2.1 Reduction of Theorem 1.6 to a positivity property for multiplicative functions

We first use a version of the Furstenberg correspondence principle (see [Reference Bergelson4]) to reformulate the results in an ergodic language.

Remarks. $\circ $ The reduction to the previous multiple recurrence statement is merely a convenience. It facilitates the purpose of getting a further reduction to a positivity property for completely multiplicative functions that we describe in Theorem 2.2. Alternatively, one could carry out this last reduction directly, as in [Reference Frantzikinakis and Host21, Section 10.2].

$\circ $ Using the terminology from [Reference Donoso, Le, Moreira and Sun16], Theorem 2.1 can be rephrased as saying that for every $\ell ,\ell '\in {\mathbb N}$ , both subsets of ${\mathbb Q}^{>0}$

$$ \begin{align*} \big\{\ell(m^2-n^2)/(\ell'mn)\colon m,n\in{\mathbb N},\ m>n\big\} \text{ and }\ \big\{\ell(m^2+n^2)/(\ell'mn)\colon m,n\in{\mathbb N}\big\} \end{align*} $$

are sets of measurable multiplicative recurrence.

A function $f\colon {\mathbb N}\to {\mathbb U}$ , where ${\mathbb U}$ is the complex unit disk, is called multiplicative if

$$ \begin{align*}f(mn)=f(m)\cdot f(n) \quad \text { whenever } (m,n)=1. \end{align*} $$

It is called completely multiplicative if the previous equation holds for all $m,n\in {\mathbb N}$ . Let

$$ \begin{align*}{\mathcal M}:=\{f\colon {\mathbb N}\to \mathbb{S}^1 \text{ is a completely multiplicative function}\}. \end{align*} $$

Wherever necessary, we extend multiplicative functions to the non-positive integers in an arbitrary way. Throughout, we assume that ${\mathcal M}$ is equipped with the topology of pointwise convergence. It easily follows that ${\mathcal M}$ is a metrizable compact space with this topology. We can identify ${\mathcal M}$ with the Pontryagin dual of the (discrete) group of positive rational numbers under multiplication. Note that the map $r/s\mapsto \mu ( T_r^{-1}A \cap T_s^{-1}A)$ , $r,s,\in {\mathbb N}$ , from $({\mathbb Q}_+,\times )$ to $[0,1]$ is well defined and positive definite. Using a theorem of Bochner-Herglotz, we get that there exists a finite Borel measure $\sigma $ on ${\mathcal M}$ such that $\sigma (\{1\})>0$ (in fact, $\sigma (\{1\})\geq \delta ^2$ , where $\delta =\mu (A)$ ) and for every $r,s\in {\mathbb N}$ ,

$$ \begin{align*}\int_{\mathcal M} f(r)\cdot \overline{f(s)}\, d\sigma(f)=\mu(T_r^{-1}A\cap T_s^{-1}A).\end{align*} $$

In particular, we have

$$ \begin{align*}\mu(T_{\ell (m^2-n^2)}^{-1}A\cap T_{\ell'mn}^{-1}A) = \int_{\mathcal M} f(\ell (m^2-n^2))\cdot \overline{f(\ell'mn)}\, d\sigma(f) \end{align*} $$

for every $m,n\in {\mathbb N}$ with $m>n$ , and

$$ \begin{align*}\mu(T_{\ell(m^2+n^2)}^{-1}A\cap T_{\ell'mn}^{-1}A) = \int_{\mathcal M} f(\ell(m^2+n^2))\cdot \overline{f(\ell'mn)}\, d\sigma(f) \end{align*} $$

for every $m,n\in {\mathbb N}$ . Therefore, Theorem 2.1 follows from the following result.

Theorem 2.2. Let $\sigma $ be a positive bounded measure on ${\mathcal M}$ such that $\sigma (\{1\})>0$ and

(2.3) $$ \begin{align} \int_{\mathcal M} f(r)\cdot \overline{f(s)}\, d\sigma(f)\geq 0\quad \text{for every } r,s\in {\mathbb N}. \end{align} $$

Then for every $\ell ,\ell '\in {\mathbb N}$ ,

1. 1. we have

   (2.4) $$ \begin{align} \lim_{N\to\infty} {\mathbb E}_{m,n\in[N], m>n}\int_{{\mathcal M}} f(\ell(m^2-n^2))\cdot \overline{f(\ell'mn)}\, d\sigma(f)>0. \end{align} $$

2. 2. we have

   (2.5) $$ \begin{align} \liminf_{N\to\infty} {\mathbb E}_{m,n\in[N]}\int_{{\mathcal M}} f(\ell (m^2+n^2))\cdot \overline{f(\ell'mn)}\, d\sigma(f)>0. \end{align} $$

The reduction up to this point is similar to that in [Reference Frantzikinakis and Host21]. The methods in [Reference Frantzikinakis and Host21] were only able to address a variant of (1) in which the expressions under the integral were products of linear factors and were ‘pairing up’ when $n=0$ and becoming nonnegative.Footnote ⁵ This positivity property is not shared by the expressions in (2.4) (and (2.5)), which is the main reason why it was not possible to deal with Pythagorean pairs in [Reference Frantzikinakis and Host21]. To overcome this obstacle, we do not use a decomposition result that covers all elements of ${\mathcal M}$ simultaneously (as was the case in [Reference Frantzikinakis and Host21]), but rather work separately with aperiodic and pretentious multiplicative functions (these notions are defined in Section 3.3). In particular, coupled with some measurability properties, this allows us to exploit the uniform concentration estimates of Propositions 2.5 and 2.11, which are not shared by all elements of ${\mathcal M}$ . We outline our approach in the next subsections.

2.2 Proof plan for part (1) of Theorem 2.2

We prove Theorem 2.2 by taking an average over the grid

$$ \begin{align*}\{(Qm+1,Qn):m,n\in{\mathbb N}\}, \end{align*} $$

where $Q\in {\mathbb N}$ is chosen depending only on $\sigma $ . In view of (2.3), it suffices to prove positivity in (2.4) when the average is taken along this subset of pairs. With $\ell ,\ell '\in {\mathbb N}$ fixed, we introduce the following notation: for $\delta>0$ , $f\in {\mathcal M}$ , and $Q,m,n\in {\mathbb N}$ , let

(2.6) $$ \begin{align} A_\delta(f,Q;m,n):=w_{\delta}(m,n)\cdot f\big(\ell\,((Qm+1)^2-(Qn)^2)\big)\cdot \overline{f\big(\ell'\, (Qm+1)Qn\big)}, \end{align} $$

where $w_\delta :{\mathbb N}^2\to [0,1]$ is the weight defined in (3.2) of Lemma 3.3 for reasons that will become clear in a moment (at a first reading, the reader could just take $w_\delta =1$ ). We also remark that the weight $w_\delta $ is supported on the set $\{m,n\in {\mathbb N}\colon m>n\}$ , so to compute $A_\delta (f,Q;m,n)$ , we only have to compute f on positive integers. Then part (1) of Theorem 2.2 follows immediately from the next result, the fact that $0\leq w_\delta (m,n)\leq 1$ , and the positivity property (2.3).

Theorem 2.3. Let $\sigma $ be a Borel probability measure on ${\mathcal M}$ such that $\sigma (\{1\})>0$ . Then there exist $\delta _0>0$ and $Q_0\in {\mathbb N}$ such that

(2.7) $$ \begin{align} \lim_{N\to\infty} {\mathbb E}_{m,n\in[N]}\int_{{\mathcal M}} A_{\delta_0}(f,Q_0;m,n)\, d\sigma(f)>0. \end{align} $$

To analyze the limit in (2.7), we use the theory of completely multiplicative functions. When f is aperiodic, the mean values of $A_\delta (f,Q;m,n)$ vanish for every Q. This is a consequence of the following result, which in turn follows from results in [Reference Frantzikinakis and Host21] (see also [Reference Matthiesen38] for related work). We shall explain later on how.

Proposition 2.4. Let $f\colon {\mathbb N}\to {\mathbb U}$ be an aperiodic completely multiplicative function. Then for every $\delta>0$ and $Q\in {\mathbb N}$ , we have

(2.8) $$ \begin{align} \lim_{N\to\infty} {\mathbb E}_{m,n\in [N]} \, A_\delta(f,Q;m,n)=0. \end{align} $$

Furthermore, for every completely multiplicative function $f\colon {\mathbb N}\to {\mathbb U}$ , the previous limit exists.

Let

(2.9) $$ \begin{align} {\mathcal M}_p=\{f\colon {\mathbb N}\to \mathbb{S}^1\colon f \, \text{ is a pretentious completely multiplicative function}\}; \end{align} $$

we show in Lemma 3.6 that ${\mathcal M}_p$ is a Borel subset of ${\mathcal M}$ . It follows from Proposition 2.4 and the bounded convergence theorem that in order to establish (2.7), it suffices to show that there exist $\delta _0>0$ and $Q_0\in {\mathbb N}$ such that

(2.10) $$ \begin{align} \lim_{N\to\infty} {\mathbb E}_{m,n\in[N]}\int_{{\mathcal M}_p} A_{\delta_0}(f,Q_0;m,n)\, d\sigma(f)>0. \end{align} $$

If f is pretentious, then it ‘pretends’ to be a twisted Dirichlet character, and thus exhibits some periodicity. We exploit this periodicity by choosing a highly divisible Q for which the averages of $A_\delta (f,Q;m,n)$ take a much simpler form. More precisely, we make use of the following concentration estimate, which is an immediate consequence of [Reference Klurman, Mangerel, Pohoata and Teräväinen35, Lemma 2.5].

Proposition 2.5. Let $f\colon \mathbb {N}\to \mathbb {U}$ be a multiplicative function such that $f\sim \chi \cdot n^{it}$ for some $t\in {\mathbb R}$ and Dirichlet character $\chi $ with period q (see Section 3.3 for definitions and notation). Let also $K\in {\mathbb N}$ be large enough so that q divides all elements of the set

(2.11) $$ \begin{align} \Phi_K :=\Big\{\prod_{p\leq K}p^{a_p}\colon K< a_p\leq 2K\Big\}. \end{align} $$

Then

$$ \begin{align*} \limsup_{N\to\infty} \max_{Q\in \Phi_K} {\mathbb E}_{n\in [N]}\big|f(Qn+1)- (Qn)^{it}\cdot \exp\big(F_N(f,K)\big)\big|\ll{\mathbb D}(f,\chi\cdot n^{it}; K,\infty)+K^{-1/2}, \end{align*} $$

where the implicit constant is absolute and

(2.12) $$ \begin{align} F_N(f,K):=\sum_{K< p\leq N} \frac{1}{p}\,\big(f(p)\cdot \overline{\chi(p)}\cdot p^{-it} -1\big). \end{align} $$

Remarks. $\circ $ It is important for our argument that the implicit constant is independent of K and the quantity $F_N(f,K)$ does not depend on Q as long as $Q\in \Phi _K$ and $q\mid Q$ .

$\circ $ We will also need the following variant from [Reference Klurman, Mangerel, Pohoata and Teräväinen35, Lemma 2.5]: For any fixed $Q\in {\mathbb N}$ such that $q\prod _{p\leq K} p\mid Q$ , we have

$$ \begin{align*}\limsup_{N\to\infty} {\mathbb E}_{n\in [N]}\big|f(Qn+1)- (Qn)^{it}\cdot \exp\big(F_N(f,K)\big)\big|\ll{\mathbb D}(f,\chi\cdot n^{it}; K,\infty)+K^{-1/2}. \end{align*} $$

$\circ $ If $f\sim \chi \cdot n^{it}$ , then the sequence $A(N):=\sum _{1< p\leq N} \frac {1}{p}\,\big |1-f(p)\cdot \overline {\chi (p)}\cdot p^{-it}\big |$ , $N\in {\mathbb N}$ , is slowly varying, in the sense that for a fixed pretentious f, we have for every $c\in (0,1)$ that $\lim _{N\to \infty }\sup _{n\in [N^c,N]}|A(n)-A(N)|=0$ .Footnote ⁶ Keeping this in mind, if we use partial summation on the interval $[N^c,N]$ and then let $c\to 0^+$ , we deduce that the main estimate of Proposition 2.5 still holds if we replace ${\mathbb E}_{n\in [N]}$ with $\mathbb {E}^{\log }_{n\in [N]}$ .

