The Additive Problem for the Number of Representations as a Sum of Two Squares

We improve a previous unconditional result about the asymptotic behavior of ∑n≤xr(n)r(n+m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{n\le x} r(n)r(n+m)$$\end{document} with r(n) the number of representations of n as a sum of two squares when m may vary with x.


Introduction
We consider the analogue of the additive divisor problem when the usual divisor function τ (n) is replaced by r(n), the number of representations of n as a sum of two squares, which is related to the divisor function for the Gaussian integers. Namely, we study the asymptotic behavior of S(x, m) = n≤x r(n)r(n + m) with m ∈ Z + . (1.1) With a broad view, this can be considered a shifted convolution of theta coefficients. Apparently, Estermann was the first author considering this problem [7]. His result implies that for m fixed and 2 k , the 2-part in the factorization of m Actually, in the original paper [7], the coefficient of x/m is expressed in the more compact form 8 d|m (−1) m+d d which arguably gives less insight about its size for large values of m and about the role of the powers of 2. Unfortunately, with our present knowledge about θ, we cannot rule out the possibility of θ belonging to a very thin interval of length 1/192 in which one can carry out a different optimization getting a slight improvement in a small range. To keep the statement simpler, we did not include it in Theorem 1.1. Theorem 1.4. If θ ∈ (5/48, 7/64], for m 11(1−4θ)/7 < x < m min(92θ−5, 46θ)/5 , the bound in Theorem 1.1 can be improved to E(x, m) x 17/23+ + (mx) 17/46+ + m (13+4θ)/28 x 1/4+ for every > 0.
To put into perspective the numerical size of the improvement of Theorem 1.1 in the range indicated in Theorem 1.4, we mention that the maximal gain is x 4/1137 attached only when θ = 7/64 and m = x 1232/1137 . For instance, for m = x, the saving is at most x 1/2208 . Note that the interval for x collapses as θ → 5/48.
When we substitute the best-known upper bound for θ, due to H. Kim and Sarnak (see Lemma 2.3 below), Theorems 1.1 and 1.4 show: Corollary 1.5. With the notation as before In the first of the following figures, it is represented the graph of the piecewise linear function β = β(α), such that x β is the corresponding bound in Corollary 1.5. The vertical dashed lines mark the change from a linear function to another. The second figure shows the detail of the range in which the improvement of Theorem 1.4 applies. The

Preliminary Results
The aim of this section is to introduce some notation and to state Proposition 2.2 that embodies all the information about the spectral expansion of S(x, m). We also recall some known results about the Hecke eigenvalues and we finish with some comments about the spectral meaning of (1.2). The material of this section is essentially included in [5] and it is recalled here for the sake of completeness and to fix the notation.
Consider the set It is plain that S(x, m) in (1.1) and #C(x, m) coincide. Consider also the allied quantity For m, even the formula is elementary (see the proof of Lemma 2.2 in [5] for a detailed discussion). Clearly, if 8 | m, then the last S(x/2, m/2) can be reduced once more. In general, we have the following.
otherwise. This is a re-formulation of the first part of [5,Lemma 2.2]. It implies that to deal with (1.2), it is enough to consider the asymptotic behavior of S(x, m) for 2 m ∈ Z + and that of A(4x, 4m) for m ∈ Z + . It turns out that both expressions have a similar spectral expansion. This is the content of the following proposition that summarizes §2 of [5] (see this reference for the proof).  where Δ is an arbitrarily chosen number 0 < Δ < min(1, y 1/2 ), and For y > 1, |t| < 1, the latter formula still holds multiplying the right-hand side by log(2y).
We recall that in this context, the Hecke operators are given by and we have Here, η t (m) = ad=m (a/d) it , and hence, |η t (m)| ≤ τ (m). In this context, the Ramanujan-Petersson conjecture claims that λ j (m) is bounded in the same way and the state of the art loses a small power using profound techniques (see [21] for the ideas leading to this breakthrough). An application of the Kuznetsov formula gives the following average result. For the proof of a stronger result, see [6,Cor.5.3] and use [10, (8.43)].
We devote the rest of this section to explain the spectral origin of the main term in the asymptotic formula (1.2).
The Fuchsian groups appearing in Proposition 2.2 do not have exceptional eigenvalues [9], and then, t j only takes real values (satisfying |t j | > 5 [14]) and t 0 = i/2 that corresponds to the zero eigenvalue 1/4 + t 2 0 of the Laplace-Beltrami operator coming from the constant eigenfunction u 0 . As If we tune Δ in such a way that h(i/2) ∼ 4πy and the contribution of the rest of the terms in the spectral expansion in Proposition 2.2 is negligible then for m fixed Lemma 2.1 complements the latter formula producing for m even the asymp- This equals 8 σ(2 k ) − 2 σ(m/2 k )x/m with standard manipulations using that σ is a multiplicative function. We can combine the three cases 2 m, 4 m with m even, and 4 | m in the somewhat artificial single formula of (1.2).

