Estimating $\pi(x)$ and related functions under partial RH assumptions

TL;DR

This paper links the validity of the Riemann hypothesis up to a certain height with bounds on prime-counting functions, providing new and improved explicit bounds for related functions under partial RH assumptions.

Contribution

It establishes direct interpretations of partial RH validity in terms of prime-counting functions and improves existing bounds for Chebyshev functions.

Findings

Explicit bounds for $ heta(x)$ and $ ext{Li}(x)$ under partial RH assumptions.

Improved Chebyshev bounds for $ ext{psi}(x)$.

Connection between RH validity range and prime-counting function estimates.

Abstract

The aim of this paper is to give a direct interpretation of the validity of the Riemann hypothesis up to a certain height $T$ in terms of the prime-counting function $\pi(x)$. This is done by proving the well-known explicit Schoenfeld bound on the RH to hold as long as $4.92 \sqrt{x/\log(x)} \leq T$. Similar statements are proven for the Riemann prime-counting function and the Chebyshov functions $\psi(x)$ and $\vartheta(x)$. Apart from that, we also improve some of the existing bounds of Chebyshov type for the function $\psi(x)$.