In order to establish (2.10), we divide the integral into two parts. The first is supported on multiplicative functions other than the Archimedean characters $(n^{it})_{n\in {\mathbb N}}$ , $t\in {\mathbb R}$ , in which case we show using Proposition 2.5 that for a highly divisible $Q_0$ , the contribution is essentially nonnegative. The second is supported on Archimedean characters. We show that this part is positive using our assumption $\sigma (\{1\})>0$ and by taking $\delta _0$ small enough so that the weight $w_{\delta _0}$ neutralizes the effect of the Archimedean characters that are different from $1$ . To carry out the first part, the key idea is to average over ‘multiplicatively large’ values of Q. More precisely, for each $K\in {\mathbb N}$ , let $\Phi _K$ be the set described in (2.11). The sequence $(\Phi _K)$ is a multiplicative Følner sequence with the property that for every $q\in {\mathbb N}$ , as soon as K is large enough, every $Q\in \Phi _K$ is divisible by q. It also has the property that for every $Q\in \Phi _K$ and a prime $p\in {\mathbb P}$ , we have $p|Q$ if and only if $p\leq K$ . Let also

(2.13) $$ \begin{align} {\mathcal A}:=\{(n^{it})_{n\in{\mathbb N}}\colon t\in {\mathbb R}\}. \end{align} $$

Note that ${\mathcal A}$ is a Borel subset of ${\mathcal M}$ since it is a countable union of compact sets (we caution the reader that ${\mathcal A}$ is not closed with the topology of pointwise convergence; in fact, it is dense in ${\mathcal M}$ ). The most important step in establishing property (2.10) is the following fact.

Lemma 2.6. Let $f\in {\mathcal M}_p\setminus {\mathcal A}$ , $\delta>0$ , $\ell ,\ell '\in {\mathbb N}$ , and $\Phi _K$ be as in (2.11). Then

$$ \begin{align*} \lim_{K\to\infty}{\mathbb E}_{Q\in\Phi_K}\lim_{N\to\infty}{\mathbb E}_{m,n\in[N]}\, A_\delta(f,Q;m,n)=0. \end{align*} $$

(Note that the inner limit exists by Proposition 2.4.)

Roughly, to prove Lemma 2.6, we use the concentration estimate of Proposition 2.5 to deduce that for $Q\in \Phi _K$ , the average ${\mathbb E}_{m,n\in [N]}\, A_\delta (f,Q;m,n)$ is asymptotically equal to $C_{\ell ,\ell '}(K)\cdot \overline {f(Q)}\cdot Q^{it}$ for some $C_{\ell ,\ell '}(K)\in {\mathbb U}$ and $t\in {\mathbb R}$ . Since $f\not \in {\mathcal A}$ , by Lemma 3.2, the average of the last expression, taken over $Q\in \Phi _K$ , converges to $0$ as $K\to \infty $ .

Using the previous result, the fact that the limit $\lim _{N\to \infty }{\mathbb E}_{m,n\in [N]}\, A_\delta (f,Q;m,n)$ exists (by Proposition 2.4), and applying the bounded convergence theorem twice, we deduce the following vanishing property.

Corollary 2.7. Let $(\Phi _K)$ and ${\mathcal A}$ be defined by (2.11) and (2.13), respectively. Let also $\sigma $ be a Borel probability measure on ${\mathcal M}_p$ . Then for every $\delta>0$ , we have

$$ \begin{align*} \lim_{K\to\infty}{\mathbb E}_{Q\in\Phi_K}\lim_{N\to\infty}{\mathbb E}_{m,n\in[N]}\,\int_{{\mathcal M}_p\setminus{\mathcal A}} A_\delta(f,Q;m,n)\, d\sigma(f)=0. \end{align*} $$

We are left to study the part of the integral supported on ${\mathcal A}$ . For such functions, the limits $\lim _{N\to \infty }{\mathbb E}_{m,n\in [N]}A_\delta (f,Q;m,n)$ do not depend on Q, and so the previous argument will not help. It is the presence of the weight $w_\delta $ that will allow us to prove the following positivity property.

Lemma 2.8. Let $\sigma $ be a Borel probability measure on ${\mathcal M}$ such that $\sigma (\{1\})>0$ and ${\mathcal A}$ be as in (2.13). Then there exist $\delta _0,\rho _0>0$ , depending only on $\sigma $ , such that

(2.14) $$ \begin{align} \liminf_{N\to\infty}\inf_{Q\in {\mathbb N}}\Re\Big( {\mathbb E}_{m,n\in[N]}\int_{{\mathcal A}} A_{\delta_0}(f,Q;m,n)\, d\sigma(f)\Big)\geq\rho_0. \end{align} $$

Remark. The weight $w_\delta (m,n)$ is introduced to force positivity in this case, since for some choices of $\ell , \ell '$ and measures $\sigma $ , the unweighted expressions have negative real parts. However, rather miraculously, if $\ell =1$ and $\ell '=2$ (which is the case to consider for Pythagorean pairs), we get positivity even in the unweighted case, and a somewhat simpler argument applies. We do not pursue this approach here though because it lacks generality.

Finally, we will see how the previous results allow us to reach our goal, which is to prove Theorem 2.3, thus completing the proof of part (1) of Theorems 1.2 and 2.2.

Proof of Theorem 2.3 assuming Proposition 2.4, Corollary 2.7 and Lemma 2.8

By combining Corollary 2.7 and Lemma 2.8, we deduce that there exist $\delta _0, \rho _0>0$ , depending only on $\sigma $ , such that

$$ \begin{align*} \liminf_{K\to\infty}{\mathbb E}_{Q\in\Phi_K} \lim_{N\to\infty} {\mathbb E}_{m,n\in[N]}\int_{{\mathcal M}_p} A_{\delta_0}(f,Q;m,n)\, d\sigma(f)\geq \rho_0. \end{align*} $$

(There is no need to take the real part on this expression since it is real.) From this, we immediately deduce that (2.10) holds for some $Q_0\in {\mathbb N}$ . As we also explained before, this fact, together with Proposition 2.4, implies (2.7) via the bounded convergence theorem, completing the proof.

To establish Theorem 2.3, it remains to prove Proposition 2.4, Lemma 2.6 (Corollary 2.7 is an immediate consequence) and Lemma 2.8. We do this in Section 4.

2.3 Proof plan for part (2) of Theorem 2.2

The general strategy is similar to that used to prove part (1) of Theorem 2.2, but there are two major differences. The first is the required concentration estimate, which is given in Proposition 2.11 below. Unlike Proposition 2.5, this result is new and of independent interest, and its proof occupies a considerable portion of the argument. The second difference is that the limit in (2.5) may not exist, which causes additional technical problems.

Arguing as before, we get that part (2) of Theorem 2.2 follows from the following positivity property.

Theorem 2.9. Let $\sigma $ be a Borel probability measure on ${\mathcal M}$ such that $\sigma (\{1\})>0$ and (2.3) holds. Then there exists $\delta>0$ such that

(2.15) $$ \begin{align} \liminf_{N\to\infty} {\mathbb E}_{m,n\in[N]}\, \tilde{w}_{\delta}(m,n)\cdot \int_{{\mathcal M}}f\big(\ell \big(m^2+n^2\big)\big)\cdot \overline{f\big(\ell'\, m n\big)}\, d\sigma(f)>0, \end{align} $$

where $\tilde {w}_\delta (m,n)$ is the weight defined in (3.3) of Lemma 3.3.

Again, to analyze the limit in (2.15), we use the theory of completely multiplicative functions. We introduce the following notation: for $\delta>0$ , $f\in {\mathcal M}$ , and $Q,m,n\in {\mathbb N}$ , let

(2.16) $$ \begin{align} B_\delta(f,Q;m,n):=\tilde{w}_{\delta}(m,n)\cdot f\big(\ell \big((Qm+1)^2+(Qn)^2\big)\big)\cdot \overline{f\big(\ell'\, (Qm+1). (Qn)\big)}. \end{align} $$

If f is aperiodic, we have the following result, which we will deduce from the results in [Reference Frantzikinakis and Host21].

Proposition 2.10. Let $f\colon {\mathbb N}\to {\mathbb U}$ be an aperiodic multiplicative function. Then for every $\delta>0$ and $Q\in {\mathbb N}$ , we have

(2.17) $$ \begin{align} \lim_{N\to\infty} {\mathbb E}_{m,n\in [N]} \, B_\delta(f,Q;m,n)=0. \end{align} $$

If f is pretentious, we will crucially use the following concentration estimate (which is a direct consequence of a more general result proved in Section 5) to analyze the average (2.15). It features a version of the pretentious distance that only considers primesFootnote ⁷ congruent to $1$ mod $4$ :

$$ \begin{align*} {\mathbb D}_1(f,\chi\cdot n^{it}; K,\infty)^2:= \sum_{\substack{K< p, \\ p\equiv 1 \quad\pmod{4}}} \frac{1}{p}\, (1-\Re(f(p)\cdot \overline{\chi(p)} \cdot p^{-it})). \end{align*} $$

Proposition 2.11. Let $f\colon \mathbb {N}\to \mathbb {U}$ be a multiplicative function such that $f\sim \chi \cdot n^{it}$ for some $t\in {\mathbb R}$ and Dirichlet character $\chi $ with period q. Let also $\Phi _K$ be as in (2.11) and suppose that K is large enough so that, say, ${\mathbb D}_1(f,\chi \cdot n^{it}; K,\infty )\leq 1$ and q divides all elements of $\Phi _K$ . Then

$$ \begin{align*} \limsup_{N\to\infty} \max_{Q\in \Phi_K} {\mathbb E}_{m,n\in [N]}\, \big|f\big((Qm+1)^2+(Qn)^2\big)- Q^{2it} \cdot (m^2+n^2)^{it}&\cdot \exp\big(G_N(f,K)\big)\big|\ll \\ & {\mathbb D}_1(f,\chi\cdot n^{it}; K,\infty)+K^{-1/2}, \end{align*} $$

where the implicit constant is absolute and

(2.18) $$ \begin{align} G_N(f,K):=2\sum_{\substack{ K< p\leq N,\\ p\equiv 1 \quad\pmod{4}}}\, \frac{1}{p}\,(f(p)\cdot \overline{\chi(p)}\cdot p^{-it} -1). \end{align} $$

As in the proof of Theorem 2.3, in order to prove Theorem 2.9, we split the integral into two parts, one that is supported on Archimedean characters and the other on its complement. To handle the second part, we use the following result, which is proved using Proposition 2.11 and can be compared to Corollary 2.7. Again, taking multiplicative averages over the variable Q is a key maneuver, but the non-convergence of the averages ${\mathbb E}_{m,n\in [N]} \, B_\delta (f,Q;m,n)$ causes considerable technical difficulties in our proofs.

