{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T00:41:09Z","timestamp":1774312869845,"version":"3.50.1"},"reference-count":24,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,12,28]],"date-time":"2021-12-28T00:00:00Z","timestamp":1640649600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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 \u03b6(s)\u2261\u2211n=1\u221en\u2212s=\u220fpprime11\u2212p\u2212s, Gauss, who estimated the rate at which prime numbers increase, and Riemann, who extended \u03b6(s) to the complex plane z and conjectured that all nontrivial zeros are in the R(z)=1\/2 axis. The nonadditive entropy Sq=k\u2211ipilnq(1\/pi)(q\u2208R;S1=SBG\u2261\u2212k\u2211ipilnpi, where BG stands for Boltzmann-Gibbs) on which nonextensive statistical mechanics is based, involves the function lnqz\u2261z1\u2212q\u221211\u2212q(ln1z=lnz). It is already known that this function paves the way for the emergence of a q-generalized algebra, using q-numbers defined as \u27e8x\u27e9q\u2261elnqx, which recover the number x for q=1. The q-prime numbers are then defined as the q-natural numbers \u27e8n\u27e9q\u2261elnqn(n=1,2,3,\u22ef), where n is a prime number p=2,3,5,7,\u22ef We show that, for any value of q, infinitely many q-prime numbers exist; for q\u22641 they diverge for increasing prime number, whereas they converge for q&gt;1; the standard prime numbers are recovered for q=1. For q\u22641, we generalize the \u03b6(s) function as follows: \u03b6q(s)\u2261\u27e8\u03b6(s)\u27e9q (s\u2208R). We show that this function appears to diverge at s=1+0, \u2200q. Also, we alternatively define, for q\u22641, \u03b6q\u2211(s)\u2261\u2211n=1\u221e1\u27e8n\u27e9qs=1+1\u27e82\u27e9qs+\u22ef and \u03b6q\u220f(s)\u2261\u220fpprime11\u2212\u27e8p\u27e9q\u2212s=11\u2212\u27e82\u27e9q\u2212s11\u2212\u27e83\u27e9q\u2212s11\u2212\u27e85\u27e9q\u2212s\u22ef, which, for q&lt;1, generically satisfy \u03b6q\u2211(s)&lt;\u03b6q\u220f(s), in variance with the q=1 case, where of course \u03b61\u2211(s)=\u03b61\u220f(s).<\/jats:p>","DOI":"10.3390\/e24010060","type":"journal-article","created":{"date-parts":[[2021,12,29]],"date-time":"2021-12-29T02:31:27Z","timestamp":1640745087000},"page":"60","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Along the Lines of Nonadditive Entropies: q-Prime Numbers and q-Zeta Functions"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0423-8895","authenticated-orcid":false,"given":"Ernesto P.","family":"Borges","sequence":"first","affiliation":[{"name":"Instituto de F\u00edsica, Universidade Federal da Bahia, Salvador 40170-115, Brazil"},{"name":"National Institute of Science and Technology of Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Takeshi","family":"Kodama","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica, Universidade Federal do Rio de Janeiro, Rio de Janeiro 21941-972, Brazil"},{"name":"Instituto de F\u00edsica, Campus da Praia Vermelha, Universidade Federal Fluminense and National Institute of Science and Technology for Nuclear Physics and Applications, Niter\u00f3i 24210-346, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9387-9194","authenticated-orcid":false,"given":"Constantino","family":"Tsallis","sequence":"additional","affiliation":[{"name":"National Institute of Science and Technology of Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil"},{"name":"Centro Brasileiro de Pesquisas F\u00edsicas, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil"},{"name":"Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA"},{"name":"Complexity Science Hub Vienna, Josefst\u00e4dter Strasse 39, 1080 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. 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