Along the Lines of Nonadditive Entropies: q-Prime Numbers and q-Zeta Functions

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 [Formula: see text] , Gauss, who estimated the rate at which prime numbers increase, and Riemann, who extended [Formula: see text] to the complex plane z and conjectured that all nontrivial zeros are in the [Formula: see text] axis. The nonadditive entropy [Formula: see text] , where BG stands for Boltzmann-Gibbs) on which nonextensive statistical mechanics is based, involves the function [Formula: see text]. It is already known that this function paves the way for the emergence of a q-generalized algebra, using q-numbers defined as [Formula: see text] , which recover the number x for [Formula: see text]. The q-prime numbers are then defined as the q-natural numbers [Formula: see text] , where n is a prime number [Formula: see text] We show that, for any value of q, infinitely many q-prime numbers exist; for [Formula: see text] they diverge for increasing prime number, whereas they converge for [Formula: see text]; the standard prime numbers are recovered for [Formula: see text]. For [Formula: see text] , we generalize the [Formula: see text] function as follows: [Formula: see text] ([Formula: see text]). We show that this function appears to diverge at [Formula: see text] , [Formula: see text]. Also, we alternatively define, for [Formula: see text] , [Formula: see text] and [Formula: see text] , which, for [Formula: see text] , generically satisfy [Formula: see text] , in variance with the [Formula: see text] case, where of course [Formula: see text].