Proposition 2.12. Let $(\Phi _K)$ , ${\mathcal A}$ , $B_{\delta }(f,Q; m,n)$ be defined by (2.11), (2.13), (2.16), respectively, and $\delta{>}0 $ . Let also $\sigma $ be a Borel probability measure on ${\mathcal M}_p$ . Then

$$ \begin{align*} \lim_{K\to\infty} \limsup_{N\to \infty}\Big|{\mathbb E}_{Q\in \Phi_{K}}\, {\mathbb E}_{m,n\in[N]} \int_{{\mathcal M}_p \setminus {\mathcal A}} \, B_{\delta}(f,Q; m,n)\, d\sigma(f)\Big|=0. \end{align*} $$

We are left to study the contribution of the set ${\mathcal A}$ of Archimedean characters in which case the presence of the weight $\tilde {w}_\delta $ allows us to establish positivity by taking $\delta $ small enough.

Lemma 2.13. Let $\sigma $ be a Borel probability measure on ${\mathcal M}_p$ such that $\sigma (\{1\})>0$ and ${\mathcal A}$ be as in (2.13). Then there exist $\delta _0,\rho _0>0$ , depending only on $\sigma $ , such that

$$ \begin{align*} \liminf_{N\to\infty}\inf_{Q\in {\mathbb N}} \Re\Big( {\mathbb E}_{m,n\in[N]}\int_{{\mathcal A}} B_{\delta_0}(f,Q;m,n)\, d\sigma(f)\Big)\geq\rho_0. \end{align*} $$

We conclude this section by noting how the previous results allow us to reach our goal, which is to prove Theorem 2.9, thus completing the proof of part (2) of Theorems 1.2 and 2.2.

Proof of Theorem 2.9 assuming Proposition 2.10, Proposition 2.12 and Lemma 2.13

We start by combining Proposition 2.12 and Lemma 2.13. We deduce that there exist $\delta _0, \rho _0>0$ , depending only on $\sigma $ , such that

$$ \begin{align*} \liminf_{K\to\infty} \liminf_{N\to\infty} {\mathbb E}_{Q\in\Phi_K} \Re\Big({\mathbb E}_{m,n\in[N]}\int_{{\mathcal M}_p} B_{\delta_0}(f,Q;m,n)\, d\sigma(f)\Big)\geq \rho_0. \end{align*} $$

In this case, it is a little bit tricky to deduce that (2.15) holds. We do it as follows. The last estimate implies that there exist $K_0\in {\mathbb N}$ and $Q_N\in \Phi _{K_0}$ , $N\in {\mathbb N}$ , such that

$$ \begin{align*} \liminf_{N\to\infty} \Re\Big({\mathbb E}_{m,n\in[N]}\int_{{\mathcal M}_p} B_{\delta_0}(f,Q_N;m,n)\, d\sigma(f)\Big)\geq \rho_0/2. \end{align*} $$

Note that since $Q_N$ belongs to a finite set, Proposition 2.10 implies that in the last expression we can replace ${\mathcal M}_p$ with ${\mathcal M}$ . Hence,

(2.19) $$ \begin{align} \liminf_{N\to\infty} {\mathbb E}_{m,n\in[N]}\int_{{\mathcal M}} B_{\delta_0}(f,Q_N;m,n)\, d\sigma(f)\geq \rho_0/2. \end{align} $$

(The real part is no longer needed since the last expression is known to be real by (2.3).) It is easy to verify (following the line of reasoning at the beginning of Proposition 4.1 below) that

$$ \begin{align*}\lim_{N\to\infty} {\mathbb E}_{m,n\in [N]}\sup_{Q\in {\mathbb N}}|\tilde{w}_\delta(Qm+1,Qn)-\tilde{w}_\delta(m,n)|=0. \end{align*} $$

We deduce that if in the definition of $B_{\delta _0}(f,Q_N;m,n)$ given in (2.16) we replace $\tilde {w}_\delta (m,n)$ with $\tilde {w}_\delta (Q_Nm+1,Q_Nn)$ , the limit on the left side of (2.19) remains unchanged. Keeping this in mind, and since $Q_N$ takes values in a finite set with upper bound, say $Q_0$ , and by the positivity property (2.3), we have

$$ \begin{align*}\tilde{w}_\delta(m,n)\cdot \int_{{\mathcal M}}f\big(\ell \big(m^2+n^2\big)\big)\cdot \overline{f\big(\ell'\, m n\big)}\, d\sigma(f)\geq 0 \end{align*} $$

for every $m,n\in {\mathbb N}$ , and we deduce that

$$ \begin{align*} \liminf_{N\to\infty} {\mathbb E}_{m,n\in[N]}\, \tilde{w}_{\delta}(m,n)\cdot \int_{{\mathcal M}}f\big(\ell \big(m^2+n^2\big)\big)\cdot \overline{f\big(\ell'\, m n\big)}\, d\sigma(f)\geq \rho_0/(2Q_0^2). \end{align*} $$

This establishes (2.15) and ends the proof.

In order to establish Theorem 2.9, it remains to prove Proposition 2.10, Proposition 2.12 and Lemma 2.13. We do this in Section 6, after having established Proposition 2.11 in Section 5, which is crucially used in the proof of Proposition 2.12.

2.4 Proof plan of Theorem 1.7

For notational convenience, when we write ${\mathbb E}^*_{k\in {\mathbb N}}$ in the following statements, we mean the limit $\lim _{K\to \infty } {\mathbb E}_{k\in \Phi _K}$ , where $(\Phi _K)$ is an arbitrary multiplicative Følner sequence, chosen so that all the limits in the following statements exist. Since our setting will always involve a countable collection of limits, such a Følner sequence always exists and can be taken as a subsequence of any given multiplicative Følner sequence.

Our argument is divided into two parts. In the first part, we reduce the problem to a positivity property of pretentious multiplicative functions, and in the second part, we verify this positivity property. To carry out the first part, we note that to prove Theorem 1.7, it is only necessary to establish the subsequent averaged version.

Theorem 2.14. Suppose that the completely multiplicative function $f\colon {\mathbb N}\to \mathbb {S}^1$ takes finitely many values and $F:=\mathbf {1}_{\{1\}}$ . Then

(2.20) $$ \begin{align} \liminf_{N\to \infty} {\mathbb E}_{m,n\in [N],m>n}\, {\mathbb E}^*_{k\in {\mathbb N}} \, F(f(k\, (m^2-n^2)))\cdot F(f(k\, 2 mn)) \cdot F(f(k\, (m^2+n^2)))>0. \end{align} $$

We write $f=gh$ , where g has aperiodicity properties and h is pretentious (see Lemma 7.3 for the exact statement). Since f is finite-valued, it follows that g takes values in d-roots of unity for some $d\in {\mathbb N}$ . Hence, we have

$$ \begin{align*}F\circ g=\mathbf{1}_{g=1}={\mathbb E}_{0\leq j <d}\, g^j. \end{align*} $$

We use the previous facts to analyze the average in (2.20). The aperiodic part is covered by the next result, which is a direct consequence of [Reference Frantzikinakis and Host21, Theorem 9.7].

Proposition 2.15. Let $f_1,f_2,f_3\colon {\mathbb N}\to {\mathbb U}$ be completely multiplicative functions and suppose that either $f_1$ or $f_2$ is aperiodic. Then

$$ \begin{align*} \lim_{N\to\infty} {\mathbb E}_{m,n\in [N],m>n} \, f_1(m^2-n^2)\cdot f_2(mn)\cdot f_3(m^2+n^2)=0. \end{align*} $$

Combining the above and some technical maneuvering, we get the following reduction, which completes the first part needed to prove Theorem 2.14.

Proposition 2.16. Suppose that for every finite-valued completely multiplicative function $h\colon {\mathbb N}\to \mathbb {S}^1$ , with $h\sim 1$ , and modified Dirichlet character $\tilde {\chi }\colon {\mathbb N}\to \mathbb {S}^1$ (see Section 3.3 for the definitions), we have

$$ \begin{align*} \liminf_{N\to \infty} {\mathbb E}_{m,n\in [N],m>n}\, {\mathbb E}^*_{k\in {\mathbb N}}\ A(k\, (m^2-n^2))\cdot A(k\, 2 mn) \cdot A(k\,(m^2+n^2))>0, \end{align*} $$

where

$$ \begin{align*} A(n):=F(h(n))\cdot F(\tilde{\chi}(n)), \quad n\in {\mathbb N}, \quad F:=\mathbf{1}_{\{1\}}. \end{align*} $$

Then for every finite-valued completely multiplicative function $f\colon {\mathbb N}\to \mathbb {S}^1$ , we have

$$ \begin{align*} \liminf_{N\to \infty} {\mathbb E}_{m,n\in [N],m>n}\, {\mathbb E}^*_{k\in {\mathbb N}} \, F(f(k\, (m^2-n^2)))\cdot F(f(k\, 2 mn)) \cdot F(f(k\, (m^2+n^2)))>0. \end{align*} $$

Therefore, it remains to verify the assumption of this result. For this purpose, we will make crucial use of the following concentration estimates, which easily follow from Propositions 2.5 and 2.11, as we will see later.

Finally, using the previous concentration estimates and the key maneuver of taking multiplicative averages over $Q\in {\mathbb N}$ , which was also a crucial element in the proof of Theorem 2.2, we verify the assumptions of Proposition 2.16.

Proposition 2.18. Let $f\colon {\mathbb N}\to \mathbb {S}^1$ be a finite-valued pretentious multiplicative function and $\tilde {\chi }\colon {\mathbb N}\to \mathbb {S}^1$ be a modified Dirichlet character. Then

$$ \begin{align*} \liminf_{N\to \infty} {\mathbb E}_{m,n\in [N],m>n}\, {\mathbb E}^*_{k\in {\mathbb N}}\ A(k\, (m^2-n^2))\cdot A(k\, 2mn) \cdot A(k\, (m^2+n^2))>0, \end{align*} $$

where

$$ \begin{align*} A(n):=F(f(n))\cdot F(\tilde{\chi}(n)), \quad n\in {\mathbb N}, \quad F:=\mathbf{1}_{\{1\}}. \end{align*} $$

Thus, to prove Theorem 2.14, it remains to verify Propositions 2.16 and 2.18. We do this in Sections 7 and 8 (the other results mentioned in this subsection are needed in the proofs of these two results and will also be verified).

3.1 Some elementary facts

We will use the following elementary property.

Lemma 3.1. Let $a\colon {\mathbb Z}\to {\mathbb U}$ be an even sequence and $l_1,l_2\in {\mathbb Z}$ , not both of them $0$ . Suppose that for some $\varepsilon>0$ and for some sequence $L_N\colon {\mathbb N}\to {\mathbb U}$ , we have

$$ \begin{align*} \limsup_{N\to\infty} {\mathbb E}_{n\in [N]} |a(n) -L_N|\leq \varepsilon. \end{align*} $$

Then

$$ \begin{align*} \limsup_{N\to\infty} {\mathbb E}_{m,n\in [N]} |a(l_1m+l_2n)-L_{lN}|\leq 2| \varepsilon, \end{align*} $$

where $l:=|l_1|+|l_2|$ .

Proof. We have

(3.1) $$ \begin{align} {\mathbb E}_{m,n\in [N]} |a(l_1m+l_2n)-L_{lN}|\leq \frac{1}{N^2}\sum_{|k|\leq l N} w_N(k)\, |a(k) -L_{lN}|, \end{align} $$

where for $k\in {\mathbb Z}$ , we let

$$ \begin{align*} w_N(k):=|\{(m,n)\in [N]^2\colon l_1m+l_2n=k\}|. \end{align*} $$

For every $k\in {\mathbb Z}$ and $m\in [N]$ , there exists at most one $n\in [N]$ for which $l_1m+l_2n=k$ . Hence, $|w_N(k)|\leq N$ for every $k\in {\mathbb Z}$ . Since a is even, we deduce that the right-hand side in (3.1) is bounded by

$$ \begin{align*} 2\, l \cdot {\mathbb E}_{k\in [lN]} |a(k) -L_{lN}|. \end{align*} $$

The asserted estimate now follows from this and our assumption.

