Correlated sums of r(n)

Abstract

We prove an asymptotic formula for P nUN rðnÞrðn þ mÞ using the spectral theory of automorphic forms and we specially study the uniformity of the error term in the asymptotic approximation when m varies. The best results are obtained under a natural conjecture about the size of a certain spectral mean of the Maass forms.We also employ large sieve type inequalities for Fourier coe‰cients of cusp forms to estimate some averages (over m) of the error term.

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

“…Our main result approximates S(x, m) improving the bound for the error term given in [5]. We prefer to establish it in terms of the optimal exponent for the Hecke eigenvalues λ j (m) of the Maass-Hecke waveforms (see the next section for more on the notation).…”

“…Here, we address the size of the error term and its uniformity in m to study to what extent m can depend on x keeping (1.2) valid. This problem was treated in [5]. Most of the results were stated there under a certain conjecture on spectral theory, but the last section includes some unconditional results.…”

“…Most of the results were stated there under a certain conjecture on spectral theory, but the last section includes some unconditional results. In connection with the asymptotic formula, it was proved [5,Cor.5.3] that (1.2) is valid if m = m(x) satisfies m = O x 17/11− for some > 0.…”

Some applications of Laplace transforms in analytic number theory

“…The arithmetic part of the proof of Theorem 4 and Theorem 5 is based on the formulas of F. Chamizo [3] and Y. Motohashi [27], respectively. These are n x…”

Sums of squares in $\mathbb{Z}[\sqrt{k}]$

“…With this idea in mind, one can prove (see [Iw2] H(X; z,w) is automorphic as a function of z or w, so it can be analysed with the spectral theory of The lack of regularity of H requires some smoothing before applying the pretrace formula. These steps were worked out in Lemma 2.3 of [Chl] (see also Proposition 2.3 of [Ch2]).…”