The $n$th+1 Prime Number Limit Formulas

TL;DR

This paper introduces a novel derivation of Golomb's limit formula for the next prime number using Euler's product representation of the zeta function, along with new variations involving logarithmic and half-prime formulas.

Contribution

It presents a new derivation method for prime limit formulas and explores variations, expanding the theoretical understanding of prime number generation.

Findings

Derived Golomb's limit formula from Euler's zeta function

Introduced new variations of prime limit formulas

Enhanced theoretical framework for prime number approximation

Abstract

A new derivation of Golomb's limit formula for generating the $n$th$+1$ prime number is presented. The limit formula is derived by extracting $p_{n+1}$ from Euler's prime product representation of the Riemann zeta function $\zeta(s)$ in the limit as $s$ approaches infinity. Also, new variations of these limit formulas are explored, such as the logarithm and a half-prime formulas for the $p_{n+1}$.