论文标题
算术进展中的素数：固定残留类别
Primes in arithmetic progressions to large moduli I: Fixed residue classes
论文作者
论文摘要
我们证明了在算术进程中对大于$ x^{1/2} $的模量进行素数的新平均值定理。我们的主要结果表明，在所有大小$ x^{1/2+δ} $的固定残基类中，将这些素数与``方便尺寸''因子进行了等分。结果，除$ o（ΔQ）$ moduli $ q \ sim q = x^{1/2+δ} $外，所有除$ O（ΔQ）$ o（ΔQ）$ o（ΔQ）$ o（ΔQ）$ Q = X} $都有预期的渐近性，我们获得的Moduli的结果大至$ x^{11/21} $。 我们的证明通过纳入了受放大方法启发的新想法，扩展了Bombieri，Fouvry，Friedlander和Iwaniec的先前技术。我们将它们与Zhang和Polymath的技术相结合，并根据我们的应用定制。特别是，我们最终依赖于来自自动形式的光谱理论（Kuznetsov痕迹公式）或代数几何（Weil和Deligne样式估计）。
We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$. Our main result shows that the primes are equidistributed for a fixed residue class over all moduli of size $x^{1/2+δ}$ with a 'convenient sized' factor. As a consequence, the expected asymptotic holds for all but $O(δQ)$ moduli $q\sim Q=x^{1/2+δ}$ and we get results for moduli as large as $x^{11/21}$. Our proof extends previous techniques of Bombieri, Fouvry, Friedlander and Iwaniec by incorporating new ideas inspired by amplification methods. We combine these with techniques of Zhang and Polymath tailored to our application. In particular, we ultimately rely on exponential sum bounds coming from the spectral theory of automorphic forms (the Kuznetsov trace formula) or from algebraic geometry (Weil and Deligne style estimates).
