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Research Article | Open Access

Asymptotic formulas for Dirichlet convolutions

Ruiyang Li, Hai Yang( )
School of Science, Xi'an Polytechnic University, Xi'an 710048, China
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Abstract

In this paper, we use convoluting prime-number-theorem-related functions (Mobius, von Mangoldt, and Liouville) by the Piltz divisor function to research an asymptotic formula for the convolution sum ( x ≥ 1 ). Our main result is Theorem 1, which separates the error terms for the PNT-related functions and the Piltz divisor function. Under the assumption of the conjectural estimate for the latter, we obtain the expected error term as in PNT.

Keywords

prime-number-theoremPiltz divisor functionasymptotic formulaDirichlet convolutionRiemann zeta-function
CLC number: 11N37, 11N64

References

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AIMS Mathematics
Volume 10 Issue 8,
August 2025
Pages 19540-19553
DOI: 10.3934/math.2025872

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Cite this article:
Li R, Yang H. Asymptotic formulas for Dirichlet convolutions. AIMS Mathematics, 2025, 10(8): 19540-19553. https://doi.org/10.3934/math.2025872

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Received: 21 September 2024
Revised: 25 June 2025
Accepted: 15 July 2025
Published: 15 August 2025
©2025 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)