Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the order of the error function fo the $ k$-free integers


Authors: D. Suryanarayana and R. Sitaramachandra Rao
Journal: Proc. Amer. Math. Soc. 28 (1971), 53-58
MSC: Primary 10.42
DOI: https://doi.org/10.1090/S0002-9939-1971-0271044-X
MathSciNet review: 0271044
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {\Delta _k}(x)$ and $ {\Delta _k}' (x)$ be the error functions in the asymptotic formulae for the number and the sum of $ k$-free integers $ \leqq x$. On the assumption of the Riemann hypothesis, we prove the following results by elementary methods:

$\displaystyle {\Delta _k}' (x) - x{\Delta _k}(x) = O({x^{1 + 3/(4k + 1) + \varepsilon }})$

and

$\displaystyle \frac{1}{x}\int_1^x {{\Delta _k}(t)dt = O({x^{3/(4k + 1)\varepsilon }}),} $

where $ \varepsilon > 0$.

References [Enhancements On Off] (What's this?)

  • [1] A. Axer, Über einige Grenzertsatze, S.-B. Akad. Wiss. Wien Ha 120 (1911), 1253-1298.
  • [2] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford, at the Clarendon Press, 1951. MR 0046485
  • [3] A. M. Vaidya, On the order of the error function of the square-free numbers, Proc. Nat. Inst. Sci. India Part A 32 (1966), 196–201. MR 0249378
  • [4] Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Mathematische Forschungsberichte, XV, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963 (German). MR 0220685

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 10.42

Retrieve articles in all journals with MSC: 10.42


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0271044-X
Keywords: $ k$-free intergers, Riemann hypothesis
Article copyright: © Copyright 1971 American Mathematical Society