Analytic properties of Dirichlet series obtained from the error term in the Dirichlet divisor problem

Jun Furuya and Yoshio Tanigawa

Vol. 245 (2010), No. 2, 239–254

DOI: 10.2140/pjm.2010.245.239

Abstract

We discuss some analytic properties of Dirichlet series Y (s) = ∑ _n=1^∞d(n)Δ(n)n^−s for Re s > 5∕4, where d(n) is the divisor function and Δ(x) is the error term in the Dirichlet divisor problem. In particular, we study an analytic continuation and an order of Y (s). As applications, we study an analytic continuation and orders of several kinds of Dirichlet series related to Δ(x).

Keywords

analytic continuation, Dirichlet series, divisor problem, double zeta function, orders of Dirichlet series

Mathematical Subject Classification 2000

Primary: 11L07, 11M41

Milestones

Received: 1 April 2009

Revised: 14 September 2009

Accepted: 4 January 2010

Published: 1 April 2010