\log p \ll \sum_{p}\frac{\log p}{p^2}< \infty.\end{align*} $$

Finally, we note that

$$ \begin{align*} \sum_{p\leqslant x}h_i(p)\log p =\sum_{p \in \mathcal{P}^{\prime}_{\leqslant x}}\frac{\varrho _i(p)}{p^2}\log p + O(1), \end{align*} $$

so that (4.49) follows by applying Lemma 4.4.

We can now evaluate the sum in (4.44) using partial summation. We obtain

$$ \begin{align*} \sum_{\substack{r\leqslant R\\ \mu^2(r) = 1}}w(r,\delta)h_i(r)&= w(R,\delta)\sum_{\substack{r\leqslant R\\ \mu^2(r)=1}}h_i(r)-\int_{1}^R \left(\sum_{\substack{r\leqslant t\\ \mu^2(r)=1}}h_i(r)\right)w'(t,\delta) \textrm{d}t\\ &=c_{h_i}\left[(\log R)^{\kappa_i}w(R,\delta)-\int_{1}^R w'(t,\delta)(\log t)^{\kappa_i}\textrm{d}t\right]\\ &\quad +o(1)\left[(\log R)^{\kappa_i}w(R,\delta)+\int_{1}^R w'(t,\delta)(\log t)^{\kappa_i}\textrm{d}t\right]\\ &= c_{h_i}\kappa_i\int_{1}^R w(t,\delta)(\log t)^{\kappa_i-1}t^{-1}\textrm{d}t+o\left((\log R)^{\kappa_i}\right)\\ &=c_{h_i}\kappa_i(\log N)^{\kappa_i}\int_{0}^{\log R/\log N}w(N^s, \delta)s^{\kappa_i -1}\textrm{d}s + o\left((\log R)^{\kappa_i}\right)\\ &\leqslant (c_{h_i}\kappa_i +o(1))(\log N)^{\kappa_i}\int_{0}^{d-\beta_i}W(s)\textrm{d}s, \end{align*} $$

where $W(s) = w(N^s, 0)s^{\kappa _i-1}.$

We now consider the contribution to (4.30) from those r which are not squarefree. Let $d_i$ denote the degree of $f_i$ . From [Reference Daniel15, Lemma 3.1], we have $\varrho _i(p^{\alpha }) \ll p^{2\alpha (1-1/d_i)}$ for all primes $p \notin S$ and any positive integer $\alpha $ . By the multiplicativity of $h_i(r)$ , it follows that $h_i(r) \ll r^{-2/d_i +\epsilon }$ .

We recall that a positive integer n is squareful if for any prime $p\mid n$ , we also have $p^2\mid n$ . Since $d_i\leqslant 3$ , we have

$$ \begin{align*}\sum_{r \textrm{ squareful}}h_i(r) \ll \sum_{r \textrm{ squareful}}r^{-2/d_i + \epsilon} \leqslant \sum_{r \textrm{ squareful}}r^{-2/3 + \epsilon}< \infty,\end{align*} $$

where the last inequality follows from partial summation together with the fact that there are $O(M^{1/2})$ squareful positive integers less than M. Since $h_i(r)$ is supported on integers with no prime factors in S, for any $\epsilon>0$ , there exists a set of primes $S_0$ , depending only on $\epsilon $ and f, such that for any $S\supseteq S_0$ , we have

$$ \begin{align*}\sum_{\substack{r \textrm{ squareful}\\ r>1}}h_i(r) <\epsilon.\end{align*} $$

For the remainder of this section, we assume that $S\supseteq S_0$ . Proceeding as in [Reference Irving27, Lemma 6.2], we use that $w(r,\delta ) \ll 1$ , and decompose each non-squarefree r into $r=r_1r_2$ , where $r_1$ is squarefree and $r_2>1$ is squareful. We have

$$ \begin{align*}\sum_{\substack{r \leqslant R\\ \mu^2(r)=0}}w(r,\delta)h_i(r) \ll \sum_{\substack{r_1 \leqslant R\\ \mu^2(r_1)=1}}h_i(r_1) \sum_{\substack{r_2 \textrm{ squareful}\\right_2>1}}h_i(r_2).\end{align*} $$

Combining with (4.46), we deduce that for any $\epsilon>0$ , we have

$$ \begin{align*}\sum_{\substack{r \leqslant R\\ \mu^2(r)=0}}w(r,\delta)h_i(r) \leqslant (\epsilon + o(1))(\log N)^{\kappa_i}.\end{align*} $$

In conclusion, we have the upper bound

(4.50) $$ \begin{align} S_4^{(i)} \leqslant \frac{(cc_iAA_ic_{h_i}\kappa_i + \epsilon + o(1))X}{(\log N)^{\kappa}}\int_{0}^{d-\beta_i}W(s)\textrm{d}s, \end{align} $$

where

(4.51) $$ \begin{align} W(s)= \left(\frac{\kappa}{\kappa+\kappa_i}\right)^{-\kappa}\left(\frac{\kappa_i}{\kappa+\kappa_i}\right)^{-\kappa_i}(2-s)^{-(\kappa+\kappa_i)}s^{\kappa_i-1}. \end{align} $$

