{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:53:38Z","timestamp":1776840818486,"version":"3.51.2"},"reference-count":55,"publisher":"American Mathematical Society (AMS)","issue":"357","license":[{"start":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T00:00:00Z","timestamp":1769558400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Many Dirichlet series of number theoretic interest can be written as a product of generating series\n \n \n \n \n \n \n \u03b6\n \n <\/mml:mi>\n \n \n d<\/mml:mi>\n ,<\/mml:mo>\n a<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n \n \n \u220f\n \n <\/mml:mo>\n \n p<\/mml:mi>\n \n \u2261\n \n <\/mml:mo>\n a<\/mml:mi>\n \u00a0<\/mml:mtext>\n (<\/mml:mo>\n \n m<\/mml:mi>\n o<\/mml:mi>\n d<\/mml:mi>\n <\/mml:mrow>\n \u00a0<\/mml:mtext>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:munder>\n (<\/mml:mo>\n 1<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n \n p<\/mml:mi>\n \n \n \u2212\n \n <\/mml:mo>\n s<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n \n )<\/mml:mo>\n \n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n \\zeta _{\\,d,a}(s)=\\prod \\limits _{p\\equiv a\\ (\\mathrm {mod}\\ d)}(1-p^{-s})^{-1}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , with\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ranging over all the primes in the primitive residue class modulo\n \n \n \n \n a<\/mml:mi>\n \u00a0<\/mml:mtext>\n (<\/mml:mo>\n \n m<\/mml:mi>\n o<\/mml:mi>\n d<\/mml:mi>\n <\/mml:mrow>\n \u00a0<\/mml:mtext>\n d<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n a\\ (\\mathrm {mod}\\ d)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and a function\n \n \n \n \n H<\/mml:mi>\n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n H(s)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n well-behaved around\n \n \n \n \n s<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n s=1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . In such a case the corresponding Euler constant can be expressed in terms of the Euler constants\n \n \n \n \n \n \u03b3\n \n <\/mml:mi>\n (<\/mml:mo>\n d<\/mml:mi>\n ,<\/mml:mo>\n a<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\gamma (d,a)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of the series\n \n \n \n \n \n \n \u03b6\n \n <\/mml:mi>\n \n \n d<\/mml:mi>\n ,<\/mml:mo>\n a<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\zeta _{\\,d,a}(s)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n involved and the (numerically more harmless) term\n \n \n \n \n \n H<\/mml:mi>\n \u2032<\/mml:mo>\n <\/mml:msup>\n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n H<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n H\u2019(1)\/H(1)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Here we systematically study\n \n \n \n \n \n \u03b3\n \n <\/mml:mi>\n (<\/mml:mo>\n d<\/mml:mi>\n ,<\/mml:mo>\n a<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\gamma (d,a)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , their numerical evaluation and discuss some examples.\n <\/p>","DOI":"10.1090\/mcom\/4057","type":"journal-article","created":{"date-parts":[[2024,12,11]],"date-time":"2024-12-11T13:18:27Z","timestamp":1733923107000},"page":"363-387","source":"Crossref","is-referenced-by-count":2,"title":["Euler constants from primes in arithmetic progression"],"prefix":"10.1090","volume":"95","author":[{"given":"Alessandro","family":"Languasco","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pieter","family":"Moree","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2025,1,28]]},"reference":[{"key":"1","series-title":"Undergraduate Texts in Mathematics","volume-title":"Introduction to analytic number theory","author":"Apostol, Tom M.","year":"1976"},{"key":"2","series-title":"Mathematical Surveys and Monographs","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/254","volume-title":"Numerical algorithms for number theory---using Pari\/GP","volume":"254","author":"Belabas, Karim","year":"[2021] \\copyright2021","ISBN":"https:\/\/id.crossref.org\/isbn\/9781470463519"},{"key":"3","doi-asserted-by":"crossref","unstructured":"B. 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