Mathematical Sciences: Mean Values and Zeros of Dirichlet Series

这项研究的目的是调查一些 关于零点分布和均值分布的未决问题 Dirichlet级数。 第一个问题是要证明 Dirichlet多项式的渐近均值估计， 长度比积分的间隔长得多，或者， 离散平均值的情况下，比求和的范围。 一个 这些结果的直接应用是研究差距 在Riemann Zeta函数的零点之间。 第二个问题 是证明由狄利克雷级数定义的某些函数 有一个类似于黎曼Zeta函数的函数方程 在临界线外有无穷多个零，但仍然 都有正比例的零。第三个 问题是要扩大对的应用范围 相关方法，从而建立（条件）平均值 zeta函数和算术函数的估计， 目前还无法获得。 另一个问题是， zeta函数分数阶矩的渐近估计。 设想的方法将需要某种假设， 零的垂直分布。 最后一个问题是 研究zeta函数的零点分布， 功能字段。

The purpose of this research is to investigate a number of open problems concerning the distribution of zeros and the mean values of Dirichlet series. The first problem is to prove asymptotic mean value estimates for Dirichlet polynomials whose lengths are much longer than the interval of integration or, in the case of discrete means, than the range of summation. An immediate application of such results is to the study of gaps between zeros of the Riemann Zeta function. The second problem is to show that certain functions defined by a Dirichlet series have a functional equation like that of the Riemann Zeta function and have infinitely many zeros off the critical line, but still have a positive proportion of their zeros on it. The third problem is to widen the range of application of the pair correlation method and thereby establish (conditional) mean value estimates for the zeta function and arithmetical functions that have so far been unobtainable. Another problem is to obtain asymptotic estimates for fractional moments of the zeta function. The approach envisioned will require some kind of assumption on the vertical distribution of zeros. The final problem is to study the distribution of zeros of the zeta functions attached to function fields.