4.7 Proof of Theorem 2.1

We now suppose that $d=2$ (i.e., $f(x,y)$ consists of irreducible factors of degree at most $2$ ). We first obtain a qualitative result by considering the limit as $\alpha \rightarrow 0$ . As $\alpha \rightarrow 0$ , we have $\kappa \rightarrow 0$ and $\kappa _i \rightarrow 1$ . Therefore, we have $A_i \rightarrow A(1)$ , which is equal to $2e^{\gamma }$ , where $\gamma = 0.57721\ldots $ is the Euler–Mascheroni constant. Moreover, by [Reference Friedlander and Iwaniec18, Equation (11.62)], we have $A(\kappa ),B(\kappa ) \rightarrow 1$ as $\kappa \rightarrow 0$ . By a very similar computation to [Reference Irving27, p. 248], we have

(4.52) $$ \begin{align} c_ic_{h_i} = \frac{e^{-\gamma \kappa_i}}{\Gamma(1+\kappa_i)}. \end{align} $$

Therefore, the ratio of the constants in the bounds for $S_1$ and $S_4^{(i)}$ is $\kappa _ic_icc_{h_i}AA_i/cB$ , which is independent of S and tends to zero as $\alpha \rightarrow 0$ . Also,

$$ \begin{align*}\lim_{\alpha \rightarrow 0}W(s) = (2-s)^{-1}.\end{align*} $$

Therefore,

(4.53) $$ \begin{align} \lim_{\alpha \rightarrow 0}S_4^{(i)} \ll \frac{(B+o(1))X}{(\log N)^{\kappa}}\left(\log 2-\log \beta_i\right). \end{align} $$

For all $\epsilon>0$ , by choosing $\beta _i$ sufficiently close to $2$ , we have the bound

(4.54) $$ \begin{align} \lim_{\alpha \rightarrow 0}S_4^{(i)} \leqslant \frac{(\epsilon B+o(1)) X}{(\log N)^{\kappa}}\ll \epsilon S_1. \end{align} $$

We recall from Sections 4.4 and 4.5 that $S_2^{(i)}$ and $S_3^{(i)}$ are also negligible compared to $S_1$ as $\alpha \rightarrow 0$ . Therefore, we see that $S(\mathcal {A},\mathcal {P},x)>0$ for sufficiently small $\alpha $ and sufficiently large N.

To obtain the best quantitative bounds, the choices of $\beta _i \in (1,2)$ should be optimised so as to minimise $S_3^{(i)}+S_4^{(i)}$ . However, in practice, numerical computations suggest that the optimal choices for $\beta _i$ are extremely close to $2$ , and little is lost in taking them arbitrarily close to 2, as above. In this case, the contributions from the sums $S_4^{(i)}$ are negligible. Taking a sum over i of the estimates from (4.37) and combining with (4.25), for any $\epsilon>0$ , we have

$$ \begin{align*}S(\mathcal{A},\mathcal{P},x) \geqslant \left(\frac{(c-\epsilon+o(1))X}{(\log N)^{\kappa}}\right)\left(B-A\alpha \sum_{i=m+1}^k \theta_i \int_{1}^{2}(2-s)^{-\kappa}\frac{\textrm{d}s}{s}\right).\end{align*} $$

Let $r(\kappa ) = B/A$ . Then we have established that $S(\mathcal {A}, \mathcal {P}, x)>0$ provided that

(4.55) $$ \begin{align} r(\kappa) - \alpha\sum_{i=m+1}^k \theta_i\int_{1}^{2}(2-s)^{-\kappa}\frac{\textrm{d}s}{s}>0. \end{align} $$

We recall that $\theta = \theta _0 + \cdots +\theta _k$ and $\kappa = \alpha \theta $ . Therefore, we may replace $\alpha \sum _{i=m+1}^k\theta _i$ with the trivial upper bound $\kappa $ , after which we find by numerical computations (see Section 4.9 for details) that the largest value of $\kappa $ we can take in (4.55) is $\kappa = 0.39000\ldots $ . Thus the condition $\alpha \theta \leqslant 0.39$ is enough to ensure that $S(\mathcal {A}, \mathcal {P},x)>0$ for sufficiently large N. This completes the proof of Theorem 2.1.

4.8 Proof of Theorem 2.3

We now discuss the case $d=3$ , where $f(x,y)$ may contain irreducible factors of degree up to $3$ . We recall in the case $d=2$ , the sums $S_4^{(i)}$ could be made negligible compared to $S_1$ by choosing $\beta _i$ arbitrarily close to $2$ , due to the factor $\log 2- \log \beta _i$ appearing in (4.53). When $d=3$ , we obtain the same bound as in (4.53), but with $\log 2-\log \beta _i$ replaced with $\log 2 - \log (\beta _i-1)$ . Consequently, $S_4^{(i)}$ is no longer negligible, even when $\alpha \rightarrow 0$ and $\beta _i$ is arbitrarily close to $2$ . In fact, it can be checked that its limit as $\alpha \rightarrow 0$ and $\beta _i \rightarrow 2$ is larger than $S_1$ , and so the above methods break down in this case.

