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June 2020 On certain integrals involving the Dirichlet divisor problem
Aleksandar Ivić, Wenguang Zhai
Funct. Approx. Comment. Math. 62(2): 247-267 (June 2020). DOI: 10.7169/facm/1819

Abstract

We prove that $$ \int_1^X\Delta(x)\Delta_3(x)dx \ll X^{13/9}\log^{10/3}X, \qquad \int_1^X\Delta(x)\Delta_4(x)\d x \ll_\varepsilon X^{25/16+\varepsilon}, $$ where $\Delta_k(x)$ is the error term in the asymptotic formula for the summatory function of $d_k(n)$, generated by $\zeta^k(s)$ ($\Delta_2(x) \equiv \Delta(x)$). These bounds are sharper than the ones which follow by the Cauchy-Schwarz inequality and mean square results for $\Delta_k(x)$. We also obtain the analogues of the above bounds when $\Delta(x)$ is replaced by $E(x)$, the error term in the mean square formula for $|\zeta(1/2+it)|$.

Citation

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Aleksandar Ivić. Wenguang Zhai. "On certain integrals involving the Dirichlet divisor problem." Funct. Approx. Comment. Math. 62 (2) 247 - 267, June 2020. https://doi.org/10.7169/facm/1819

Information

Published: June 2020
First available in Project Euclid: 26 October 2019

zbMATH: 07225512
MathSciNet: MR4113988
Digital Object Identifier: 10.7169/facm/1819

Subjects:
Primary: 11M06, 11N36

Rights: Copyright © 2020 Adam Mickiewicz University

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Vol.62 • No. 2 • June 2020
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