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The rich history of prime numbers includes great names such as Euclid, whofirst analytically studied the prime numbers and proved that there is aninfinite 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, whoextended $\zeta(s)$ to the complex plane $z$ and conjectured that allnontrivial 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 $\ln_qz\equiv\frac{z^{1-q}-1}{1-q}\;(\ln_1 z=\ln z)$ that is interconnected to a$q$-generalized algebra, using $q$-numbers defined as $\langle x\rangle_q\equive^{\ln_q x}$ ($\langle x\rangle_1\equiv x$). The $q$-prime numbers are thendefined as the $q$-natural numbers $\langle n\rangle_q\equiv e^{\ln_qn}\;(n=1,2,3,\dots)$, where $n$ is a prime number $p=2,3,5,7,\dots$ We showthat, 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 $q\le 1$, wegeneralize the $\zeta(s)$ function as follows:$\zeta_q(s)\equiv\langle\zeta(s)\rangle_q$ ($s\in\mathbb{R}$). We show thatthis function appears to diverge at $s=1+0$, $\forall q$. Also, wealternatively define, for $q\le 1$,$\zeta_q^{\Sigma}(s)\equiv\sum_{n=1}^\infty\frac{1}{\langlen\rangle_q^s}=1+\frac{1}{\langle 2\rangle_q^s}+\dots $ and$\zeta_q^{\Pi}(s)\equiv\prod_{p\,prime}\frac{1}{1-\langlep\rangle_q^{-s}}=\frac{1}{1-\langle 2\rangle_q^{-s}}\frac{1}{1-\langle3\rangle_q^{-s}}\frac{1}{1-\langle 5\rangle_q^{-s}}\cdots$, which, for $q<1$,generically satisfy $\zeta_q^\Sigma(s)<\zeta_q^\Pi(s)$, in variance with the$q=1$ case, where of course $\zeta_1^\Sigma(s)=\zeta_1^\Pi(s)$.

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