The next well-known property of multiplicative functions will also be used several times.

Lemma 3.2. Let $(\Phi _K)$ be a multiplicative Følner sequence. If $f\colon {\mathbb N}\to {\mathbb U}$ is a completely multiplicative function and $f\neq 1$ , then

$$ \begin{align*} \lim_{K\to\infty}{\mathbb E}_{n\in \Phi_K}\, f(n)=0. \end{align*} $$

Proof. Since $f\neq 1$ , there exists $p\in \mathbb {P}$ such that $f(p)\neq 1$ . By the definition of $\Phi _K$ , we have

$$ \begin{align*} \lim_{K\to\infty}\frac{|\Phi_K\cap (p\cdot \Phi_K)|}{|\Phi_K|}=1. \end{align*} $$

From this and the fact that $f(pn)=f(p)\cdot f(n)$ , we get

$$ \begin{align*} {\mathbb E}_{n\in \Phi_K}\, f(n)= {\mathbb E}_{n\in p\cdot\Phi_K}\, f(n) +o_{K\to\infty}(1) = f(p)\cdot {\mathbb E}_{n\in \Phi_K}\, f(n) +o_{K\to\infty}(1). \end{align*} $$

Since $f(p)\neq 1$ , we deduce that ${\mathbb E}_{n\in \Phi _K}\, f(n)=o_{K\to \infty }(1)$ .

3.2 Some useful weights

In the proof of Theorems 1.1 and 1.2, we will utilize weighted averages. The weights are employed to ensure that the averages ${\mathbb E}_{m,n\in [N]}\, A_\delta (f,Q,m,n)$ and ${\mathbb E}_{m,n\in [N]}\, B_\delta (f,Q,m,n)$ , where $A_\delta , B_\delta $ are as in (2.6), (2.16), respectively, have a positive real part if f is an Archimedean character and $\delta $ is sufficiently small.

We will now define these weights. If $\delta \in (0,1/2)$ , we consider the circular arc with center $1$ given by

$$ \begin{align*} I_\delta:=\{e(\phi)\colon \phi \in (-\delta,\delta)\}. \end{align*} $$

Lemma 3.3. For every $\delta \in (0,1/2)$ , let $F_\delta \colon \mathbb {S}^1\to [0,1]$ be the trapezoid function that is equal to $1$ on the arc $I_{\delta /2}$ and $0$ outside the arc $I_\delta $ . Let also

(3.2) $$ \begin{align} w_\delta(m,n):=F_\delta \big((\ell (m^2-n^2) )^i\cdot (\ell'mn)^{-i}\big) \cdot \mathbf{1}_{m> n}, \quad m,n\in{\mathbb N}, \end{align} $$

and

(3.3) $$ \begin{align} \tilde{w}_\delta(m,n):=F_\delta \big((\ell(m^2+n^2))^i\cdot (\ell'mn)^{-i}\big), \quad m,n\in{\mathbb N}. \end{align} $$

Then

$$ \begin{align*} \lim_{N\to\infty} {\mathbb E}_{m,n\in [N]}\, w_\delta(m,n)>0 \ \text{ and }\ \lim_{N\to\infty} {\mathbb E}_{m,n\in [N]}\, \tilde{w}_\delta(m,n)>0. \end{align*} $$

Proof. We first cover the weight in (3.2). Note that the limit we want to evaluate is equal to

$$ \begin{align*} \lim_{N\to\infty} {\mathbb E}_{m,n\in [N]}\, F_\delta \big((\ell((m/N)^2-(n/N)^2))^i\cdot (\ell'(m/N)\cdot (n/N))^{-i}\big) \cdot \mathbf{1}_{m/N> n/N}. \end{align*} $$

Let $\tilde {F}_\delta \colon [0,1]\times [0,1]\to [0,1]$ be given by

$$ \begin{align*} \tilde{F}_\delta(x,y):=F_\delta\big((\ell (x^2-y^2))^i\cdot (\ell'xy)^{-i}\big) \cdot \mathbf{1}_{x> y}, \quad x,y\in [0,1]. \end{align*} $$

Then $\tilde {F}_\delta $ is Riemann integrable on $[0,1]\times [0,1]$ as it is bounded and continuous except for a set of Lebesgue measure $0$ . Hence, the limit we aim to compute exists and is equal to the Riemann integral

$$ \begin{align*} \int_0^1\int_0^1 \tilde{F}_\delta(x,y)\, dx\, dy. \end{align*} $$

It remains to show that this integral is positive, and since $\tilde {F}_\delta $ is nonnegative, it suffices to show that $\tilde {F}_\delta $ does not vanish almost everywhere.

To verify the nonvanishing property, note that if $x= ay$ where $a:=\frac {\ell '+\sqrt {(\ell ')^2+4\ell ^2}}{2\ell }>1$ , then $x>y$ and $\ell (x^2-y^2)= \ell 'xy$ , and as a consequence, $(\ell (x^2-y^2))^i\cdot (\ell 'xy)^{-i}= 1$ . Hence, $\tilde {F}_\delta (x,y)=F_\delta (1)=1$ on the line $x= ay$ . Since $\tilde F_\delta $ is continuous in the region $x>y$ , this proves that it stays bounded away from zero in a neighborhood of the line $x=ay$ in that region, and hence, it does not vanish almost everywhere. This completes the proof for the weight (3.2).

The argument for the second weight (3.3) is very similar, so we only summarize it. Let $\tilde {F}_\delta \colon [0,1]\times [0,1]\to [0,1]$ be given by

$$ \begin{align*} \tilde{F}_\delta(x,y):=F_\delta\big((\ell(x^2+y^2))^i\cdot (\ell'xy)^{-i}\big)\cdot \mathbf{1}_{(0,1]\times (0,1]}(x,y). \end{align*} $$

Then the limit we want to evaluate exists and is equal to the Riemann integral

$$ \begin{align*} \int_0^1\int_0^1 \tilde{F}_\delta(x,y)\, dx\, dy. \end{align*} $$

The integral is positive because $\tilde {F}_\delta $ is nonnegative and does not vanish almost everywhere. To verify the nonvanishing property, we argue as follows. Pick $k\in {\mathbb Z}_+$ such that $b:=\ell '/\ell \cdot e^{2k\pi }>2$ and let $a:=\frac {b+\sqrt {b^2-4}}{2}$ . If $x,y\in [0,1]$ are such that $x= ay$ , then $x>y$ and $\ell (x^2+y^2)=e^{2k\pi }\, \ell 'xy $ , which implies $(\ell (x^2+y^2))^i\cdot (\ell 'xy)^{-i}= e^{2k\pi i}=1$ .

3.3 Multiplicative functions

We record here some basic notions and facts about multiplicative functions that will be used throughout the article.

3.3.1 Dirichlet characters

A Dirichlet character $\chi $ is a periodic completely multiplicative function and is often thought of as a multiplicative function on ${\mathbb Z}_m$ for some $m\in {\mathbb N}$ . In this case, $\chi $ takes the value $0$ on integers that are not coprime to m and takes values on $\phi (m)$ -roots of unity on all other integers, where $\phi $ is the Euler totient function. If $\chi $ is a Dirichlet character, we define the modified Dirichlet character $\tilde {\chi }\colon {\mathbb N}\to \mathbb {S}^1$ to be the completely multiplicative function satisfying

$$ \begin{align*} \tilde{\chi}(p):= \begin{cases} \chi(p), &\quad \text{if } \chi(p)\neq 0\\ 1, &\quad \text{if } \chi(p)=0. \end{cases} \end{align*} $$

We note in passing that the level sets of modified Dirichlet characters $\tilde {\chi }$ , which can be seen as finite colorings of ${\mathbb N}$ , are precisely the colorings that appear in Rado’s theorem when showing that certain systems of linear equations are not partition regular. In particular, a system of linear equations is partition regular if and only if it has a monochromatic solution in any coloring realized by a modified Dirichlet character.

3.3.2 Distance between multiplicative functions

Following Granville and Soundararajan [Reference Granville and Soundararajan25, Reference Granville and Soundararajan27], in this and the next subsection, we define a distance and a related notion of pretentiousness between multiplicative functions. If $f,g\colon {\mathbb N}\to {\mathbb U}$ are multiplicative functions and $x,y\in {\mathbb R}_+$ with $x<y$ , we let

(3.4) $$ \begin{align} {\mathbb D}(f,g; x,y)^2:= \sum_{x< p \leq y} \frac{1}{p}\, (1-\Re(f(p)\cdot \overline{g(p)})). \end{align} $$

We also let

(3.5) $$ \begin{align} {\mathbb D}(f,g)^2:=\sum_{p\in \mathbb{P}} \frac{1}{p}\, (1-\Re (f(p)\cdot \overline{g(p)})). \end{align} $$

Note that if $|f|=|g|=1$ , then

$$ \begin{align*} {\mathbb D}(f,g)^2=\frac{1}{2}\cdot \sum_{p\in \mathbb{P}} \frac{1}{p}\, |f(p)-g(p)|^2. \end{align*} $$

It can be shown (see [Reference Granville and Soundararajan26] or [Reference Granville and Soundararajan27, Section 2.1.1]) that ${\mathbb D}$ satisfies the triangle inequality

$$ \begin{align*} {\mathbb D}(f, g) \leq {\mathbb D}(f, h) + {\mathbb D}(h, g) \end{align*} $$

for all $f,g,h\colon {\mathbb P}\to {\mathbb U}$ . Also, for all $f_1, f_2, g_1, g_2\colon {\mathbb P}\to {\mathbb U}$ , we have (see [Reference Granville and Soundararajan25, Lemma 3.1])

(3.6) $$ \begin{align} {\mathbb D}(f_1f_2, g_1g_2) \leq {\mathbb D}(f_1, g_1) + {\mathbb D}(f_2, g_2). \end{align} $$

3.3.3 Pretentious multiplicative functions

If $f,g\colon {\mathbb N}\to {\mathbb U}$ are multiplicative functions, we say that f pretends to be g, and write $f\sim g$ , if ${\mathbb D}(f,g)<+\infty $ . It follows from (3.6) that if $f_1\sim g_1$ and $f_2\sim g_2$ , then $f_1f_2\sim g_1g_2$ . We say that f is pretentious if $f\sim \chi \cdot n^{it}$ for some $t\in {\mathbb R}$ and Dirichlet character $\chi $ , in which case

$$ \begin{align*} \sum_{p\in \mathbb{P}} \frac{1}{p}\, (1-\Re (f(p)\cdot \overline{\chi(p)}\cdot p^{-it}))<+\infty. \end{align*} $$

The value of t is uniquely determined; this follows from (3.6) and the fact that $n^{it}\not \sim \chi $ for every nonzero $t\in {\mathbb R}$ and Dirichlet character $\chi $ (see, for example, [Reference Granville and Soundararajan27, Corollary 11.4] or [Reference Granville and Soundararajan26, Proposition 7]).

Although real-valued or finite-valued multiplicative functions always have a mean value, we caution the reader that this is not the case for general multiplicative functions with values on the unit circle. For example, we have

$$ \begin{align*} {\mathbb E}_{n\in [N]} \, n^{it}=N^{it}/(1+it)+o_N(1), \end{align*} $$

so we have non-convergent means when $t\neq 0$ . But even multiplicative functions satisfying $f\sim 1$ can have non-convergent means. In particular, if $f\sim 1$ is a completely multiplicative function, then it is known (see, for example, [Reference Elliott17, Theorems 6.2]) that there exists $c\neq 0$ such that

$$ \begin{align*} {\mathbb E}_{n\in[N]}\, f(n)= c\cdot e(A(N)) +o_N(1), \end{align*} $$

where $A(N):= \sum _{p\leq N} \frac {1}{p}\, \Im (f(p))$ , $N\in {\mathbb N}$ . Hence, we have non-convergent means when, for example,

$$ \begin{align*} \sum_{p\in \mathbb{P}} \frac{1}{p}\, \Im (f(p))=+\infty, \end{align*} $$

which is the case if $f(p):=e(1/ \log \log {p})$ , $p\in {\mathbb P}$ . This oscillatory behavior of the mean values of some complex-valued multiplicative functions has to be taken into account and will cause problems in the proofs of some of our main results.

