Several new relations for the $n^{th}$ prime number

TL;DR

This paper introduces new mathematical relations for the nth prime number using properties of the prime counting function and logarithms, contributing to the theoretical understanding of prime distribution.

Contribution

It presents novel relations for the nth prime number derived from the properties of the prime counting function and logarithmic functions, expanding existing mathematical frameworks.

Findings

Derived new relations for the nth prime number

Utilized properties of the prime counting function and logarithms

Enhanced theoretical understanding of prime distribution

Abstract

We arrive at some new relations for the prime number $P_n$, based on the logarithmic and absolute-value properties of the function $\pi(x)$.