New estimates for some integrals of functions defined over primes

TL;DR

This paper provides new estimates for integrals involving prime number functions like π(x) and θ(x), some of which depend on the Riemann hypothesis, advancing understanding of prime distribution.

Contribution

It introduces novel bounds for integrals of prime-related functions, improving previous estimates and exploring implications of the Riemann hypothesis.

Findings

New bounds for integrals involving π(x) and θ(x)

Results depend on the Riemann hypothesis correctness

Enhanced understanding of prime distribution estimates

Abstract

In this paper we give new estimates for integrals involving some arithmetic functions defined over prime numbers. The main focus here is on the prime counting function $\pi(x)$ and the Chebyshev $\vartheta$-function. Some of these estimates depend on the correctness of the Riemann hypothesis on the nontrivial zeros of the Riemann zeta function $\zeta(s)$.