The Error Term In The Primes Counting Function

TL;DR

This paper improves the bounds on the error term of the prime counting function by leveraging recent results on prime densities in short intervals, and discusses implications for the zeros of the zeta function and Mertens conjecture.

Contribution

It introduces an improved estimate of the error term in the prime counting function using recent prime density results, connecting to zeta zeros and Mertens conjecture.

Findings

Error term improved from subexponential to fractional exponential size

Derived equivalent results for zeta zeros

Discussed implications for Mertens conjecture

Abstract

This article considers the error term of the primes counting function. It applies some recent results on the densities of prime numbers in short intervals to derive an improvement of the error term from subexponential size to fractional exponential size. The corresponding equivalent results for the zeros of the zeta function and Mertens conjecture are also discussed.