The Error in a Smooth Weighted Prime Number Formula and Zero-free Regions for the Riemann Zeta Function

TL;DR

This paper investigates the error bounds in a smooth weighted prime number theorem, explores implications for the Riemann zeta function's zero-free regions, and applies findings to Goldbach representations.

Contribution

It introduces new error bounds for smooth weighted prime number formulas and extends their application to Goldbach representations.

Findings

Improved error bounds for smooth weighted prime number theorem

Established connections between error bounds and zero-free regions of the Riemann zeta function

Extended results to average weighted Goldbach representations

Abstract

We study the error bound for a smooth weighted prime number theorem, and its implication to the zero-free region for the Riemann zeta function using the method of Pintz. We also give an application to the average number of smooth weighted Goldbach representations and generalize the result to the case of smooth weighted average k-Goldbach representations.