Finally, we record an observation that will only be used in the proof of Theorem 1.5.

Lemma 3.4. Let $f\colon {\mathbb N}\to {\mathbb U}$ be a pretentious finite-valued multiplicative function. Then $f\sim \chi $ for some Dirichlet character $\chi $ and

(3.7) $$ \begin{align} \sum_{p\in {\mathbb P}} \frac{1}{p}|1-f(p)\cdot \overline{\chi(p)}|<+\infty. \end{align} $$

Remark. It can be shown using (3.7) that finite-valued pretentious multiplicative functions always have convergent means.

Proof. Since f is pretentious, we have $f\sim \chi \cdot n^{it}$ for some $t\in {\mathbb R}$ and Dirichlet character $\chi $ . Then ${\mathbb D}(n^{it}, g)<+\infty $ , where $g:=f\cdot \overline {\chi }$ is a finite-valued multiplicative function. In particular, there exists $d\in {\mathbb N}$ for which $g^d$ is the constant $1$ , so from (3.6), it follows that ${\mathbb D}(n^{idt}, 1)<+\infty $ , which in turn implies that $t=0$ (hence, $f\sim \chi $ ) and

$$ \begin{align*} \sum_{p\in {\mathbb P}\colon f(p)\cdot \overline{\chi(p)}\neq 1}\frac{1}{p}<+\infty. \end{align*} $$

Hence,

$$ \begin{align*} \sum_{p\in {\mathbb P}}\frac{1}{p} |\Im (f(p)\cdot \overline{\chi(p)})|<+\infty. \end{align*} $$

If we combine this with ${\mathbb D}(f,\chi )<+\infty $ , we deduce that (3.7) holds.

3.3.4 Aperiodic multiplicative functions

We say that a multiplicative function $f\colon {\mathbb N}\to {\mathbb U}$ is aperiodic if for every $a,b\in {\mathbb N}$ ,

$$ \begin{align*}\lim_{N\to\infty}\frac1N\sum_{n=1}^N\, f(an+b)=0.\end{align*} $$

The following well-known result of Daboussi-Delange [Reference Daboussi and Delange13, Corollary 1] states that a multiplicative function is aperiodic if and only if it is non-pretentious.

Lemma 3.5. Let $f\in {\mathcal M}$ . Then either $f\sim \chi \cdot n^{it}$ for some Dirichlet character $\chi $ and $t\in {\mathbb R}$ , or f is aperiodic.

In our arguments, we typically distinguish two cases – one where a multiplicative function is aperiodic; then we show that the expressions we are interested in vanish. The complementary one where the multiplicative function is pretentious is treated using concentration estimates.

3.4 Some Borel measurability results

Recall that ${\mathcal M}$ is equipped with the topology of pointwise convergence. In the proof of Theorem 2.2, we require certain Borel measurability properties of subsets of ${\mathcal M}$ and related maps. The second property proved below will only be used in the proof of part (2) of Theorem 2.2.

Recall that if f is pretentious, then there exist a unique $t=t_f\in {\mathbb R}$ and a Dirichlet character $\chi $ such that $f\sim \chi \cdot n^{it}$ .

Proof. We prove (1). For $a,b\in {\mathbb N}$ , we let $M_{a,b}$ be the set of $f\in {\mathcal M}$ such that

$$ \begin{align*} \limsup_{N\to\infty} |{\mathbb E}_{n\in [N]} \, f(an+b)|>0. \end{align*} $$

Clearly, $M_{a,b}$ is a Borel subset of ${\mathcal M}$ . By Lemma 3.5, we have ${\mathcal M}_p=\bigcup _{a,b\in {\mathbb N}}M_{a,b}$ , and the result follows.

We prove (2). By [Reference Kechris32, Theorem 14.12], it suffices to show that the graph

$$ \begin{align*} \Gamma:=\{(f,t_f)\in {\mathcal M}_p\times {\mathbb R} \} \end{align*} $$

is a Borel subset of ${\mathcal M}_p\times {\mathbb R}$ . If $\chi _k$ , $k\in {\mathbb N}$ , is an enumeration of all Dirichlet characters, and

$$ \begin{align*} \Gamma_k:=\{(f,t_f)\in {\mathcal M}_p\times {\mathbb R}\colon f\sim \chi_k\cdot n^{it_f}\}, \end{align*} $$

then

$$ \begin{align*} \Gamma=\bigcup_{k\in{\mathbb N}} \Gamma_k. \end{align*} $$

Hence, it suffices to show that for every $k\in {\mathbb N}$ , the set $\Gamma _k$ is Borel. Note that

$$ \begin{align*} \Gamma_k:=\{(f,t)\in{\mathcal M}_p\times {\mathbb R}\colon {\mathbb D}(f,\chi_k\cdot n^{it})<\infty\}. \end{align*} $$

Since for $k\in {\mathbb N}$ the map $(f,t)\mapsto {\mathbb D}(f,\chi _k\cdot n^{it})$ is clearly Borel, the set $\Gamma _k$ is Borel. This completes the proof.

5.1 Preparatory counting arguments

The following lemma will be used multiple times subsequently.

Lemma 5.2. For $Q,N\in {\mathbb N}$ , $a,b\in {\mathbb Z}$ , and primes $p, q$ such that $p,q\equiv 1 \ \pmod {4}$ and $(pq, Q)=1$ , let

(5.4) $$ \begin{align} w_{N,Q}(p,q):=\frac{1}{N^2}\, \sum_{\substack{m,n\in [N],\\ p,q\, \mid\mid\, (Qm+a)^2+(Qn+b)^2} } 1. \end{align} $$

Then

(5.5) $$ \begin{align} w_{N,Q}(p,p)=\frac{2}{p}\Big(1-\frac{1}{p}\Big)^2 + O\Big(\frac{1}{N}\Big), \end{align} $$

and if $p\neq q$ , we have

(5.6) $$ \begin{align} w_{N,Q}(p,q)=\frac{4}{pq}\Big(1-\frac{1}{p}\Big)^2 \Big(1-\frac{1}{q}\Big)^2 +O\Big(\frac{1}{N}\Big), \end{align} $$

where the implicit constants are absolute.

Remark. We deduce the approximate identity

$$ \begin{align*} w_{N,Q}(p,q)= w_{N,Q}(p,p)\cdot w_{N,Q}(q,q)+O\Big(\frac{1}{N}\Big), \end{align*} $$

which is crucial for the proof of the concentration estimates. However, because of the $O\big (\frac {1}{N}\big )$ errors, these approximate identities will only be useful to us for sums that contain $o(N)$ terms.

Proof. Throught the discussion, we use $\epsilon $ to designate a number in $\{0,1,2,3,4\}$ .

We first establish (5.5). Let p satisfy the assumptions. Note first that if $p\mid Qn+b$ and $p\mid (Qm+a)^2+(Qn+b)^2$ , then also $p^2\mid (Qm+a)^2+(Qn+b)^2$ ; hence, we get no contribution to the sum (5.4) in this case. So we can assume that $p\nmid Qn+b$ . Since $p\equiv 1 \ \pmod 4$ , the number $-1$ is a quadratic residue $\! \! \! \mod {p}$ , and we have exactly two solutions $m \ \pmod {p}$ to the congruence

(5.7) $$ \begin{align} (Qm+a)^2+(Qn+b)^2\equiv 0 \quad\pmod {p}. \end{align} $$

Hence, for those $n\in [N]$ , we have $2[N/p]+\epsilon $ solutions in the variable $m\in [N]$ to (5.7). Since there are $N-[N/p]+\epsilon $ integers $n\in [N]$ with $p\nmid Qn+b$ (we used that $(p,Q)=1$ here), we get a total of

$$ \begin{align*} 2[N/p]\, (N-[N/p]) +O(N)=2N^2/p-2N^2/p^2+O(N) \end{align*} $$

solutions of $m,n\in [N]$ to the congruence (5.7). Similarly, we get that if $p\nmid Qn+b$ , then the number of solutions $m,n\in [N]$ to the congruence $(Qm+a)^2+(Qn+b)^2\equiv 0 \ \pmod {p^2}$ is

$$ \begin{align*} 2[N/p^2]\, (N-[N/p])+O(N)=2N^2/p^2-2N^2/p^3+O(N). \end{align*} $$

(We used that $-1$ is also a quadratic residue $\! \! \! \mod {p^2}$ .) These solutions should be subtracted from the previous solutions of (5.7) in order to count the number of solutions of $m,n\in [N]$ for which $p\, \mid \mid \, (Qm+a)^2+(Qn+b)^2$ . We deduce that

(5.8) $$ \begin{align} \frac{1}{N^2}\, \sum_{\substack{m,n\in [N],\\ p\, \mid\mid\, (Qm+a)^2+(Qn+b)^2} } 1=\frac{2}{p}-\frac{4}{p^2}+\frac{2}{p^3}+O\Big(\frac{1}{N}\Big)=\frac{2}{p}\Big(1-\frac{1}{p}\Big)^2 + O\Big(\frac{1}{N}\Big), \end{align} $$

which proves (5.5).

Next, we establish (5.6). Let $p,q $ satisfy the assumptions. As explained in the previous case, those $n\in [N]$ for which $p\mid Qn+b$ or $q\mid Qn+b$ do not contribute to the sum (5.4) defining $w_{N,Q}(p,q)$ ; hence, we can assume that $(pq,Qn+b)=1$ . Let

$$ \begin{align*} A_{r,s}:=\frac{1}{N^2}\, \sum_{\substack{m,n\in [N],\\ r,s\mid (Qm+a)^2+(Qn+b)^2,\, (rs,Qn+b)=1 }} 1 \end{align*} $$

and note that

(5.9) $$ \begin{align} w_{N,Q}(p,q)=A_{p,q}-A_{p^2,q}-A_{p,q^2}+A_{p^2,q^2}. \end{align} $$

We first compute $A_{p,q}$ . Since $p{\kern-1pt}\equiv{\kern-1pt} q{\kern-1pt}\equiv{\kern-1pt} 1 \ \pmod 4$ , the number $-1$ is a quadratic residue $\! \! \! \mod {p}$ and $\! \! \! \mod {q}$ , and we get by the Chinese remainder theorem, that for each $n\in [N]$ with $p,q\nmid Qn+b$ , we have $4$ solutions $m \ \pmod {pq}$ to the congruence

(5.10) $$ \begin{align} (Qm+a)^2+(Qn+b)^2\equiv 0 \quad\pmod {pq}. \end{align} $$

WeFootnote ⁸ deduce that for each $n\in [N]$ with $(pq,Qn+b)=1$ , we have $4[N/(pq)]+\epsilon $ solutions in the variable $m \in [N]$ to the congruence (5.10). Since the number of $n\in [N]$ for which $(pq,Qn+b)=1$ is $N-[N/p]-[N/q]+[N/pq]$ , we get that the total number of solutions to the congruence (5.10) with $m,n\in [N]$ and $(pq,n)=1$ is

(5.11) $$ \begin{align} &4 [N/(pq)] \, (N-[N/p]-[N/q]+[N/(pq)])+ O(N)= \notag\\ &\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad N^2 \cdot (4/(pq))\cdot (1-1/p-1/q+1/(pq)) +O(N). \end{align} $$

Hence,

$$ \begin{align*} A_{p,q}:= \frac{4}{pq}\Big(1-\frac{1}{p}\Big) \Big(1-\frac{1}{q}\Big) +O\Big(\frac{1}{N}\Big). \end{align*} $$

Similarly, using that $-1$ is also a quadratic residue $\mod {p^k}$ and $\mod {q^k}$ for $k=1,2$ , we find that

$$ \begin{align*} A_{p^2,q}= \frac{4}{p^2q}\Big(1-\frac{1}{p}\Big) \Big(1-\frac{1}{q}\Big) +O\Big(\frac{1}{N}\Big), \end{align*} $$

and

$$ \begin{align*} A_{p,q^2}= \frac{4}{pq^2}\Big(1-\frac{1}{p}\Big) \Big(1-\frac{1}{q}\Big) +O\Big(\frac{1}{N}\Big). \end{align*} $$

Also,

$$ \begin{align*} A_{p^2,q^2}= \frac{4}{p^2q^2}\Big(1-\frac{1}{p}\Big) \Big(1-\frac{1}{q}\Big) +O\Big(\frac{1}{N}\Big). \end{align*} $$

Using the last four identities and (5.9), we deduce that (5.6) holds. This completes the proof.

