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

  • JUN FURUYA (a1) and YOSHIO TANIGAWA (a2)

Abstract

In our previous paper [2], we derived an explicit representation of the integral ∫1t−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.

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References

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1.Apostol, T. M., Mathematical analysis, 2nd edn. (Addison-Wesley, Reading, MA, 1974).
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), 2446.
3.Furuya, J. and Tanigawa, Y., On the integrals containing the error term in the circle problem, preprint.
4.Heath-Brown, D. R. and Tsang, K. M., Sign changes of E(T), Δ(x), and P(x), J. Num. Th. 49 (1994), 7383.
5.Lang, S., Undergraduate analysis, 2nd Edn. (Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1997).
6.Lavrik, A. F., Israilov, M. I. and Edgorov, Ž., On integrals containing the error term in the divisor problem, Acta Arith. 37 (1980), 381389. (in Russian)
7.Sitaramachandrarao, R., An integral involving the remainder term in the Piltz divisor problem, Acta Arith. 48 (1) (1987), 8992.
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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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