Hostname: page-component-cb9f654ff-rkzlw Total loading time: 0 Render date: 2025-08-21T11:28:10.279Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  09 December 2011

JUN FURUYA  and
YOSHIO TANIGAWA
JUN FURUYA
Affiliation:
Department of Integrated Arts and Science, Okinawa National College of Technology, Nago, Okinawa, 905-2192, Japan e-mail: jfuruya@okinawa-ct.ac.jp
YOSHIO TANIGAWA
Affiliation:
Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan e-mail: tanigawa@math.nagoya-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

11N37

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 1 , January 2012 , pp. 133 - 147
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Apostol, T. M., Mathematical analysis, 2nd edn. (Addison-Wesley, Reading, MA, 1974).Google Scholar
2.Furuya, J. and Tanigawa, Y., Explicit representations of the integral containing the error term in the divisor problem, Acta Math. Hungar. 129 (1–2) (2010), 24–46.CrossRefGoogle Scholar
3.Furuya, J. and Tanigawa, Y., On the integrals containing the error term in the circle problem, preprint.Google Scholar
4.Heath-Brown, D. R. and Tsang, K. M., Sign changes of E(T), Δ(x), and P(x), J. Num. Th. 49 (1994), 73–83.CrossRefGoogle Scholar
5.Lang, S., Undergraduate analysis, 2nd Edn. (Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1997).CrossRefGoogle Scholar
6.Lavrik, A. F., Israilov, M. I. and Edgorov, Ž., On integrals containing the error term in the divisor problem, Acta Arith. 37 (1980), 381–389. (in Russian)Google Scholar
7.Sitaramachandrarao, R., An integral involving the remainder term in the Piltz divisor problem, Acta Arith. 48 (1) (1987), 89–92.CrossRefGoogle Scholar