## Dirichlet series as a generalization of power series

James M. Osborn

James M. Osborn

Published in Proceedings of the National Academy of Sciences of the United States of America

In this note, we construct certain families of Dirichlet series that satisfy functional equations. We shall also indicate an arithmetical application of our results. The work presented here depends upon the theory of prehomogeneous vector spaces.

This is a sequel (Part II) to an earlier article with the same title. There are reasons to expect that the estimates proved in Part I without the factor $(\log\log H)^{-C}$ represent the real truth, and this is indeed proved in part II on the assumption that in the first estimate $2k$ is an integer. %This is of great interest, for little has been k...

The main theme of this paper is to systematize the Hardy-Landau $\Omega$ results and the Hardy $\Omega_{\pm}$ results on the divisor problem and the circle problem. The method of ours is general enough to include the abelian group problem and the results of Richert and the later modifications by Warlimont, and in fact theorem 6 of ours is an improv...

In this paper the number of the zeros of generalised Dirichlet series satisfying some reasonable conditions is discussed.

Published in Mathematical Notes

For the Dirichlet series corresponding to a functionF with positive exponents increasing to ∞ and with abscissa of absolute convergenceA ∈ (−∞, +∞], it is proved that the sequences (μ(σ, F(m))) of maximal terms and (Λ(σ, F(m))) of central exponents are nondecreasing to ∞ asm → ∞ for any givenσ

Published in Compositio Mathematica

We consider the Dirichlet series Z(P,A;s) = ∑\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \sum _{m \in A \cap {\Bbb Z}^n } $$ \end{document} P−s(m) (s ∈ C) where P...

In a recent paper K. Ramachandra states some conjectures, and gives consequences in the theory of the Riemann zeta function. In this paper we will present two different disproofs of them. The first will be an elementary application of the Szasz-M\"unto theorem. The second will depend on a version of the Voronin universality theorem, and is also sli...

Published in Lithuanian Mathematical Journal

The joint limit distribution of functions given by Dirichlet series is studied. The necessary and sufficient condition when this distribution is a product of marginal distributions is found. An example of such Dirichlet series with linear independent systems of exponents is presented.

For ``good Dirichlet series'' $F(s)$ we prove that there are infinitely many poles $p_1+ip_2$ in $\Im (s)>C$ for every fixed $C>0$. Also we study the gaps between the numbers $p_2$ arranged in the non-decreasing order.