An analogue of the Bombieri–Vinogradov Theorem for Fourier coefficients of cusp forms

Acharya, Ratnadeep

doi:10.1007/s00209-017-1875-2

Cited by 2 publications

(4 citation statements)

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“…In our setting, it is unclear to us how to construct such an L(s) with a pole of odd order at s = 1 which is known to have good analytic properties. 1 It was suggested by Molteni [19,] that an auxiliary L-function L(s) with a pole of order two at s = 1 could be used, but the strategy outlined therein has a serious gap. We fill this gap using work of Li, especially [15,Corollary 7].…”

Section: Remarkmentioning

confidence: 99%

“…Remark. The case with n = 2 is known by the work of Acharya [1]. The first two authors handled the cases with n = 2, 3 in [13] and obtained a stronger upper bound than that in Corollary 1.4 under GRC.…”

Section: Remarkmentioning

confidence: 99%

“…Subject to this hypothesis, we can choose χ ′ ∈ {ν 1 , ν 2 } − {χ}. If we choose β ∈ (1 − ε 4 , 1) to be the point at which L(s, π ⊗ χ ′ ) vanishes, then we may conclude that for all 0 < ε < 1 2 , there exist χ ′ and β ∈ (1 − ε 4 , 1) (depending at most on ε and π) such that L(β, Π ⋆ ) = 0. With these choices of χ ′ (mod q ′ ) and β, the bound (3.3) reduces to Corollary 3.7.…”

Section: Zero-free Regions and The Initial Rangementioning

confidence: 99%

“…Actually, Perelli's approach still works for any holomorphic cusp form on SL 2 (Z). Recently, Acharya [1] and the first two authors [13] improved independently the level to 1/2 for any holomorphic or Maass cusp form on SL 2 (Z). For any automorphic form π on higher-rank group SL n (Z) with n 3, let λ π (m) to be the m-th Dirichlet coefficient of the associated L-function L(s, π), one can also show a result of Bombieri-Vinogradov type For instance, the first two authors [13] established (1.2) with Q = x 2 n+1 (log x) −B under the generalized Ramanujan conjecture (GRC) and a certain condition concerning Siegel's zeros of the twisted L-functions L(s, π ⊗χ).…”

Section: Introductionmentioning

confidence: 99%

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

Jiang

Lü

2019

Can. J. Math.-J. Can. Math.

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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 $\unicode[STIX]{x1D70C}(n)d(n-a)$ .

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Section: Remarkmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Zero-free Regions and The Initial Rangementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Jiang

Lü

2019

Can. J. Math.-J. Can. Math.

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A Bombieri–Vinogradov Theorem for Higher-Rank Groups

Jiang

Lü

Thorner

et al. 2021

International Mathematics Research Notices

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We establish a result of Bombieri–Vinogradov type for the Dirichlet coefficients at prime ideals of the standard $L$-function associated to a self-dual cuspidal automorphic representation $\pi $ of $\operatorname {GL}_n$ over a number field $F$ when $\pi$ is not a quadratic twist of itself. Our result does not rely on any unproven progress towards the generalized Ramanujan conjecture or the nonexistence of Landau–Siegel zeros. In particular, when $\pi $ is fixed and not equal to a quadratic twist of itself, we prove the first unconditional Siegel-type lower bound for the twisted $L$-values $|L(1,\pi \otimes \chi )|$ in the $\chi $-aspect, where $\chi $ is a primitive quadratic Hecke character over $F$. Our result improves the levels of distribution in other works that relied on these unproven hypotheses. As applications, when $n=2,3,4$, we prove a $\textrm {GL}_n$ analogue of the Titchmarsh divisor problem and a nontrivial bound for a certain $\textrm {GL}_n\times \textrm {GL}_2$ shifted convolution sum.

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An analogue of the Bombieri–Vinogradov Theorem for Fourier coefficients of cusp forms

Cited by 2 publications

References 10 publications

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

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

A Bombieri–Vinogradov Theorem for Higher-Rank Groups

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