On the error term in the explicit formula of Riemann--von Mangoldt

TL;DR

This paper refines the explicit error term in the Riemann--von Mangoldt formula, making previous results explicit and applying them to prime distribution problems and classical inequalities.

Contribution

It provides an explicit $O(x ext{log} x/T)$ error term for the Riemann--von Mangoldt formula, building on Wolke and Ramaré's work.

Findings

Explicit error term $O(x ext{log} x/T)$ derived

Applications to primes between powers demonstrated

Improved bounds in prime number theorem and Ramanujan's inequality

Abstract

We provide an explicit $O(x\log x/T)$ error term for the Riemann--von Mangoldt formula by making results of Wolke (1983) and Ramar\'e (2016) explicit. We also include applications to primes between consecutive powers, the error term in the prime number theorem and an inequality of Ramanujan.