当前位置: X-MOL 学术Mathematika › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
BOMBIERI–VINOGRADOV THEOREMS FOR MODULAR FORMS AND APPLICATIONS
Mathematika ( IF 0.8 ) Pub Date : 2019-12-05 , DOI: 10.1112/mtk.12014
Peng‐Jie Wong 1
Affiliation  

In this article, we consider a prime number theorem for arithmetic progressions “weighted” by Fourier coefficients of modular forms, and we develop Siegel‐Walfisz type and Bombieri–Vinogradov type estimates for such a modular analogue. As an application, we have a Turán type estimate for modular forms asserting that for any δ > 0 and non‐CM normalised Hecke eigenform f,
P f ( a , q ) q 2 + δ ,
with a possible exceptional set of q of density 0 (depending at most on f and δ), where ( a , q ) = 1 , P f ( a , q ) denotes the least prime p, with λ f ( p ) 0 , congruent to a ( mod q ) , and λ f ( p ) is the pth Fourier coefficient of f. Moreover, we show the existence of a positive absolute constant C0, independent of f, such that there are infinitely many pairs ( p 1 , p 2 ) of distinct primes satisfying
| p 1 p 2 | C 0 and λ f ( p 1 ) λ f ( p 2 ) 0 ,
which presents a modular analogue of the recent work of Maynard and Zhang on bounded gaps between primes.


中文翻译:

模块化形式的BOMBIERI–VINGRADVOV定理和应用

在本文中,我们考虑了算术级数的素数定理,该算术级数由模块化形式的傅立叶系数“加权”,并且我们为此类模块化类似物开发了Siegel-Walfisz型和Bombieri-Vinogradov型估计。作为应用程序,我们对模块形式有一个Turán类型的估计,断言对于任何形式 δ > 0 和非CM归一化的Hecke特征形f
P F 一种 q q 2 + δ
可能有一组特殊的密度为0的q(最多取决于f和δ),其中 一种 q = 1个 P F 一种 q 表示最小素数p,其中 λ F p 0 ,与 一种 q λ F p fp傅里叶系数。此外,我们证明了存在独立于f的正绝对常数C 0,从而存在无数对 p 1个 p 2 满足的不同素数
| p 1个 - p 2 | C 0 λ F p 1个 λ F p 2 0
给出了Maynard和Zhang最近关于素数之间有限差距的工作的模块化模拟。
更新日期:2019-12-05
down
wechat
bug