New bounds for the prime counting function \pi(x)

TL;DR

This paper presents new bounds for the prime counting function (x), improving existing estimates and applying these results to demonstrate the existence of prime numbers within smaller intervals.

Contribution

It introduces improved bounds for (x) and applies them to establish prime existence in smaller numerical intervals, advancing prime distribution knowledge.

Findings

New bounds for (x) established

Improved estimates over previous literature

Prime existence in smaller intervals proven

Abstract

In this paper we establish a number of new estimates concerning the prime counting function \pi(x), which improve the estimates proved in the literature. As an application, we deduce a new result concerning the existence of prime numbers in small intervals.