We will also need to give upper bounds for $w_{N,Q}(p,q)$ when $p,q$ are not necessarily primes, and also give upper bounds that do not involve the error terms $O(1/N)$ that cause us problems in some cases (this is only relevant when $pq\geq N$ ). The next lemma is crucial for us and gives an upper bound that is good enough for our purposes.

Lemma 5.3. For $l, Q,N \in {\mathbb N}$ and $a,b\in {\mathbb Z}$ with $-Q\leq a,b\leq Q$ , let

$$ \begin{align*} w_{N,Q}(l):=\frac{1}{N^2} \, \sum_{\substack{m,n\in [N],\\ l\mid (Qm+a)^2+(Qn+b)^2} } 1. \end{align*} $$

If l is a sum of two squares, then

(5.12) $$ \begin{align} w_{N,Q}(l)\ll \frac{Q^2}{l}, \end{align} $$

where the implicit constant is absolute. In particular, if $w_{N,Q}(p,q)$ is as in (5.4), taking $l=p$ and $l=pq$ where $p, q$ are distinct primes of the form $1 \ \pmod {4}$ , we get

(5.13) $$ \begin{align} w_{N,Q}(p,p)\ll \frac{Q^2}{p}, \quad w_{N,Q}(p,q)\ll \frac{Q^2}{pq}. \end{align} $$

Proof. Recall that an integer is a sum of two squares if and only if in its factorization as a product of primes, all prime factors congruent to $3 \ \pmod {4}$ occur with even multiplicity. It follows that if l is a sum of two squares and $l\mid (Qm+a)^2+(Qn+b)^2$ , then the ratio $\big ((Qm+a)^2+(Qn+b)^2\big )/l$ is also a sum of two squares. We deduce from this and our assumption $|a|,|b|\leq Q$ that if l is a sum of two squares, then

$$ \begin{align*} w_{N,Q}(l)\leq \frac{1}{N^2}\, \sum_{k\leq 3Q^2N^2/l}r_2(k)\ll \frac{Q^2}{l}, \end{align*} $$

where $r_2(k)$ denotes the number of representations of k as a sum of two squares, and to get the second estimate, we used the well-known fact $\sum _{k\le n}r_2(k)\ll n$ . This completes the proof.

5.2 Concentration estimate for additive functions

We start with a concentration estimate for additive functions that will eventually get lifted to a concentration estimate for multiplicative functions.

Lemma 5.4 (Turán-Kubilius inequality for sums of squares)

Let $K_0,N\in {\mathbb N}$ , $a,b\in {\mathbb Z}$ with $-Q\leq a,b,\leq Q$ , and $h\colon \mathbb {N}\to \mathbb {C}$ be an additive function that is bounded by $1$ on primes and such that

1. 1. $h(p)=0$ for all primes $p\leq K_0$ and $p>N$ ;

2. 2. $h(p)=0$ for all primes $p\equiv 3 \ \pmod 4$ ;

3. 3. $h(p^k)=0$ for all primes p and $k\geq 2$ .

Let also $Q= \prod _{p\leq K_0}p^{a_p}$ for some $a_p\in {\mathbb N}$ . Then for all large enough N, depending only on $K_0$ , we have

(5.14) $$ \begin{align} {\mathbb E}_{m,n\in [N]}\, \big|h\big((Qm+a)^2+(Qn+b)^2\big)- H_N(h,K_0) \big|^2\ll {\mathbb D}^2(h; K_0,\sqrt{N})+Q^2\cdot {\mathbb D}^2(h; \sqrt{N},N)+K_0^{-1}, \end{align} $$

where the implicit constant is absolute,

(5.15) $$ \begin{align} H_N(h,K_0):=2\, \sum_{K_0< p\leq N}\, \frac{h(p)}{p} \end{align} $$

and

$$ \begin{align*} {\mathbb D}^2(h; K_0,N):=\sum_{K_0< p\leq N} \frac{|h(p)|^2}{p}. \end{align*} $$

Proof. We consider the additive functions $h_1,h_2$ , which are the restrictions of h to the primes ${K_0<p\le \sqrt {N}}$ and $\sqrt {N}<p\le N$ .Footnote ⁹ More precisely,

$$ \begin{align*} h_1(p^k):= \begin{cases}h(p), \quad &\text{if }\ k=1 \text{ and } K_0<p\leq \sqrt{N}\\ 0, \quad &\text{otherwise} \end{cases} \end{align*} $$

and

$$ \begin{align*} h_2(p^k):=\begin{cases}h(p), \quad &\text{if }\ k=1 \text{ and } \sqrt{N}<p\leq N\\ 0, \quad &\text{otherwise} \end{cases}. \end{align*} $$

We also define

(5.16) $$ \begin{align} H_{i,N}(h_i,K_0):=2\sum_{K_0< p\leq N}\, \frac{h_i(p)}{p}, \quad i=1,2, \end{align} $$

and the technical variant

(5.17) $$ \begin{align} H^{\prime}_{1,N}(h_i,Q,K_0):=\sum_{K_0< p\leq N }\, w_{N,Q}(p)\cdot h_1(p), \end{align} $$

where

(5.18) $$ \begin{align} w_{N,Q}(p):= \frac{1}{N^2}\, \sum_{\substack{m,n\in [N],\\ p\, \mid\mid\, (Qm+a)^2+(Qn+b)^2} } 1. \end{align} $$

(Note that $w_{N,Q}(p)=w_{N,Q}(p,p)$ , where $w_{N,Q}(p,q)$ is as in (5.4).) The reason for introducing this variant is because it gives the mean value of $h_1$ along sums of squares. Indeed, using properties (1)–(3), we have

(5.19) $$ \begin{align} {\mathbb E}_{m,n\in [N]}\, h_1 ((Qm+a)^2 &+(Qn+b)^2)={\mathbb E}_{m,n\in [N]}\, \sum_{p\, \mid\mid\, (Qm+a)^2+(Qn+b)^2}h_1(p)\notag\\ &\qquad =\frac{1}{N^2} \sum_{K_0< p \leq N} h_1(p) \, \sum_{\substack{m,n\in [N],\\ p\, \mid\mid\, (Qm+a)^2+(Qn+b)^2} } 1 = H^{\prime}_{1,N}(h_1,Q,K_0). \end{align} $$

Using (5.5) of Lemma 5.2 and that $h_1(p)=0$ for $p> \sqrt {N}$ and $h_1(p)$ is bounded by $1$ , we get

(5.20) $$ \begin{align} |H_{1,N}(h_1,K_0)-H^{\prime}_{1,N}(h_1,Q,K_0)|\ll \sum_{K_0< p\leq \sqrt{N}} \frac{1}{p^2}+\frac{1}{\sqrt{N}} \leq \frac{1}{K_0} +\frac{1}{\sqrt{N}}. \end{align} $$

Hence, in order to prove (5.14), it suffices to estimate

(5.21) $$ \begin{align} {\mathbb E}_{m,n\in [N]}\big|h_1((Qm+a)^2+(Qn+b)^2)- H^{\prime}_{1,N}(h_1,Q,K_0)\big|^2 \end{align} $$

and

(5.22) $$ \begin{align} {\mathbb E}_{m,n\in [N]}\big|h_2((Qm+a)^2+(Qn+b)^2)\big|^2+ |H_{2,N}(h_1,K_0)|^2. \end{align} $$

We first deal with the expression (5.21). Using (5.19) and expanding the square below, we get

(5.23) $$ \begin{align} {\mathbb E}_{m,n\in [N]} \, \big|h_1((Qm+a)^2 &+(Qn+b)^2)- H^{\prime}_{1,N}(h_1,Q,K_0)\big|^2=\notag\\ &\quad {\mathbb E}_{m,n\in [N]}\, \big|h_1((Qm+a)^2+(Qn+b)^2)\big|^2 - |H^{\prime}_{1,N}(h_1,Q,K_0)|^2. \end{align} $$

To estimate this expression, first note that since $h_1$ is additive and $h_1(p^k)=0$ for $k\geq 2$ , we have

(5.24) $$ \begin{align} {\mathbb E}_{m,n\in [N]}\, \big|h_1((Qm+a)^2+(Qn+b)^2)\big|^2= {\mathbb E}_{m,n\in [N]}\, \Big|\sum_{p\, \mid \mid \,(Qm+a)^2+(Qn+b)^2} h_1(p)\Big|^2. \end{align} $$

Expanding the square, using the fact that $h_1(p)=0$ unless $K_0<p\leq \sqrt {N}$ , and the definition of $w_{N,Q}(p,q)$ given in (5.4), we get that the right-hand side is equal to

$$ \begin{align*} \sum_{K_0<p\leq \sqrt{N}} |h_1(p)|^2\cdot w_{N,Q}(p,p)+ \sum_{K_0<p,q\leq \sqrt{N}, \, p\neq q} h_1(p)\cdot \overline{h_1(q)}\cdot w_{N,Q}(p,q). \end{align*} $$

Using equation (5.5) of Lemma 5.2, we get that the first term is at most

$$ \begin{align*} 2\cdot \sum_{K_0<p\leq \sqrt{N}}\frac{|h_1(p)|^2}{p}+ O(N^{-1/2}). \end{align*} $$

Using equations (5.5) and (5.6) of Lemma 5.2, we get that the second term is equal to (we crucially use the bound $p,q\leq \sqrt {N}$ here and the prime number theorem)

$$ \begin{align*} \sum_{K_0<p,q\leq \sqrt{N}, \, p\neq q} h_1(p)\cdot \overline{h_1(q)} \cdot w_{N,Q}(p,p)\cdot w_{N,Q}(q,q) &+O((\log{N})^{-2})\leq \\ &\quad (H^{\prime}_{1,N}(h_1,Q,K_0))^2+O((\log{N})^{-2}), \end{align*} $$

where to get the last estimate, we added to the sum the contribution of the diagonal terms $p=q$ (which is nonnegative), and used (5.17) and the fact that $h_1(p)=0$ for $p>\sqrt {N}$ . Combining (5.23) with the previous estimates, we are led to the bound

(5.25) $$ \begin{align} {\mathbb E}_{m,n\in [N]} \, \big|h_1((Qm+a)^2+(Qn+b)^2)- H^{\prime}_{1,N}(h_1,Q,K_0)\big|^2 &\ll \notag\\ &{\mathbb D}^2(h_1; K_0, \sqrt{N})+O((\log{N})^{-2}). \end{align} $$

