Along the lines of nonadditive entropies: $q$-prime numbers and $q$-zeta functions

TL;DR

This paper explores the generalization of prime numbers and the Riemann zeta function using nonadditive entropy and q-generalized algebra, revealing new properties and divergences for different q-values.

Contribution

It introduces the concept of q-prime numbers and q-zeta functions, extending classical number theory with nonadditive entropy frameworks and analyzing their properties.

Findings

q-prime numbers diverge for q≤1 and converge for q>1

q-zeta functions diverge at s=1 for all q

q-generalized zeta functions satisfy inequalities different from classical case

Abstract

The rich history of prime numbers includes great names such as Euclid, who first analytically studied the prime numbers and proved that there is an infinite number of them, Euler, who introduced the function $\zeta(s)\equiv\sum_{n=1}^\infty n^{-s}=\prod_{p\,prime} \frac{1}{1- p^{-s}}$, Gauss, who estimated the rate at which prime numbers increase, and Riemann, who extended $\zeta(s)$ to the complex plane $z$ and conjectured that all nontrivial zeros are in the $\mathbb{R}(z)=1/2$ axis. The nonadditive entropy $S_q=k\sum_ip_i\ln_q(1/p_i)\;(q\in\mathbb{R})$ involves the function $\ln_q z\equiv\frac{z^{1-q}-1}{1-q}\;(\ln_1 z=\ln z)$ that is interconnected to a $q$-generalized algebra, using $q$-numbers defined as $\langle x\rangle_q\equiv e^{\ln_q x}$ ($\langle x\rangle_1\equiv x$). The $q$-prime numbers are then defined as the $q$-natural numbers $\langle n\rangle_q\equiv e^{\ln_q n}\;(n=1,2,3,\dots)$, where $n$ is a prime number $p=2,3,5,7,\dots$ We show that, for any value of $q$, infinitely many $q$-prime numbers exist; for $q\le1$ they diverge for increasing prime number, whereas they converge for $q>1$; the standard prime numbers are recovered for $q=1$. For $q\le 1$, we generalize the $\zeta(s)$ function as follows: $\zeta_q(s)\equiv\langle\zeta(s)\rangle_q$ ($s\in\mathbb{R}$). We show that this function appears to diverge at $s=1+0$, $\forall q$. Also, we alternatively define, for $q\le 1$, $\zeta_q^{\Sigma}(s)\equiv\sum_{n=1}^\infty\frac{1}{\langle n\rangle_q^s}=1+\frac{1}{\langle 2\rangle_q^s}+\dots $ and $\zeta_q^{\Pi}(s)\equiv\prod_{p\,prime}\frac{1}{1-\langle p\rangle_q^{-s}}=\frac{1}{1-\langle 2\rangle_q^{-s}}\frac{1}{1-\langle 3\rangle_q^{-s}}\frac{1}{1-\langle 5\rangle_q^{-s}}\cdots$, which, for $q<1$, generically satisfy $\zeta_q^\Sigma(s)<\zeta_q^\Pi(s)$, in variance with the $q=1$ case, where of course $\zeta_1^\Sigma(s)=\zeta_1^\Pi(s)$.