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\ge 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.
Citation: Ruiyang Li, Hai Yang. Asymptotic formulas for Dirichlet convolutions[J]. AIMS Mathematics, 2025, 10(8): 19540-19553. doi: 10.3934/math.2025872
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\ge 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.
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Ruiyang Li, Hai Yang. Asymptotic formulas for Dirichlet convolutions[J]. AIMS Mathematics, 2025, 10(8): 19540-19553. doi: 10.3934/math.2025872
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