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arXiv · 2412.11069

On the sum of $\Delta_{k}(n)$ in the Piltz divisor problem for $k=3$ and $k=4$

T. Makoto Minamide·Yoshio Tanigawa·Nigel Watt
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Abstract

Let $\Delta_{k}(x)$ be the error term in the classical asymptotic formula for the sum $\sum_{n\leq x}d_{k}(n)$, where $d_{k}(n)$ is the number of ways $n$ can be written as a product of $k$ factors. We study the analytic properties of the Dirichlet series $\sum_{n=1}^{\infty}\Delta_{k}(n)n^{-s}$ and use Perron's formula to estimate the sums $\sum_{n\leq x}\Delta_{3}(n)$ and $\sum_{n\leq x}\Delta_{4}(n)$ for large $x>0$.

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T. Makoto Minamide, Yoshio Tanigawa, Nigel Watt. 2024-12-15. On the sum of $\Delta_{k}(n)$ in the Piltz divisor problem for $k=3$ and $k=4$. https://doi.org/10.4064/aa230223-28-3

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