However, we recall the additional hypothesis in Theorem 2.3 that Assumption 2.2 holds. By enlarging $\mathcal {P}$ if necessary, we may assume that equality holds in (2.9) – namely,

$$\begin{align*}\mathcal{P}\backslash S = \bigcup_{j=1}^n \{p \equiv t_i \ (\mathrm{mod}\ q_i)\}. \end{align*}$$

By Dirichlet’s theorem on primes in arithmetic progressions, (2.4) holds with $\alpha \leqslant \sum _{j=1}^n \frac {1}{q_j-1} \leqslant 0.32380$ . Moreover, it follows from Lemma 5.8 (a version of the Chebotarev density theorem) that (2.5) holds for some $\theta _i \leqslant 3$ . We now explain why this choice of $\mathcal {P}$ is easier to handle than arbitrary choices of $\mathcal {P}$ of the same density $\alpha $ .

When applying the switching principle for $S_4^{(i)}$ , we wrote $f_i(a,b) = pr$ for $p \in \mathcal {P}$ . Since we may assume $q_1, \ldots , q_n \in S_0$ , the congruence condition $C(a,b)$ forces $f_i(a,b)$ to lie in a particular congruence class modulo $q_1, \ldots , q_n$ . Combined with the fact that $p \equiv t_j \ (\mathrm {mod}\ q_j)$ for some j, we see that r lies in a particular congruence class modulo $q_j$ for some $j\in \{1, \ldots , n\}$ , and this congruence class depends only on $t_j, C(a,b)$ and f. Moreover, by (4.26), we have that $\gcd (r, \Delta ) = \gcd (f_i(a_0,b_0),\Delta )$ depends only on $C(a,b)$ and f. We deduce that there exist $r_1,\ldots , r_n$ , depending only on $t_1, \ldots , t_n, C(a,b)$ and f, such that $r':= |r|/\gcd (r,\Delta )$ lies in the set

$$\begin{align*}\mathcal{T} := \{r' \in \mathbb{N}: r'\equiv r_j \ (\mathrm{mod}\ q_j) \textrm{ for some }j\in\{1, \ldots, n\}\}. \end{align*}$$

Adding this condition into (4.43) and (4.44) and changing notation from $r'$ back to r, we obtain

(4.56) $$ \begin{align} S_4^{(i)}\leqslant \frac{(cc_iAA_i+o(1))X}{(\log N)^{\kappa_i+\kappa}}\sum_{r \in \mathcal{T}_{\leqslant x}}w(r,\delta)h_i(r). \end{align} $$

As demonstrated by Irving [Reference Irving27, Lemma 6.1], the argument based on [Reference Friedlander and Iwaniec18, Theorem A.5] we used to obtain (4.46) can be generalised to give

(4.57) $$ \begin{align} \sum_{\substack{r \in \mathcal{T}_{\leqslant x}\\ \mu^2(r)=1}}h_i(r) \leqslant \sum_{j=1}^n \sum_{\substack{r \leqslant x\\ r\equiv r_j \ (\mathrm{mod}\ q_j)}}h_i(r) = c_{h_i}(\log x)^{\kappa_i}\sum_{j=1}^n \frac{1}{q_j-1} + O((\log x)^{\kappa_i-1}). \end{align} $$

Proceeding as before, we deduce the same estimate for $S_4^{(i)}$ as in (4.50), but with an additional factor $\alpha _0 := \sum _{j=1}^n \frac {1}{q_j-1}$ . It is now clear qualitatively that for sufficiently large $q_1, \ldots , q_n$ (i.e., as $\alpha _0 \rightarrow 0$ ), the sums $S_4^{(i)}$ are once again negligible compared to $S_1$ .

We now make the above discussion more quantitative in order to complete the proof of Theorem 2.3. Combining everything, we see that $S(\mathcal {A},\mathcal {P},x)>0$ for sufficiently large N provided that

(4.58) $$ \begin{align}r(\kappa)-\sum_{i\,:\, \deg f_i =2}\alpha \theta_i \int_{1}^2(2-s)^{-\kappa}\frac{\textrm{d}s}{s}- \sum_{\substack{i \,:\, \deg f_i =3}}\alpha_0 A_i\kappa_ic_ic_{h_i}\int_{0}^1 W(s) \textrm{d}s>0. \end{align} $$

We denote the left-hand side of (4.58) by $F(\boldsymbol {\theta })$ . Recalling (4.51) and (4.52), we have

