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.
