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arXiv · 2303.12347

On a variant of the prime number theorem

Wei Zhang
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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.

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Wei Zhang. 2023-03-22. On a variant of the prime number theorem. https://doi.org/10.1016/j.jnt.2023.10.011

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