Next, we estimate the expression (5.22). Since $h_2$ is additive and satisfies properties (1)–(3), we get using (5.24) (with $h_2$ in place of $h_1$ ) and expanding the square

$$ \begin{align*} {\mathbb E}_{m,n\in [N]} \, \big|h_2((Qm+a)^2+(Qn+b)^2)\big|^2= \sum_{\sqrt{N}< p, q \leq N} h_2(p)\, \overline{h_2(q)} \, w_{N,Q}(p,q). \end{align*} $$

Since $h_2(p)\neq 0$ only when $p\equiv 1 \ \pmod {4}$ , using (5.13) of Lemma 5.3, we get that the right-hand side is bounded by

$$ \begin{align*} \ll Q^2&\cdot \Big(\sum_{\sqrt{N}< p,q\leq N} \frac{|h_2(p)| \, |h_2(q)|}{pq}+ \sum_{\sqrt{N}< p\leq N} \frac{|h_2(p)|^2}{p}\Big)= \\ &\qquad\qquad\qquad\qquad Q^2\cdot \Big(\Big(\sum_{\sqrt{N}< p\leq N} \frac{|h_2(p)|}{p}\Big)^2+{\mathbb D}^2(h_2; \sqrt{N},N) \Big)\leq \\ &\qquad\qquad Q^2 \cdot \Big(\sum_{\sqrt{N}< p\leq N} \frac{|h_2(p)|^2}{p}\cdot \sum_{\sqrt{N}< p\leq N} \frac{1}{p}+{\mathbb D}^2(h_2; \sqrt{N},N) \Big)\ll Q^2\cdot {\mathbb D}^2(h_2; \sqrt{N},N), \end{align*} $$

where we crucially used the estimate

$$ \begin{align*} \sum_{\sqrt{N}< p\leq N} \frac{1}{p}\ll 1. \end{align*} $$

Similarly, we find

$$ \begin{align*} (H_{2,N}(h_2,K_0))^2=4\, \Big(\sum_{\sqrt{N}< p\leq N} \frac{|h_2(p)|}{p}\Big)^2\ll {\mathbb D}^2(h_2; \sqrt{N},N). \end{align*} $$

Combining the previous estimates, we get the following bound for the expression in (5.22):

(5.26) $$ \begin{align} {\mathbb E}_{m,n\in [N]}\big(h_2((Qm+a)^2+(Qn+b)^2)\big)^2+ (H_{2,N}(h_1,K_0))^2\ll Q^2\cdot {\mathbb D}^2(h_2; \sqrt{N},N). \end{align} $$

Combining the bounds (5.20), (5.25), (5.26), we get the asserted bound (5.14), completing the proof.

5.3 Concentration estimates for multiplicative functions

Next we use Lemma 5.4 to get a variant that deals with multiplicative functions.

Lemma 5.5. Let $K_0,N\in {\mathbb N}$ , $a,b\in {\mathbb Z}$ with $-Q\leq a,b\leq Q$ , and $f\colon \mathbb {N}\to \mathbb {U}$ be a multiplicative function that satisfies

1. 1. $f(p)=1$ for all primes $p\leq K_0$ and $p> N$ ;

2. 2. $f(p)=1$ for all primes $p\equiv 3 \ \pmod 4$ ;

3. 3. $f(p^k)=1$ for all primes p and $k\geq 2$ .

Let also $Q= \prod _{p\leq K_0}p^{a_p}$ with $a_p\in {\mathbb N}$ . If N is large enough, depending only on $K_0$ , then

(5.27) $$ \begin{align} {\mathbb E}_{m,n\in [N]}\, \big|f\big((Qm+a)^2+(Qn+b)^2\big)- &\exp\big(G_N(f,K_0)\big)\big|\ll \notag\\ &({\mathbb D}+{\mathbb D}^2)(f,1; K_0,\sqrt{N})+ Q\cdot {\mathbb D}(f,1; \sqrt{N},N)+K_0^{-\frac{1}{2}}, \end{align} $$

where the implicit constant is absolute and

(5.28) $$ \begin{align} G_N(f,K_0):=2\sum_{ K_0< p\leq N}\, \frac{1}{p}\,(f(p) -1). \end{align} $$

Proof. Let $h:\mathbb {N}\to \mathbb {C}$ be the additive function given on prime powers by

$$ \begin{align*} h(p^k):=f(p^k)-1. \end{align*} $$

We note that due to our assumptions on $f,$ properties (1)–(3) of Lemma 5.4 are satisfied for $h/2$ , which is bounded by $1$ on primes.

Using that $z=e^{z-1}+O(|z-1|^2)$ for $|z|\leq 1$ and property (3), we have

$$ \begin{align*} f(m^2+n^2)=\prod_{p^k\, \mid \mid \, m^2+n^2}f(p^k)= \prod_{p\, \mid \mid \, m^2+n^2}\big(\exp(h(p))+O(|h(p)|^2)\big). \end{align*} $$

Applying the estimate $|\prod _{i\leq k}z_i-\prod _{i\leq k}w_i|\leq \sum _{i\leq k}|z_i-w_i|$ , we deduce that for all $m,n\in {\mathbb N}$ , we have

$$ \begin{align*} f(m^2+n^2)=\exp(h(m^2+n^2))+O\Big(\sum_{p \, \mid \mid\, m^2+n^2}|h(p)|^2\Big). \end{align*} $$

Using this and since $G_N(f,K_0)=H_N(h, K_0)$ , where $H_N(h,K_0)$ is given by (5.15), we get

(5.29) $$ \begin{align} {\mathbb E}_{m,n\in [N]}\, &\big|f\big((Qm+a)^2+(Qn+b)^2\big)- \exp\big(G_N(f,K_0)\big)\big| \ll \notag\\ &\qquad {\mathbb E}_{m,n\in [N]}|\exp\big(h((Qm+a)^2+(Qn+b)^2)\big)-\exp(H_N(h, K_0))|+\notag\\ &\qquad\qquad\qquad\quad\qquad\qquad\qquad\qquad\qquad {\mathbb E}_{m,n\in [N]}\sum_{p\, \mid \mid\, (Qm+a)^2+(Qn+b)^2}|h(p)|^2. \end{align} $$

Next we use the inequality $|e^{z_1}-e^{z_2}|\le |z_1-z_2|$ , which is valid for $\Re z_1,\Re z_2\le 0$ , to bound the last expression by

(5.30) $$ \begin{align} {\mathbb E}_{m,n\in [N]}|h((Qm+a)^2+(Qn+b)^2)-H_N(h,K_0)|+{\mathbb E}_{m,n\in [N]}\sum_{p \, \mid \mid \, (Qm+a)^2+(Qn+b)^2}|h(p)|^2. \end{align} $$

To bound the first term, we use Lemma 5.4. It gives that for all large enough N, depending on $K_0$ only, we have

(5.31) $$ \begin{align} &{\mathbb E}_{m,n\in [N]}\, \big|h\big((Qm+a)^2+(Qn+b)^2\big)- H_N(h,K_0) \big|\ll\notag\\ &\quad {\mathbb D}(h; K_0,\sqrt{N})+Q\cdot {\mathbb D}(h; \sqrt{N},N)+K_0^{-\frac{1}{2}} \ll {\mathbb D}(f,1; K_0,\sqrt{N})+ Q\cdot {\mathbb D}(f,1; \sqrt{N},N)+K_0^{-\frac{1}{2}}, \end{align} $$

where to get the last bound, we used that $|h(p)|^2\leq 2-2\,\Re (f(p))$ , which holds since $|f(p)|\leq 1$ . To bound the second term in (5.30), we note that using properties (1)–(3) of Lemma 5.4, we have

(5.32) $$ \begin{align} {\mathbb E}_{m,n\in [N]}&\sum_{p\, \mid \mid\, (Qm+a)^2+(Qn+b)^2} |h(p)|^2 = \sum_{ K_0< p \leq N}\ |h(p)|^2\, w_{N,Q}(p)\ll \notag \\ &\qquad\qquad \sum_{ K_0< p \leq N}\frac{|h(p)|^2}{p}+O((\log{N})^{-1}) \ll \mathbb{D}^2(f, 1;K_0,N)+O((\log{N})^{-1}) , \end{align} $$

where $w_{N,P}(p)$ is as in (5.18) and we used equation (5.5) of Lemma 5.2 and the prime number theorem to get the first bound. Combining (5.29)–(5.32), we get the asserted bound.

We use the previous result to deduce the following improved version.

Lemma 5.6. Let $K_0,N\in {\mathbb N}$ and $f\colon \mathbb {N}\to \mathbb {U}$ be a multiplicative function such that $f(p)= 1$ for all primes $p> N$ with $p\equiv 1 \ \pmod {4}$ . Let also $Q= \prod _{p\leq K_0}p^{a_p}$ with $a_p\in {\mathbb N}$ . If N is large enough, depending only on $K_0$ , then for all $a,b\in {\mathbb Z}$ with $-Q\leq a,b\leq Q$ and $(a^2+b^2,Q)=1$ , we have

(5.33) $$ \begin{align} {\mathbb E}_{m,n\in [N]}\, \big|f\big((Qm+a)^2+(Qn+b)^2\big) &- \exp\big(G_N(f,K_0)\big)\big|\ll \notag \\ &\ ({\mathbb D}_1+{\mathbb D}_1^2)(f,1; K_0,\sqrt{N})+ Q\cdot {\mathbb D}_1(f,1; \sqrt{N},N)+K_0^{-\frac{1}{2}}, \end{align} $$

where the implicit constant is absolute and

(5.34) $$ \begin{align} G_N(f,K_0):=2\sum_{ \substack{K_0< p\leq N; \\ p\equiv 1 \quad\pmod{4}}}\, \frac{1}{p}\,(f(p) -1),\end{align} $$

$$ \begin{align*} {\mathbb D}_1(f,1; x,y)^2:= \sum_{\substack{x< p\leq y;\\ p\equiv 1 \quad\pmod{4}}} \frac{1}{p}\, (1-\Re(f(p))) \end{align*} $$

for $x<y$ .

Proof. We first define the multiplicative function $\tilde {f}\colon {\mathbb N}\to {\mathbb U}$ on prime powers as follows

$$ \begin{align*} \tilde{f}(p^k):= \begin{cases}f(p^k), \quad &\text{if }\ p> K_0\\ 1, \quad &\text{otherwise} \end{cases}. \end{align*} $$

Since $p\leq K_0$ implies $p\mid Q$ and $(a^2+b^2,Q)=1$ , we get that $p\nmid (Qm+a)^2+(Qn+b)^2$ for every $ p\leq K_0$ ; hence,

$$ \begin{align*} f((Qm+a)^2+(Qn+b)^2)=\tilde{f}((Qm+a)^2+(Qn+b)^2) \quad \text{for every } m,n\in{\mathbb N}. \end{align*} $$

Note also that $G_N(\tilde {f},Q)=G_N(f,Q)$ and ${\mathbb D}_1(\tilde {f},1; K_0,N)={\mathbb D}_1(f,1;K_0,N)$ . It follows that in order to establish (5.33), it is enough to show that for all large enough N, depending only on $K_0$ , we have

(5.35) $$ \begin{align} {\mathbb E}_{m,n\in [N]}|\tilde{f}((Qm+a)^2+(Qn+b)^2)&- \exp(G_N(\tilde{f},Q))|\ll \notag\\ &({\mathbb D}_1+{\mathbb D}_1^2)(\tilde{f},1; K_0,\sqrt{N})+ Q\cdot {\mathbb D}_1(\tilde{f},1; \sqrt{N},N)+ K_0^{-\frac{1}{2}}. \end{align} $$

In order to establish (5.35), we make a series of further reductions that will eventually allow us to apply Lemma 5.5. For every $p\equiv 3 \ \pmod 4,$ we have that $p\mid m^2+n^2$ implies that $p\mid m$ and $p\mid n$ . Consequently, the contribution to the average of those $m,n\in [N]$ for which $(Qm+a)^2+(Qn+b)^2$ is divisible by some prime $p\equiv 3 \ \pmod 4$ is (note that $(Qm+a)^2+(Qn+b)^2$ is only divisible by primes $p>K_0$ )

$$ \begin{align*} \ll \frac{1}{N^2}\sum_{K_0< p\leq N} \left[\frac{N}{p}\right]^2\ll \frac{1}{K_0}, \end{align*} $$

which is acceptable.