(4.59) $$ \begin{align} A_i\kappa_ic_ic_{h_i}\int_{0}^1 W(s) \textrm{d}s = \frac{A_i\kappa_ie^{-\gamma \kappa_i}\kappa^{-\kappa}\kappa_i^{-\kappa_i}(\kappa+\kappa_i)^{\kappa+\kappa_i}}{\Gamma(1+\kappa_i)}\int_0^1 (2-s)^{-(\kappa+\kappa_i)}s^{\kappa_i-1}\textrm{d}s. \end{align} $$

For a fixed choice of $\kappa <1/2$ , the integrand is a decreasing function of $\kappa _i$ because $0\leqslant \frac {s}{(2-s)}\leqslant 1$ for any $s\in [0,1]$ . The functions $\Gamma (1+\kappa _i)^{-1}$ , $\kappa _i^{-\kappa _i}$ and $e^{-\gamma \kappa _i}$ are also decreasing in $\kappa _i$ in the range $\kappa _i\in (1/2,1)$ . Let $t = \alpha _0 \deg f$ . Since $\kappa _i \geqslant 1-\kappa \geqslant 1-t$ , we therefore obtain an upper bound by replacing $\kappa _i$ with $1-t$ in all these terms. The remaining terms in (4.59) are all increasing in $\kappa _i$ , and we apply the trivial bound $\kappa _i \leqslant 1$ . Finally, we note that $\kappa ^{-\kappa }(\kappa +\kappa _i)^{\kappa +\kappa _i}$ is an increasing function in $\kappa $ for $\kappa <1/2$ , and so we may replace $\kappa $ by t in this expression. Therefore, (4.59) can be bounded by

(4.60) $$ \begin{align} \frac{A(1)e^{\gamma(t-1)}t^{-t}(t+1)^{t+1}}{\Gamma(2-t)}\int_{0}^1 (2-s)^{-1}s^{-\kappa}\textrm{d}s. \end{align} $$

We denote the factor outside the integral in (4.60) by $H(t)$ . The integral in (4.60) is equal to the first integral in (4.58), as can be seen by making the substitution $u=2-s$ . Therefore,

$$ \begin{align*} F(\boldsymbol{\theta})&\geqslant r(\kappa)-\left(\sum_{i\,:\,\deg f_i =2}\alpha\theta_i + \sum_{i\,:\, \deg f_i = 3} \alpha_0 H(t)\right) \int_0^1 (2-s)^{-1}s^{-\kappa}\textrm{d}s. \end{align*} $$

Recalling that $A(1)=2e^{\gamma }$ , it can be checked that $H(t)>4$ for all $t>0.2$ , and so in particular, $H(t)>2\theta _i$ whenever $\deg f_i =2$ . Moreover, r is a decreasing function of $\kappa $ , so $r(\kappa )\geqslant r(t)$ . We obtain

(4.61) $$ \begin{align} F(\boldsymbol{\theta})&\geqslant r(t)-H(t)\int_{0}^1 (2-s)^{-1}s^{-t}\textrm{d}s\left(\sum_{i\,:\,\deg f_i=2}\alpha/2 +\sum_{\substack{i:\deg f_i =3}}\alpha_0\right)\nonumber\\ &\geqslant r(t)-\frac{tH(t)}{3}\int_{0}^1 (2-s)^{-1}s^{-t}\textrm{d}s \end{align} $$

for any $t>0.2$ . We find by numerical computations that the above expression is positive provided that $t\leqslant 0.32380.$ . Since $t= \alpha _0\deg f$ , this is implied by the condition (2.9) assumed in Theorem 2.3.

4.9 Details of the numerical computations

We now explain how we obtained numerical values with guaranteed error bounds in the proofs of Theorem 2.1 and Theorem 2.3. From [Reference Friedlander and Iwaniec18, Equations (11.62), (11.63)], we have for $0< \kappa < 1/2$ that

$$\begin{align*}A(\kappa) = \frac{2e^{\gamma\kappa}}{\Gamma(1-\kappa)}\frac{q(\kappa)}{p(\kappa)+ q(\kappa)}, \qquad B(\kappa) = \frac{2e^{\gamma \kappa}}{p(\kappa) + q(\kappa)}, \end{align*}$$

where

$$ \begin{align*} p(\kappa) &= \int_{0}^{\infty}z^{-\kappa}\exp(-z+\kappa\operatorname{Ei}(-z))\operatorname{d}z \\ q(\kappa) &=\int_{0}^{\infty}z^{-\kappa}\exp(-z-\kappa\operatorname{Ei}(-z))\operatorname{d}z \end{align*} $$

and where $\operatorname {Ei}(-z) = -\int _{z}^{\infty }\frac {e^{-u}}{u}\operatorname {d}u$ is the standard exponential integral. For any $0<\kappa < 1/2$ , the integrand defining $p(\kappa )$ is monotonically decreasing, decays exponentially as $z \rightarrow \infty $ , and is bounded above by $2$ . Similarly, the integrand defining $q(\kappa )$ is monotonically decreasing and decays exponentially as $z \rightarrow \infty $ , but diverges as $z \rightarrow 0$ . To analyse its behaviour near $0$ , we note that for $\delta \in (0,1)$ and $\kappa < 0.4$ , we have

