Averages of exponential twists of the von Mangoldt function

TL;DR

This paper improves bounds on exponential sums involving the von Mangoldt function, which are related to the quasi-Riemann hypothesis, extending previous work by Vinogradov and Murty-Srinivas.

Contribution

It provides enhanced results for exponential sums of the von Mangoldt function with fractional powers, advancing understanding in analytic number theory.

Findings

Improved bounds for exponential sums with von Mangoldt function

Extensions of previous results by Vinogradov and Murty-Srinivas

Relevance to the quasi-Riemann hypothesis

Abstract

In this paper, we obtain some improved results for the exponential sum $\sum_{x<n\leq 2x}\Lambda(n)e(\alpha k n^{\theta})$ with $\theta\in(0,5/12),$ where $\Lambda(n)$ is the von Mangoldt function. Such exponential sums have relations with the so-called quasi-Riemann hypothesis and were considered by Vinogradov \cite{Va} and Murty-Srinivas \cite{Mu}.