Estimates of Some Functions Over Primes without R.H.

TL;DR

This paper discusses methods to estimate prime-related functions without relying on the Riemann Hypothesis, focusing on zero-free regions and zeroes of the zeta function to improve classical number theory estimates.

Contribution

It provides new effective estimates of prime number functions without assuming the Riemann Hypothesis, based on computations related to zeta zeroes and zero-free regions.

Findings

Improved bounds for prime counting functions without R.H.

Verification of zeta zeroes on the critical line.

Extension of zero-free regions for the zeta function.

Abstract

Some computations made about the Riemann Hypothesis and in particular, the verification that zeroes of zeta belong on the critical line and the extension of zero-free region are useful to get better effective estimates of number theory classical functions which are closely linked to zeta zeroes like psi(x), theta(x), pi(x) or the k-th prime number.