$$ \begin{align*} &\int_{0}^{\delta}z^{-\kappa}e^{-z}\exp(-\kappa\operatorname{Ei}(-z))\operatorname{d}z\\ &\leqslant \int_{0}^{\delta}z^{-0.4}\operatorname{exp}\left(0.4\int_{z}^{\infty}\frac{e^{-u}}{u}\operatorname{d}u\right)\operatorname{d}z\\ &\leqslant \int_{0}^{\delta}z^{-0.4} \operatorname{exp}\left(0.4\int_{z}^{1}\frac{\operatorname{d}u}{u} + 0.4\int_{1}^{\infty}e^{-u}\operatorname{d}u\right)\operatorname{d}z\\ &\leqslant \int_{0}^{\delta}z^{-0.4}\exp(-0.4\log(z) + 0.4e^{-1})\operatorname{d}z\\ &\leqslant 2 \int_{0}^{\delta}z^{-0.8}\operatorname{d}z\\ &\leqslant 10\delta^{0.2}. \end{align*} $$

Therefore, we can obtain estimates for $p(\kappa ), q(\kappa )$ (and hence also $A(\kappa ), B(\kappa )$ with provable error bounds using standard numerical integration methods such as the trapezoid rule. The functions from (4.55) and (4.61) are bounded, continuous, and monotonically decreasing in $\kappa $ , and so it is straightforward to find their roots to the desired precision. This was implemented in Sage, which produced the values $0.39000, 0.32380$ correct to five decimal places in $12.3$ seconds on a standard laptop with $2$ cores.

5.1 Algebraic reduction of the problem

Let K be a number field of degree n satisfying the Hasse norm principle. In [Reference Irving27, Lemma 2.6], Irving turns the problem of establishing the Hasse principle for

(5.1) $$ \begin{align} f(t) = \mathbf{N}(x_1, \ldots, x_n) \neq 0 \end{align} $$

into a sieve problem. Irving assumes that $f(t) \in \mathbb {Z}[t]$ is an irreducible cubic polynomial. However, in the following result, we demonstrate that Irving’s strategy can be applied to establish a similar result for an arbitrary polynomial $f\in \mathbb {Z}[t]$ . We recall that $f(x,y)$ denotes the homogenisation of f. Throughout this section, we make the choice

(5.2) $$ \begin{align} \mathcal{P} = \{p \notin S: \textrm{ the inertia degrees of }p\textrm{ in }K/\mathbb{Q}\textrm{ are not coprime}\} \end{align} $$

for a finite set of primes S containing all ramified primes in $K/\mathbb {Q}$ .

By multiplicativity of norms, it suffices to find integers $a,b$ such that b and $f(a,b)$ are in $N_{K/\mathbb {Q}}(K^*)$ . Since K satisfies the Hasse norm principle, we have $\mathbb {Q}^*\cap N_{K/\mathbb {Q}}(I_K) = N_{K/\mathbb {Q}}(K^*)$ , where $I_K$ denotes the ideles of K. Consequently, to show that $c\in \mathbb {Q}^*$ is a norm from K, it suffices to find elements $x_v \in K_v^{*}$ for each place v of K, such that

1. (1) For all but finitely many places v of K, we have $x_v\in \mathcal {O}_v^*$ (this ensures that $(x_v) \in I_K$ ).

2. (2) For all places w of $\mathbb {Q}$ , we have

   (5.3) $$ \begin{align} \prod_{v\mid w}N_{K_v/\mathbb{Q}_w}(x_v)=c. \end{align} $$

The arguments in [Reference Irving27, Lemma 2.2, Lemma 2.3, Lemma 2.4] go through without changes. We summarise them below.

We now give a slight generalisation of [Reference Irving27, Lemma 2.5] to the case of an arbitrary polynomial f.

Example 5.4. Let $f(t) = (t^2-2)(-t^2+3)$ and $K = \mathbb {Q}(i)$ , so that

$$ \begin{align*} f(x,y) &= (x^2-2y^2)(-x^2+3y^2)\\ N_{K/\mathbb{Q}}(u,v) &= u^2+v^2. \end{align*} $$

It is known that there is a Brauer–Manin obstruction to the Hasse principle for the equation $(t^2-2)(-t^2+3) = u^2+v^2 \neq 0$ by the work of Iskovskikh [Reference Iskovskikh28]. However, Proposition 5.1 still applies.

