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The Bombieri–Vinogradov Theorem on Higher Rank Groups and its Applications

Part of: Multiplicative number theory Discontinuous groups and automorphic forms Exponential sums and character sums

Published online by Cambridge University Press:  07 March 2019

Yujiao Jiang  and
Guangshi Lü
Yujiao Jiang
Affiliation:
School of Mathematics and Statistics, Shandong University, Weihai, Weihai, Shandong 264209, China Email: [email protected]
Guangshi Lü
Affiliation:
School of Mathematics, Shandong University, Jinan, Shandong 250100, China Email: [email protected]
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Abstract

We study the analogue of the Bombieri–Vinogradov theorem for $\operatorname{SL}_{m}(\mathbb{Z})$ Hecke–Maass form $F(z)$. In particular, for $\operatorname{SL}_{2}(\mathbb{Z})$ holomorphic or Maass Hecke eigenforms, symmetric-square lifts of holomorphic Hecke eigenforms on $\operatorname{SL}_{2}(\mathbb{Z})$, and $\operatorname{SL}_{3}(\mathbb{Z})$ Maass Hecke eigenforms under the Ramanujan conjecture, the levels of distribution are all equal to $1/2,$ which is as strong as the Bombieri–Vinogradov theorem. As an application, we study an automorphic version of Titchmarch’s divisor problem; namely for $a\neq 0,$

$$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D6EC}(n)\unicode[STIX]{x1D70C}(n)d(n-a)\ll x\log \log x,\end{eqnarray}$$
where $\unicode[STIX]{x1D70C}(n)$ are Fourier coefficients $\unicode[STIX]{x1D706}_{f}(n)$ of a holomorphic Hecke eigenform $f$ for $\operatorname{SL}_{2}(\mathbb{Z})$ or Fourier coefficients $A_{F}(n,1)$ of its symmetric-square lift $F$. Further, as a consequence, we get an asymptotic formula
$$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D6EC}(n)\unicode[STIX]{x1D706}_{f}^{2}(n)d(n-a)=E_{1}(a)x\log x+O(x\log \log x),\end{eqnarray}$$
 where $E_{1}(a)$ is a constant depending on $a$. Moreover, we also consider the asymptotic orthogonality of the Möbius function against the arithmetic function $\unicode[STIX]{x1D70C}(n)d(n-a)$.

Keywords

Bombieri–Vinogradov theoremFourier coefficientMaass formshifted convolution sumprime

MSC classification

Primary: 11F30: Fourier coefficients of automorphic forms
Secondary: 11L05: Gauss and Kloosterman sums; generalizations 11L20: Sums over primes 11N37: Asymptotic results on arithmetic functions 11N75: Applications of automorphic functions and forms to multiplicative problems
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 4 , August 2020 , pp. 928 - 966
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

Author Y. J. is supported by the China Postdoctoral Science Foundation (No. 2017M620285), and G. L. is supported in part by NSFC (Nos. 11771252, 11531008), IRT16R43, and Taishan Scholars Project.

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