arXiv 2107.043482021-07math.NT

一个具有素幂模的Bennieri-Vinogradov型定理

A Bombieri-Vinogradov-type theorem with prime power moduli

Stephan Baier, Sudhir Pujahari

备注：: 22 pages

摘要：

在2020年，Roger Baker \cite{Bak}证明了下列算术级数的素数定理中关于模的例外集的一个结果。令$\mathcal{S}$是一组成对互质模$q\le x^{9/40}$。然后素数$l\le x$按期望分布在算术级数mod $q$中，除了$\mathcal{S}$的一个子集，其基数由$\log x$的幂限定。我们使用$p$ -adic变体Harman筛将Baker的范围扩展到$q\le x^{1/4-\varepsilon}$，如果$\mathcal{S}$被限制为素数幂$p^N$，其中$p\le (\log x)^C$对于一些固定但任意的$C>0$。对于足够大的$C$，我们得到了几乎所有的结果。在此之前，Barban、Linnik和Chudakov \cite{BLC}在更宽的范围$p^N\le x^{3/8-\varepsilon}$内建立了$π(x;p^N,a)$的期望类型的渐近估计，其中$p$是奇素数。Gallagher \cite{Gal}将这个范围扩展到了$p^N\le x^{2/5-\varepsilon}$，而Huxley \cite{Hux2}将Gallagher的指数改进为$5/12$。Banks和Shparlinski \cite{BaS}最近为更宽的范围$p^N\le x^{0.4736}$建立了正确数量级的下限。然而，所有这些结果都适用于{ \it固定}素数$p$，并且相关估计中的$O$常数依赖于$p$。因此，它们不包含我们的结果。在我们文章的一部分中，我们描述了我们的方法与这些结果的关系。

英文摘要：

In 2020, Roger Baker \cite{Bak} proved a result on the exceptional set of moduli in the prime number theorem for arithmetic progressions of the following kind. Let $\mathcal{S}$ be a set of pairwise coprime moduli $q\le x^{9/40}$. Then the primes $l\le x$ distribute as expected in arithmetic progressions mod $q$, except for a subset of $\mathcal{S}$ whose cardinality is bounded by a power of $\log x$. We use a $p$-adic variant Harman's sieve to extend Baker's range to $q\le x^{1/4-\varepsilon}$ if $\mathcal{S}$ is restricted to prime powers $p^N$, where $p\le (\log x)^C$ for some fixed but arbitrary $C>0$. For large enough $C$, we thus get an almost all result. Previously, an asymptotic estimate for $π(x;p^N,a)$ of the expected kind, with $p$ being an odd prime, was established in the wider range $p^N\le x^{3/8-\varepsilon}$ by Barban, Linnik and Chudakov \cite{BLC}. Gallagher \cite{Gal} extended this range to $p^N\le x^{2/5-\varepsilon}$ and Huxley \cite{Hux2} improved Gallagher's exponent to $5/12$. A lower bound of the correct order of magnitude was recently established by Banks and Shparlinski \cite{BaS} for the even wider range $p^N\le x^{0.4736}$. However, all these results hold for {\it fixed} primes $p$, and the $O$-constants in the relevant estimates depend on $p$. Therefore, they do not contain our result. In a part of our article, we describe how our method relates to these results.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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