When $p\equiv 1 \ (\mathrm {mod}\ 4)$ , the prime p splits in $K/\mathbb {Q}$ , and so the inertia degrees of p in $K/\mathbb {Q}$ are coprime. However, when $p \equiv 3 \ (\mathrm {mod}\ 4)$ , the prime p is inert in $K/\mathbb {Q}$ and has degree 2, so the inertia degrees are not coprime. Therefore, we have $\mathcal {P} = \{p: p\equiv 3 \ (\mathrm {mod}\ 4)\}$ . We choose $S=\{2\}$ . In this example, it can be checked that the congruence conditions $a \equiv 8 \ (\mathrm {mod}\ 16)$ and $b \equiv 1 \ (\mathrm {mod}\ 16)$ are sufficient. Finally, for the infinite place, we just have the condition $f(a,b)>0$ . The sieve problem we obtain is to find integers $a,b$ such that

1. (1) $a \equiv 8 \ (\mathrm {mod}\ 16), b \equiv 1 \ (\mathrm {mod}\ 16)$ ,

2. (2) $f(a,b)>0$ ,

3. (3) $f(a,b)$ has no prime factors $p \equiv 3 \ (\mathrm {mod}\ 4)$ .

We remark that $f(a/b) = b^{-4}f(a,b)$ , and since $2=[K:\mathbb {Q}]$ divides $4$ , $b^{-4}$ is automatically a norm from K. This explains why in (3) above, we can consider prime factors of $f(a,b)$ rather than $bf(a,b)$ .

An integer is the sum of two squares if and only if it is non-negative and all prime factors $p \equiv 3 \ (\mathrm {mod}\ 4)$ occur with an even exponent. The above conditions are stronger than this, so the algebraic reduction performed in Proposition 5.1 is consistent with what we already knew for this example.

Since the Hasse principle fails for this example, we know that the above sieve problem cannot have a solution. This is indeed the case since condition (1) implies that $-a^2+3b^2 \equiv 3 \ (\mathrm {mod}\ 4)$ , and so $f(a,b)$ must contain a prime factor $p \equiv 3 \ (\mathrm {mod}\ 4)$ .

The fact that the above sieve problem has no solutions also does not contradict Theorem 2.1. For a prime $p>3$ , we have that $\nu _1(p)$ is 2 if $p \equiv \pm 1 \ (\mathrm {mod}\ 8)$ and zero otherwise, and $\nu _2(p)$ is 2 if $p \equiv \pm 1\ (\mathrm {mod}\ 12)$ and zero otherwise. Consequently, by Dirichlet’s theorem on primes in arithmetic progressions, for $i\in \{1,2\}$ , asymptotically one half of the primes $p\in \mathcal {P}$ have $\nu _i(p) = 2$ and half have $\nu _i(p) = 0$ . We conclude that $\theta _1 = \theta _2 = 1$ , so that $\theta = 2$ . Theorem 2.1 therefore requires the density of $\mathcal {P}$ to be less than $0.39000\ldots /\theta =0.19503\ldots $ . However, the density of $\mathcal {P}$ here is $1/2$ , and so Theorem 2.1 does not apply to this example.

5.2 The Chebotarev density theorem

In order to apply the sieve results from Section 2 to prove Theorem 1.1 and Theorem 1.3, we need to compute the densities (2.4) and (2.5) for the choice of $\mathcal {P}$ given in (5.2). For more complicated number fields K, such as when $K/\mathbb {Q}$ is non-abelian, it is not possible to write down the set $\mathcal {P}$ explicitly in terms of congruence conditions. However, we can still compute the densities (2.4) and (2.5) by appealing to the Chebotarev density theorem, which can be viewed as a vast generalisation of Dirichlet’s theorem on primes in arithmetic progressions.

Let $K/\mathbb {Q}$ be a number field of degree n. Let p be a prime, unramified in $K/\mathbb {Q}$ . We can factorise the ideal $(p)$ as $(p) = \mathfrak {p}_1\cdots \mathfrak {p}_r$ , where $\mathfrak {p}_1, \ldots , \mathfrak {p}_r$ are distinct prime ideals in $\mathcal {O}_K$ . The splitting type of p in $K/\mathbb {Q}$ is the partition $(a_1, \ldots ,a_r)$ of n, where $a_i$ is the inertia degree of $\mathfrak {p}_i$ (i.e., $N(\mathfrak {p}_i) = p^{a_i}$ ). Equivalently, the splitting type is the list of degrees of the irreducible factors of the minimum polynomial of $K/\mathbb {Q}$ , when factorised modulo p.

Suppose first that $K/\mathbb {Q}$ is Galois, with Galois group G. Then G acts transitively on $\{\mathfrak {p}_1, \ldots , \mathfrak {p}_r\}$ . Fix $i \in \{1, \ldots , r\}$ . The Decomposition group $D_{\mathfrak {p}_i}$ is the stabilizer of $\mathfrak {p}_i$ under this action. Note that $D_{\mathfrak {p}_i}$ is cyclic, and there is an isomorphism

$$ \begin{align*}\psi_i : D_{\mathfrak{p}_i} \rightarrow \mathrm{Gal}((\mathcal{O}_K/\mathfrak{p}_i)/(\mathbb{Z}/(p))).\end{align*} $$

The group $\mathrm {Gal}((\mathcal {O}_K/\mathfrak {p}_i)/(\mathbb {Z}/(p)))$ is generated by the Frobenius element defined by $x\mapsto x^p$ , which has order $a_i$ . Let $\sigma _i$ denote the preimage of the Frobenius element under $\psi _i$ . We define the Artin symbol

$$ \begin{align*}\left[\frac{K/\mathbb{Q}}{p}\right] = \{\sigma_1, \ldots, \sigma_r\}.\end{align*} $$

The Artin symbol is a conjugacy class of G. Indeed, all the $\mathfrak {p}_i$ ’s lie in the same orbit of G (there is only one orbit as G acts transitively). Stabilisers of points in the same orbit of an action are conjugate, and so all the $D_{\mathfrak {p}_i}$ are conjugate.

