  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 $\zeta_1^\Sigma(s)=\zeta_1^\Pi(s)$.
