The ratios conjecture for real Dirichlet characters and multiple Dirichlet series

IF 1.2 2区数学 Q1 MATHEMATICS

Transactions of the American Mathematical Society Pub Date : 2024-01-03 DOI:10.1090/tran/9113

Martin Čech

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引用次数: 0

Abstract

Conrey, Farmer and Zirnbauer introduced a recipe to find asymptotic formulas for the sum of ratios of products of shifted L-functions. These ratios conjectures are very powerful and can be used to determine many statistics of L-functions, including moments or statistics about the distribution of zeros.

We consider the family of real Dirichlet characters, and use multiple Dirichlet series to prove the ratios conjectures with one shift in the numerator and denominator in some range of the shifts. This range can be improved by extending the family to include non-primitive characters. All of the results are conditional under the Generalized Riemann hypothesis.

This extended range is good enough to enable us to compute an asymptotic formula for the sum of shifted logarithmic derivatives near the critical line. As an application, we compute the one-level density for test functions whose Fourier transform is supported in ( − 2 , 2 ) \left (-2,2\right ) , including lower-order terms.

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实狄利克特字符和多重狄利克特数列的比率猜想

Conrey、Farmer 和 Zirnbauer 提出了一种方法，可以找到移位 L 函数乘积比率之和的渐近公式。这些比率猜想非常强大，可用于确定 L 函数的许多统计数据，包括矩或有关零点分布的统计数据。我们考虑实 Dirichlet 字符族，并使用多重 Dirichlet 级数来证明比率猜想，其分子和分母在移位的某个范围内有一次移位。这个范围可以通过扩展该族以包括非原始字符来改进。所有结果都是广义黎曼假设条件下的结果。这一扩展范围足以让我们计算出临界线附近移位对数导数之和的渐近公式。作为应用，我们计算了傅里叶变换在 ( - 2 , 2 ) \left (-2,2\right ) 中得到支持的检验函数的单级密度，包括低阶项。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Transactions of the American Mathematical Society 数学-数学

CiteScore

2.30

自引率

7.70%

发文量

171

审稿时长

3-6 weeks

期刊介绍： All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are. This journal is devoted to research articles in all areas of pure and applied mathematics. To be published in the Transactions, a paper must be correct, new, and significant. Further, it must be well written and of interest to a substantial number of mathematicians. Piecemeal results, such as an inconclusive step toward an unproved major theorem or a minor variation on a known result, are in general not acceptable for publication. Papers of less than 15 printed pages that meet the above criteria should be submitted to the Proceedings of the American Mathematical Society. Published pages are the same size as those generated in the style files provided for AMS-LaTeX.