We now come to the statement of the Chebotarev density theorem. For a conjugacy class C of G, we let $\pi _C(x)$ denote the number of primes $p\leqslant x$ whose Artin symbol is equal to C. We define the (natural) density of a set of primes $\mathcal {P}$ to be

$$ \begin{align*}\lim_{x \rightarrow \infty}\left( \frac{\#\{p \in \mathcal{P}: p \leqslant x\}}{\pi(x)} \right),\end{align*} $$

if such a limit exists.

In order to obtain the explicit error terms from (2.4) and (2.5), we need an effective version of the Chebotarev density theorem. The following lemma is a straightforward consequence of a more refined result due to Lagarias and Odlyzko [Reference Lagarias and Odlyzko33, Theorem 1].

Theorem 5.6 (Effective Chebotarev density theorem)

For any $A\geqslant 1$ , we have

$$ \begin{align*}\pi_C(x) =\pi(x)\left(\frac{\#C}{\#G} + O_A((\log x)^{-A})\right),\end{align*} $$

where the implied constant may depend on $K,C$ and A.

We now consider the non-Galois case. As above, let $(p) = \mathfrak {p}_1\cdots \mathfrak {p}_r$ be the factorisation of $(p)$ in $\mathcal {O}_K$ . Let $\widehat {K}$ denote the Galois closure of K, and let $G= \mathrm {Gal}(\widehat {K}/\mathbb {Q})$ . This time, there is no action of G on $\{\mathfrak {p}_1, \ldots , \mathfrak {p}_r\}$ because the $\mathfrak {p}_i$ ’s could split further in $\widehat {K}$ , and elements of G could permute prime factors which are above different $\mathfrak {p}_i$ ’s.

To get around this, we define $H=\mathrm {Gal}(\widehat {K}/K)$ , and instead consider the action of G on the set X of left cosets of H in G. For an element $\sigma \in G$ , the cyclic group $\langle \sigma \rangle $ generated by $\sigma $ acts by left multiplication on X. The sizes of the orbits of this action form a partition of $[G:H] =[K:\mathbb {Q}] = n$ . Moreover, it can be checked that conjugate elements of G give the same orbit sizes, so we can associate a single partition of n with each Artin symbol $\left ([\frac {\widehat {K}/\mathbb {Q}}{p}\right ]$ . The following fact relating this partition with the splitting type of p can be found in [Reference Janusz30, Ch. 3, Proposition 2.8].

Let $K=\mathbb {Q}(\alpha )$ , and let g be the minimum polynomial of $\alpha $ . We can also view $\langle \sigma \rangle $ as acting on the set of n roots of g in $\widehat {K}$ . By definition of H, we have that $\sigma \alpha = \sigma '\alpha $ if and only if $\sigma H = \sigma ' H$ . It follows that the orbit sizes of $\langle \sigma \rangle $ acting on X are the same as the orbit sizes of $\langle \sigma \rangle $ acting on the roots of g, which in turn are the cycle lengths of $\sigma $ viewed as a permutation on the n roots of g in $\widehat {K}$ . The set of $\sigma \in G$ with cycle lengths $(a_1, \ldots , a_r)$ is a union $\bigcup _{i=1}^s C_i$ of conjugacy classes $C_i$ . We may now apply Theorem 5.6 to each of these conjugacy classes separately. Putting everything together, we have the following result on densities of splitting types in non-Galois extensions.

Lemma 5.8. Let K be a number field of degree n over $\mathbb {Q}$ , and let $\widehat {K}$ denote its Galois closure. Let $G= \mathrm {Gal}(\widehat {K}/\mathbb {Q})$ , viewed as a permutation group on the n roots of the minimum polynomial of K in $\widehat {K}$ . For a partition $\mathbf {a} = (a_1, \ldots , a_r)$ of n, let $\mathcal {P}(\mathbf {a})$ denote the set of primes with splitting type $\mathbf {a}$ in $K/\mathbb {Q}$ , and let $T(\mathbf {a})$ denote the proportion of elements of G with cycle shape $\mathbf {a}$ . Then for any $A\geqslant 1$ ,

$$ \begin{align*}\#\{p \in \mathcal{P}(\mathbf{a}): p \leqslant x\}= \pi(x)\left(T(\mathbf{a}) + O_A((\log x)^{-A})\right).\end{align*} $$