Proof of the Main Result
After Proposition 2.2 and knowing that the involved Eisenstein series behave essentially as the square of the Riemann zeta function, a fundamental problem to get uniform asymptotic formulas for S(x, m) is to find good upper bounds for Even if we employ exactly the same techniques as in [5], the reduction given in [13] for the upper bound of θ changes substantially the way in which the optimization can be made to obtain Theorem 1.1, by this reason, the unconditional results in [5] do not correspond to substitute θ ≤ 5/28, the best upper bound available at that time, in Theorem 1.1. There is also a new technique here not appearing in [5]. It consists in using the Hecke relation [10, (8.39)] (λ j ∈ R because the Hecke operators are self-adjoint) to get via Cauchy's inequality The advantage of this expression is that for d = m now |u j (z 0 )| 2 is multiplied by a changing sign coefficient that we can exploit using spectral theory to quantify the cancellation if T and m are not very large. In [5], Cauchy's inequality was applied directly to (3.1) and it required some knowledge about |u j (z 0 )| 4 . The alternative use of Cauchy's inequality described above was suggested by Raphael Steiner (personal communication) and we fully credit him for this important remark that conveniently developed is responsible for the range m 3/2−θ ≤ x < m 3/2+3θ in Theorem 1.1. for every > 0.
Proof. We vary a little the previous scheme, smoothing the sum and completing the spectrum. Namely, we start noting that e 4 S(m, T ) is less than Using Cauchy's inequality, (3.2), and recalling that η t (m) are also Hecke eigenvalues obeying this relation, we have Note that λ j (1) = η t (1) = and k(t) is the inverse Selberg-Harish-Chandra transform of h(t) = e −t 2 /T 2 . As indicated, the Hecke operator T d 2 acts on z.
Recall that the matrices corresponding to the maps z → (az+b)/d in the definition of the Hecke operators (2.2) are representatives of Γ\Γ m where Γ m formally has the same definition as Γ, but imposing that the determinant of its elements is m instead of 1. Then In [6, §5], it is shown a general estimate for the inverse Selberg-Harish-Chandra transform that gives in this case In particular, k(t) decays faster than any power and as sinh 2 (t/2) ∼ t 2 /4 for t small, the main contribution comes from u(γz 0 , z 0 ) T −2+ . Let us define In [11, (A.9), (A.10)] (see [4, §10] for a more explicit statement), there is a general bound for M (t) based on a counting argument. We will show that we can take advantage of the special form of z 0 , as in [10, Lemma 13.1], to get Combining this with (3.4), we deduce

And since (3.3)
which gives the result. It remains to prove (3.5). The points z 0 = i and z 0 = (i − 1)/2 are in the same orbit under PSL 2 (Z).
Whence to prove (3.5), we can restrict ourselves to the case z 0 = i, Γ = PSL 2 (Z). A calculation shows that if γ has determinant d 2 To ease references, we state here the bound for S(m, T ) obtained combining Lemma 2.4 and a convexity bound coming from Lemma 2.5 and Bessel inequality.  We divide the proof of Theorem 1.1 in two parts studying separately the spectral contribution in the cases m ≤ x and m > x.

Proof. We have
We first state an auxiliary result bringing Proposition 2.2 closer to the estimation of E(x, m). Let E(x, m) be the result of subtracting 8πσ −1 (m)x/|Γ\H| to the quantities S(x, m) with 2 m or A(4x, 4m). Then, for x ≥ 1 Our bounds for S(m, T ) are greater than T 2 , even under Ramanujan-Petersson conjecture; hence, the term T 2 in E T is irrelevant in practice.
and this is O x 1/2 log(2x) using the crude bound Proof. We have √ mH(T ) Clearly, min 1, (T Δ) −3/2 is not increasing in T and T 1/2 min(m θ , m 1/2 T −1 ) is greater for T = m 1/2−θ than for T = m 1/2 . It assures F (Δ, m 1/2 ) F (Δ, m 1/2−θ ) and we have If x ≥ m 3/2−θ , we choose the first arguments in the minima to get Hence, inf G(Δ) ≤ G(x −1/3 ) x 2/3 + m 1/2 x 1/3 + m (1+2θ)/4 x 1/2 and the central term is negligible, because x 2/3 > m 1/2 x 1/3 if x > m 3/2 , and we have m 1/2 x 1/3 < m (1+2θ)/4 x 1/2 otherwise. If x < m 3/2−θ , we choose the second arguments in the minima to deduce x, and in our range, the last term is negligible. Namely, we have to prove The bound Δ < (x/m) 1/2 is required by Proposition 2.2 and it is less important for the optimization. Note that F (Δ, T ) − m 1/2 x 1/2 Δ is like in the proof of Proposition 3.4 except for a coefficient not depending on Δ and T . Hence, the same argument applies to show that the supremum on T in (3.7), say G(Δ), satisfies In the last range of the case m ≤ x in the proof of Proposition 3.4, the term m θ gave the minimum and was of greater order than m 1/2 Δ 1/4 , and then, it is natural to consider by continuity that this is still the situation. With this idea in mind, let us take Δ −1 = x 1/6 m 1/6−2θ/3 to balance the corresponding terms. Note that in our range, it satisfies the assumption Δ −1 > m 1/2 x −1/2 . The result is that (3.7) holds with m (1+2θ)/3 x 1/3 accompanied with the extra terms In our range, the minima are m θ and m (θ−1)/2 x 1/4 , respectively. Hence, it only remains to check that m (1+2θ)/3 x 1/3 ≥ m θ x 1/2 which follows from m ≥ x.
Finally, we prove the estimate required for Theorem 1.4.