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A simple expression for the remainder of divisor problem

  • Vivek V Rane
Article

Abstract

By elementary methods, we obtain with ease a highly simple expression for \(\Delta (x)\), the remainder term of Dirichlet’s divisor problem. Incidentally, we give an elegant expression for \(\int _0^1 {\zeta (s_1 ,\alpha )(\zeta (s_2 ,\alpha )+\zeta (s_2 ,1-\alpha )} )\hbox {d}\alpha \) for \(\hbox {Re}\,s_1,\) \(\hbox {Re }s_2\), Re \((s_1 +s_2 )<1\). Here \(\zeta (s,\alpha )\) is the Hurwitz zeta function.

Keywords

Hurwitz/Riemann zeta function Bernoulli polynomial divisor problem 

2010 Mathematics Subject Classification

11M06 

References

  1. 1.
    Ivic A, The Riemann Zeta-Function (Chapter 3 and Chapter 13) (1985) (Wiley-Interscience Publication) p. 517Google Scholar
  2. 2.
    Titchmarsh E C, The Theory of Riemann Zeta Function (Chapter 12) (1951) (Oxford: Clarendon Press)Google Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.MumbaiIndia

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