Mathematical Sciences: Mean Values and Zeros of Dirichlet Series

本研究的目的是探讨狄利克雷级数的零分布和均值的若干开放问题。第一个问题是证明狄利克雷多项式的渐近均值估计，它的长度远远大于积分的区间，或者在离散均值的情况下，远远大于求和的范围。这些结果的一个直接应用是研究黎曼ζ函数零点之间的间隙。第二个问题是证明某些由狄利克雷级数定义的函数有一个类似黎曼ζ函数的泛函方程并且在临界线上有无穷多个零，但是在临界线上仍然有正比例的零。第三个问题是扩大对相关方法的应用范围，从而为迄今为止无法获得的zeta函数和算术函数建立（有条件的）均值估计。另一个问题是得到zeta函数的分数阶矩的渐近估计。设想的方法需要对零的垂直分布进行某种假设。最后一个问题是研究附在函数场上的函数的零点分布。

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.