Mean values of the error term with shifted arguments in the circle problem

Jun Furuya and Yoshio Tanigawa
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 20, 2014, Number 2, Pages 44—51
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Authors and affiliations

Jun Furuya
¹ Department of Integrated Human Sciences (Mathematics), Hamamatsu University School of Medicine
Handayama 1-20-1, Higashi-ku, Hamamatsu city, Shizuoka, 431-3192, Japan
² Department of Integrated Arts and Science, Okinawa National College of Technology
Nago, Okinawa, 905-2192, Japan

Yoshio Tanigawa
Graduate School of Mathematics, Nagoya University
Nagoya, 464-8602, Japan

Abstract

In this paper, we show the relation between the shifted sum of a number-theoretic error term and its continuous mean (integral). We shall obtain a certain expression of the shifted sum as a linear combination of the continuous mean with the Bernoulli polynomials as their coefficients. As an application of our theorem, we give better approximations of the continuous mean by a shifted sum.

Keywords

• The circle problem
• Mean value of error terms
• Shifted sum
• Bernoulli polynomial

AMS Classification

• 11N37

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