Glasgow Mathematical Journal

Table of Contents - January 2012 - Volume 54, Issue 01  

Research Article

EXPLICIT REPRESENTATIONS OF THE INTEGRAL CONTAINING THE ERROR TERM IN THE DIVISOR PROBLEM II

JUN FURUYAa1 and YOSHIO TANIGAWAa2*

a1 Department of Integrated Arts and Science, Okinawa National College of Technology, Nago, Okinawa, 905-2192, Japan e-mail: jfuruya@okinawa-ct.ac.jp

a2 Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan e-mail: tanigawa@math.nagoya-u.ac.jp

Abstract

In our previous paper [2], we derived an explicit representation of the integral ∫1∞t−θΔ(t)logjtdt by differentiation under the integral sign. Here, j is a fixed natural number, θ is a complex number with 1 < θ ≤ 5/4 and Δ(x) denotes the error term in the Dirichlet divisor problem. In this paper, we shall reconsider the same formula by an alternative approach, which appeals to only the elementary integral formulas concerning the Riemann zeta- and periodic Bernoulli functions. We also study the corresponding formula in the case of the circle problem of Gauss.

(Received June 28 2010)

(Revised December 10 2010)

(Accepted August 15 2011)

2000 Mathematics Subject Classification

  • 11N37

Footnotes

* The second author is supported by Grant-in-Aid for Scientific Research No. 21540012.