Functiones et Approximatio Commentarii Mathematici

On certain integrals involving the Dirichlet divisor problem

Aleksandar Ivić and Wenguang Zhai

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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 $|\zeta(1/2+it)|$.

Article information

Funct. Approx. Comment. Math., Volume 62, Number 2 (2020), 247-267.

First available in Project Euclid: 26 October 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11N36: Applications of sieve methods 11M06: $\zeta (s)$ and $L(s, \chi)$

(general) Dirichlet divisor problem moments Riemann zeta-function


Ivić, Aleksandar; Zhai, Wenguang. On certain integrals involving the Dirichlet divisor problem. Funct. Approx. Comment. Math. 62 (2020), no. 2, 247--267. doi:10.7169/facm/1819.

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