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