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Omega results for the error term in the square-free divisor problem for square-full integers

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  20 March 2024

Debika Banerjee  and
Makoto T. Minamide
Debika Banerjee*
Affiliation:
Department of Mathematics, Indraprastha Institute of Information Technology Delhi (IIIT-Delhi), Okhla, Phase III, New Delhi 110020, India
Makoto T. Minamide
Affiliation:
Department of Mathematics, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 753-8511, Japan e-mail: minamide@yamaguchi-u.ac.jp
*
e-mail: debika@iiitd.ac.in
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Abstract

In this paper, we investigate the distributive properties of square-free divisors over square-full integers. We first compute the mean value of the number of such divisors and obtain the error term which appears in its asymptotic formula. We then show that if one assumes the Riemann Hypothesis, then the omega estimate of such an error term can be drastically improved. Finally, we compute the omega estimate of the mean square of such an error term.

Keywords

Divisor functionRiemann zeta functionzero-free regionzero density estimatesRiemann Hypothesis

MSC classification

Primary: 11M06: $zeta (s)$ and $L(s, chi)$ 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses
Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 4 , December 2024 , pp. 1011 - 1028
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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