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
Full paper (PDF, 196 Kb)

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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

References

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