5.3 Proof of Theorem 1.1 and Theorem 1.3

Proof of Theorem 1.1

Since K satisfies the Hasse norm principle, we may apply the algebraic reduction from Proposition 5.1. Therefore, it suffices to show that conditions (1)–(3) from Proposition 5.1 hold for $\mathcal {P}$ as defined in (5.2), and for some choice of S containing all the ramified primes in $K/\mathbb {Q}$ . Let $F(x,y)$ be the binary form obtained from $yf(x,y)$ after removing any repeated factors. Clearly, to prove $bf(a,b)$ is free from prime factors in $\mathcal {P}$ , it suffices to prove $F(a,b)$ is, and so we may replace $bf(a,b)$ with $F(a,b)$ in Condition (3) of Proposition 5.1. We apply Theorem 2.1 to the binary form $F(x,y)$ , which has nonzero discriminant. We choose S to be the union of the ramified primes in $K/\mathbb {Q}$ and the set $S_0$ from Theorem 2.1, and we make the choice $\mathcal {R}=\mathcal {B}N$ , where $\mathcal {B}$ takes the form (2.7) for the parameters $r,\xi>0$ coming from the application of Proposition 5.1.

It remains only to check that (2.4) and (2.5) hold with $\alpha \theta \leqslant 0.39$ . By Lemma 5.8, (2.4) holds with $\alpha = T(G)$ . We now compute the quantity $\theta $ . We claim that $\theta \leqslant k+j+1$ . We recall that $\theta $ is a sum over the quantities $\theta _i$ associated to each irreducible factor of f, plus an additional term $\theta _0=1$ coming from the homogenising factor $f_0(x,y) = y$ . We may therefore reduce to the case where f is itself irreducible of degree at most $2$ , with the goal of proving that

$$ \begin{align*}\theta \leqslant \begin{cases} 3, & \textrm {if }f\textrm{ is quadratic and }L\subseteq \widehat{K},\\ 2, & \textrm{if }f\textrm{ otherwise}, \end{cases} \end{align*} $$

where L denotes the number field generated by f.

If f is linear, then $\nu _f(p)=1$ for all $p\notin S$ , and so $\theta = \theta _0+ 1 = 2$ , as required. We now consider the case where f is an irreducible quadratic. Let

$$ \begin{align*}\nu_f(p) = \#\{t \ (\mathrm{mod}\ p): f(t) \equiv 0 \ (\mathrm{mod}\ p)\}.\end{align*} $$

If $L\subseteq \widehat {K}$ , then Lemma 5.8 could be applied to compute $\theta $ , with the desired error terms from (2.6). However, we apply the trivial bound $\nu _f(p)\leqslant 2$ for $p \notin S$ since it is not possible to improve on this in general. We therefore obtain $\theta \leqslant \theta _0+ 2 = 1+2 = 3$ , which is satisfactory.

We now assume that $L\not \subseteq \widehat {K}$ . We want to show that $\theta =2$ , or equivalently that $\nu _f(p)=1$ on average over $p \in \mathcal {P}$ . Let $M = \widehat {K}L$ be the compositum of $\widehat {K}$ and L. Since $\widehat {K}\cap L =\mathbb {Q}$ , we have by [Reference Lang34, Ch. VI, Theorem 1.14] that $M/\mathbb {Q}$ is Galois, and

$$ \begin{align*}\mathrm{Gal}(M/\mathbb{Q}) \cong \mathrm{Gal}(L/\mathbb{Q}) \times \mathrm{Gal}(\widehat{K}/\mathbb{Q}) \cong \mathbb{Z}/2\mathbb{Z} \times \mathrm{Gal}(\widehat{K}/\mathbb{Q}).\end{align*} $$

We have $\nu _f(p) = 2$ if p is split in L, and $\nu _f(p) = 0$ if p is inert in L, and so

(5.4) $$ \begin{align} \theta = 1+ 2\lim_{x \rightarrow \infty}\left(\frac{\#\{p\leqslant x: p \in \mathcal{P}, \,p \textrm{ split in }L \}}{\#\{p\leqslant x: p \in \mathcal{P}\}}\right). \end{align} $$

Let $\sigma '=(\tau , \sigma )$ be an element of $\mathrm {Gal}(M/\mathbb {Q})$ , where $\tau \in \mathrm {Gal}(L/\mathbb {Q})$ and $\sigma \in \mathrm { Gal}(\widehat {K}/\mathbb {Q})$ . Applying Lemma 5.7, the primes $p\in \mathcal {P}$ correspond to the $\sigma '$ for which $\sigma $ has non-coprime cycle lengths (so these primes have density $T(G)$ as mentioned above). If in addition, the prime p is split in L, then we require that $\tau = \textrm {id}$ . Therefore, by Lemma 5.8, asymptotically as $x\rightarrow \infty $ , one half of the primes in $\mathcal {P}$ are also split in L. We conclude from (5.4) that (2.6) holds with $\theta = 1+ 2(1/2) = 2$ .