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 $\delta >0$ and non‐CM normalised Hecke eigenform f,

$${\mathcal{P}}_f(a,q)\le q^{2+\delta },$$

with a possible exceptional set of q of density 0 (depending at most on f and δ), where $(a,q)=1$, ${\mathcal{P}}_f(a,q)$ denotes the least prime p, with ${\lambda }_f(p)\ne 0$, congruent to $a(\mathrm{mod}q)$, and ${\lambda }_f(p)$ 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

$$|p_1-p_2|\le C_0\text{and}{\lambda }_f(p_1){\lambda }_f(p_2)\ne 0,$$

which presents a modular analogue of the recent work of Maynard and Zhang on bounded gaps between primes.

中文翻译：

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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$\delta >0$和非CM归一化的Hecke特征形f，

$${\mathcal{P}}_F（\mathrm{一种}，q）\le q^{2+\delta }，$$

可能有一组特殊的密度为0的q（最多取决于f和δ），其中$（\mathrm{一种}，q）=\mathrm{1个}$， ${\mathcal{P}}_F（\mathrm{一种}，q）$表示最小素数p，其中${\lambda }_F（p）\ne 0$，与 $\mathrm{一种}（模q）$和 ${\lambda }_F（p）$是f的p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$（p_{\mathrm{1个}}，p_2）$ 满足的不同素数

$$|p_{\mathrm{1个}}-p_2|\le C_0\text{和}{\lambda }_F（p_{\mathrm{1个}}）{\lambda }_F（p_2）\ne 0，$$

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。