Next, we show that the contribution to the average in (5.35) of those $m,n\in [N]$ for which $ (Qm+a)^2+(Qn+b)^2$ is divisible by $p^2$ for some prime $p\equiv 1 \ \pmod 4$ with $p>P_0$ (hence $p\nmid Q$ ) is also acceptable. Indeed, for fixed $n\in [N]$ such that $p\nmid Qn+b$ , there exist at most $2[N/p^2]+2$ values of $m\in [N]$ such that $p^2\mid (Qm+a)^2+(Qn+b)^2.$ However, if $p\mid Qn+b$ and $p\mid (Qm+a)^2+(Qn+b)^2$ , then also $p\mid Qm+a$ . Hence, the contribution to the average in (5.35) of those $m,n\in [N]$ for which $ (Qm+a)^2+(Qn+b)^2$ is divisible by $p^2$ for some prime $p\equiv 1 \ \pmod 4$ is bounded by (note again that $(Qm+a)^2+(Qn+b)^2$ is only divisible by primes $p> K_0$ )

$$ \begin{align*} \ll \frac{1}{N^2} \Big(\sum_{K_0< p\leq N}\Big(\left[\frac{N}{p^2}\right]+1\Big)N+\sum_{K_0< p\leq N}\left[\frac{N}{p}\right]^2\Big)\ll \frac{1}{K_0}+\frac{1}{\log{N}}, \end{align*} $$

where we used the prime number theorem to bound $\frac {1}{N}\sum _{K_0<p\leq N} 1$ .

Combining the above reductions, we deduce that in order to establish the estimate (5.35), we may further assume that

(5.36) $$ \begin{align} \tilde{f}(p^k)=1 \ \text{ for all } \ p\in {\mathbb P},\, k\ge 2, \ \text{ and }\ \tilde{f}(p^k)=1 \ \text{ for all } \ p\equiv 3 \quad\pmod 4, \, k\in {\mathbb N}. \end{align} $$

We are now in a situation where Lemma 5.5 is applicable and gives that for all large enough N, depending only on $K_0$ , if ${\mathbb D}_1(f,1;K_0,N)\leq 1$ , we have (note that (5.36) implies that $\mathbb {D}_1(\tilde {f}, 1;K_0,N)=\mathbb {D}(\tilde {f}, 1;K_0,N)$ )

$$ \begin{align*} {\mathbb E}_{m,n\in [N]}|\tilde{f}((Qm+a)^2+(Qn+b)^2)&- \exp(G_N(\tilde{f},K_0))|\ll \\ &\quad\quad ({\mathbb D}_1+{\mathbb D}_1^2)(\tilde{f},1; K_0,\sqrt{N})+ Q\cdot {\mathbb D}_1(\tilde{f},1; \sqrt{N},N)+K_0^{-\frac{1}{2}}. \end{align*} $$

Combining this bound with the bounds we got in order to arrive to this reduction, we get that (5.35) is satisfied. This completes the proof.

5.4 Proof of Proposition 5.1

We start with some reductions. Suppose that the statement holds when $\chi =1$ and $t=0$ . We will show that it holds for arbitrary $\chi $ and t. Let $\tilde {f}:=f \cdot \overline {\chi }\cdot n^{-it}$ , and apply the conclusion for $\chi =1, t=0$ . We get the following bound for $\tilde {f}$ :

(5.37) $$ \begin{align} &{\mathbb E}_{m,n\in [N]}\, \big|\tilde{f}\big((Qm+a)^2+(Qn+b)^2\big)- \exp\big(G_N(\tilde{f},K_0)\big)\big|\ll \notag\\ &\qquad\quad ({\mathbb D}_1+{\mathbb D}_1^2)(f,1; K_0,\sqrt{N})+Q^2\cdot{\mathbb D}_1(f,1;N,3Q^2N^2)+ Q\cdot {\mathbb D}_1(f,1; \sqrt{N},N)+K_0^{-1/2}. \end{align} $$

Note that since $\chi $ is periodic with period q and $q\mid Q$ , we have $\chi ((Qm+a)^2+(Qn+b)^2)=\chi (a^2+b^2)$ for every $m,n\in {\mathbb N}$ . Furthermore, since by assumption $(a^2+b^2, Q)=1$ and $q\mid Q$ , we have $(a^2+b^2, q)=1$ ; hence, $|\chi (a^2+b^2)|=1$ . Also, $\lim _{m,n\to \infty }((Qm+a)^2+(Qn+b)^2)^{it}-Q^{2it}\cdot (m^2+n^2)^{it}=0$ and ${\mathbb D}_1(\tilde {f}, 1;x,y)={\mathbb D}_1(f, \chi \cdot n^{it};x,y)$ . Lastly, note that

$$ \begin{align*} G_N(\tilde{f},K_0)=2\sum_{\substack{ K_0< p\leq N,\\ p\equiv 1 \quad\pmod{4}}}\, \frac{1}{p}\,(\tilde{f}(p) -1)= 2\sum_{\substack{ K_0< p\leq N,\\ p\equiv 1 \quad\pmod{4}}}\, \frac{1}{p}\,(f(p)\cdot \overline{\chi(p)}\cdot n^{-it} -1)=G_N(f,K_0). \end{align*} $$

After inserting this information in (5.37), we get that (5.3) is satisfied.

So it suffices to show that if $Q= \prod _{p\leq K_0}p^{a_p}$ for some $a_p\in {\mathbb N}$ , then if N is large enough, depending only on Q, and ${\mathbb D}_1(f, 1;K_0,N)\leq 1$ , we have

(5.38) $$ \begin{align} &{\mathbb E}_{m,n\in [N]}\, \big|f\big((Qm+a)^2+(Qn+b)^2\big)- \exp\big(G_N(f,K_0)\big)\big|\ll \notag\\ &\qquad ({\mathbb D}_1+{\mathbb D}_1^2)(f,1; K_0,\sqrt{N})+ Q^2\cdot {\mathbb D}_1(f,1;N,3Q^2N^2)+ Q\cdot {\mathbb D}_1(f,1; \sqrt{N},N)+K_0^{-1/2}. \end{align} $$

For every $N\in {\mathbb N}$ , we decompose f as $f=f_{N,1} \cdot f_{N,2}$ , where the multiplicative functions $f_{N,1},f_{N,2}\colon {\mathbb N}\to {\mathbb U}$ are defined on prime powers as follows:

$$ \begin{align*} f_{N,1}(p^k):=& \begin{cases}f(p), \quad &\text{if }\ k=1 \text{ and } p> N,\, p\equiv 1 \quad\pmod{4} \\ 1, \quad &\text{otherwise} \end{cases}, \\ f_{N,2}(p^k):=& \begin{cases}1, \quad &\text{if }\ k=1 \text{ and } p> N, \, p\equiv 1 \quad\pmod{4} \\ f(p^k), \quad &\text{otherwise} \end{cases}. \end{align*} $$

We first study the function $f_{N,1}$ . Following the notation of Lemma 5.3 for $l,Q,N\in {\mathbb N}$ , we let

$$ \begin{align*} w_{N,Q}(l):=\frac{1}{N^2} \, \sum_{\substack{m,n\in [N],\\ l\mid (Qm+a)^2+(Qn+b)^2} } 1. \end{align*} $$

Lemma 5.3 implies that if l is a sum of two squares, then

(5.39) $$ \begin{align} w_{N,Q}(l)\ll \frac{Q^2}{l}. \end{align} $$

Since for $N\gg Q$ we have $f_{N,1}((Qm+a)^2+(Qn+b)^2)-1\neq 0$ only if $(Qm+a)^2+(Qn+b)^2$ is divisible by one or two primes $p>N$ ,Footnote ¹⁰ we get

(5.40) $$ \begin{align} {\mathbb E}_{m,n\in[N]} &|f_{N,1}\big((Qm+a)^2+(Qn+b)^2\big)-1| \leq \notag\\ &\qquad \sum_{\substack{N<p\le 3Q^2N^2,\\ p\equiv 1 \quad\pmod{4}}} |f(p)-1|\, w_{N,Q}(p) + \sum_{\substack{N<p,q\le 3Q^2N^2,\, p\neq q\\ p,q\equiv 1 \quad\pmod{4}}} |f(pq)-1|\, w_{N,Q}(pq), \end{align} $$

where we used that $f_{N,1}(p)=f(p)$ for all $p>N$ , and in the second sum, we have ignored the contribution of the diagonal terms $p=q$ since, by construction, $f_{N,1}(p^2)=1$ for all primes p. Using (5.39) for $l:=p$ , which is a sum of two squares since $p\equiv 1 \ \pmod {4}$ , we estimate the first term as follows:Footnote ¹¹

$$ \begin{align*} \sum_{\substack{N<p\le 3Q^2N^2,\\ p\equiv 1 \quad\pmod{4}}} &|f(p)-1|\, w_{N,Q}(p) \ll Q^2\sum_{\substack{N<p\le 3Q^2N^2,\\ p\equiv 1 \quad\pmod{4}}} \frac{|f(p)-1|}{p}\leq \\[5pt] &\quad Q^2 \cdot \Big(\sum_{\substack{N<p\le 3Q^2N^2,\\ p\equiv 1 \quad\pmod{4}}} \frac{|f(p)-1|^2}{p}\Big)^{\frac{1}{2}}\cdot \Big(\sum_{\substack{N<p\le 3Q^2N^2,\\ p\equiv 1 \quad\pmod{4}}} \frac{1}{p} \Big)^{\frac{1}{2}} \ll Q^2\cdot {\mathbb D}_1(f,1;N,3Q^2N^2), \end{align*} $$

where we used that $\sum _{N\le p\le 3Q^2N^2}\frac {1}{p}\ll 1$ for $N\geq Q$ . Similarly, using (5.39) for $l:=pq$ , which is a sum of two squares since $pq\equiv 1 \ \pmod {4}$ , we estimate the second term in (5.40) as follows (note that since $p\neq q$ , we have $f(pq)=f(p)f(q)$ ):

$$ \begin{align*} &\sum_{\substack{N<p,q\le 3Q^2N^2,\, p\neq q, \\ p,q\equiv 1 \quad\pmod{4}}} |f(pq)-1|\, w_{N,Q}(pq) \ll Q^2\sum_{\substack{N<p,q\le 3Q^2N^2,\\ p,q\equiv 1 \quad\pmod{4}}}\frac{|f(p)-1|+|f(q)-1|}{pq}\leq \\[5pt] &\quad\qquad 2\, Q^2 \cdot \Big(\sum_{\substack{N<p,q\le 3Q^2N^2,\\ p,q\equiv 1 \quad\pmod{4}}} \frac{|f(p)-1|^2}{pq}\Big)^{\frac{1}{2}}\cdot \Big(\sum_{\substack{N<p,q\le 3Q^2N^2,\\ p,q\equiv 1 \quad\pmod{4}}} \frac{1}{pq} \Big)^{\frac{1}{2}} \ll Q^2\cdot \mathbb{D}_1(f,1;N,3Q^2N^2), \end{align*} $$

where we used that $\sum _{N\le p\le 3Q^2N^2}\frac {1}{p}\ll 1$ for $N\geq Q$ . Combining the above estimates and (5.40), we deduce that for $N\gg Q$ , we have