Bombieri–vinogradov Theorems for Modular Forms and Applications

Wong, Peng‐Jie

doi:10.1112/mtk.12014

Cited by 4 publications

(3 citation statements)

References 45 publications

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“…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). Wong [28] showed (1.2) with Q = x min{ 1 n−2 , 1 2 }−ε under two similar conditions. The main tools of Jiang and Lü are the generalized Vaughan identity and the distribution of λ π (m) in arithmetic progressions, while that of Wong is Gallagher's technique as in [5].…”

Section: Introductionmentioning

confidence: 86%

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: Introductionmentioning

confidence: 86%

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

Jiang

Lü

2019

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

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“…Let be the von Mangoldt function, and define the numbers by Note that . For fixed , Wong [Won20, Theorem 9] proved that if satisfies GRC and has no Landau–Siegel zero for all Dirichlet characters , then for any , This conditionally endows with a positive level of distribution . The hypotheses for (2.8) hold for attached to non-CM holomorphic cuspidal newforms on congruence subgroups of .…”

Section: Applicationsmentioning

confidence: 99%

Zeros of Rankin–Selberg L-functions in families

Humphries,

Thorner

2024

Compositio Math.

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Let $\mathfrak {F}_n$ be the set of all cuspidal automorphic representations $\pi$ of $\mathrm {GL}_n$ with unitary central character over a number field $F$ . We prove the first unconditional zero density estimate for the set $\mathcal {S}=\{L(s,\pi \times \pi ')\colon \pi \in \mathfrak {F}_n\}$ of Rankin–Selberg $L$ -functions, where $\pi '\in \mathfrak {F}_{n'}$ is fixed. We use this density estimate to establish: (i) a hybrid-aspect subconvexity bound at $s=\frac {1}{2}$ for almost all $L(s,\pi \times \pi ')\in \mathcal {S}$ ; (ii) a strong on-average form of effective multiplicity one for almost all $\pi \in \mathfrak {F}_n$ ; and (iii) a positive level of distribution for $L(s,\pi \times \widetilde {\pi })$ , in the sense of Bombieri–Vinogradov, for each $\pi \in \mathfrak {F}_n$ .

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“…Note that a π (p) = λ π (p). For fixed θ < 1 n 2 −2 , Wong [71,Theorem 9] proved that if π satisfies GRC and L(s, π × ( π ⊗ χ)) has no Landau-Siegel zero for all primitive Dirichlet characters χ, then for any A > 0, (2.4)…”

Section: Effective Multiplicity Onementioning

confidence: 99%

Zeros of Rankin-Selberg $L$-functions in families

Humphries,

Thorner

2021

Preprint

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Let F n be the set of all cuspidal automorphic representations π of GL n with unitary central character over a number field F . We prove the first unconditional zero density estimate for the set S = {L(s, π × π ′ ) : π ∈ F n } of Rankin-Selberg L-functions, where π ′ ∈ F n ′ is fixed. We use this density estimate to prove:(i) a strong average form of effective multiplicity one for GL n ;(ii) that given π ∈ F n defined over Q, the convolution π × π has a positive level of distribution in the sense of Bombieri-Vinogradov; (iii) that almost all L(s, π×π ′ ) ∈ S have a hybrid-aspect subconvexity bound on Re(s) = 1 2 ; (iv) a hybrid-aspect power-saving upper bound for the variance in the discrepancy of the measures |ϕ(x + iy)| 2 y −2 dxdy associated to GL 2 Hecke-Maaß newforms ϕ with trivial nebentypus, extending work of Luo and Sarnak for level 1 cusp forms; and (v) a nonsplit analogue of quantum ergodicity: almost all restrictions of Hilbert Hecke-Maaß newforms to the modular surface dissipate as their Laplace eigenvalues grow.

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Bombieri–vinogradov Theorems for Modular Forms and Applications

Cited by 4 publications

References 45 publications

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

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

Zeros of Rankin–Selberg L-functions in families

Zeros of Rankin-Selberg $L$-functions in families

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