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On the sum of $Δ_{k}(n)$ in the Piltz divisor problem for $k=3$ and $k=4$ (2412.11069v1)
Published 15 Dec 2024 in math.NT
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 $x>0$.