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On a variant of the prime number theorem (2303.12347v4)
Published 22 Mar 2023 in math.NT
Abstract: In this paper, we can show that \begin{align*} S_{\Lambda}(x)=\sum_{1\leq n\leq x}\Lambda \left(\left[\frac{x}{n}\right]\right)= \sum_{n=1}{\infty} \frac{\Lambda(n)}{n(n+1)}x +O\left(x{7/15+1/195+\varepsilon}\right), \end{align*} where $\Lambda(n)$ is the von Mangdolt function. Moreover, we can also give a similar result related to the divisor function, which improves previous results.