BOMBIERI–VINOGRADOV THEOREMS FOR MODULAR FORMS AND APPLICATIONS
BOMBIERI–VINOGRADOV THEOREMS FOR MODULAR FORMS AND APPLICATIONS
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DOI：
10.1112/mtk.12014
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发表时间：
2019
期刊：
影响因子：
0.8
通讯作者：
PENG-JIE Wong
中科院分区：
文献类型：
--
作者：
PENG-JIE Wong
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) q2+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 (p1, p2) of distinct primes satisfying |p1 − p2| C0 and λ f (p1)λ f (p2) = 0, which presents a modular analogue of the recent work of Maynard and Zhang on bounded gaps between primes. §
