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Lobachevskii Journal of Mathematics

, Volume 39, Issue 1, pp 77–83 | Cite as

On an Asymptotic Property of Divisor τ-Function

  • T. Hakobyan
  • S. Vostokov
Article
  • 14 Downloads

Abstract

In this paper for μ > 0 we study an asymptotic behavior of the sequence defined as T n (μ) = (τ(n))−1\({\max _{1 \leqslant t \leqslant \left[ {{n^{1/\mu }}} \right]}}\) {τ(n + t)}, where τ(n) denotes the number of natural divisors of given positive integer n. The motivation of this observation is to explore whether τ-function oscillates rapidly.

Keywords and phrases

τ-function divisor Stirling’s formula Prime Number Theorem Derichlet’s divisor problem 

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References

  1. 1.
    K. Chandrasekharan, Introduction to Analytic Number Theory (Springer, Berlin, Heidelberg, 1968).CrossRefMATHGoogle Scholar
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    E. RamMurty, Problems in Analytical Number Theory (Springer, Berlin, Heidelberg, 1998).Google Scholar
  3. 3.
    M. A. Korolev, “On Karatsuba’s problem concerning the divisor function τ(n),” Monatsh. Math. 168, 403–441 (2012).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    A. A. Karatsuba, “Uniform approximation of the remainder term in the Dirichlet divisor problem,” Math. USSR Izv. 6, 467–475 (1972).CrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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