Some Mean Value Theorems for the Riemann Zeta-function and Dirichlet L-functions

The theory of the Riemann zeta-function ζ(s) and Dirichlet L-functions L(s, χ) abounds with unsolved problems. Chronologically the first of these, now known as the Riemann Hypothesis (RH), originated from Riemann’s remark that it is very probable that all non-trivial zeros of ζ(s) lie on the line < s = 12 . Later on Piltz conjectured the same for all of the functions L(s, χ) (GRH). The vertical distribution of the zeta zeros is the subject of Montgomery’s pair correlation conjecture, which can also be generalized for L(s, χ). In this theory there are many other questions, most of them still open, about value distribution, non-existence of linear relations among zeros of a function, non-existence of common zeros of these functions etc. The results mentioned in this article may be seen as a first step in addressing some of these matters.