{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:10:30Z","timestamp":1776791430152,"version":"3.51.2"},"reference-count":25,"publisher":"American Mathematical Society (AMS)","issue":"270","license":[{"start":{"date-parts":[[2010,9,25]],"date-time":"2010-09-25T00:00:00Z","timestamp":1285372800000},"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 Taking\n \n \n \n \n r<\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n r>0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , let\n \n \n \n \n \n \n \u03c0\n \n <\/mml:mi>\n \n 2<\/mml:mn>\n r<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\pi _{2r}(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n denote the number of prime pairs\n \n \n \n \n (<\/mml:mo>\n p<\/mml:mi>\n ,<\/mml:mo>\n \n p<\/mml:mi>\n +<\/mml:mo>\n 2<\/mml:mn>\n r<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n (p,\\,p+2r)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with\n \n \n \n \n p<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n x<\/mml:mi>\n <\/mml:mrow>\n p\\le x<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The prime-pair conjecture of Hardy and Littlewood (1923) asserts that\n \n \n \n \n \n \n \u03c0\n \n <\/mml:mi>\n \n 2<\/mml:mn>\n r<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \n \u223c\n \n <\/mml:mo>\n 2<\/mml:mn>\n \n C<\/mml:mi>\n \n 2<\/mml:mn>\n r<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n \n \n l<\/mml:mi>\n i<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msub>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\pi _{2r}(x)\\sim 2C_{2r}\\,\\mathrm {li}_2(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with an explicit constant\n \n \n \n \n \n C<\/mml:mi>\n \n 2<\/mml:mn>\n r<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n C_{2r}>0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . There seems to be no good conjecture for the remainders\n \n \n \n \n \n \n \u03c9\n \n <\/mml:mi>\n \n 2<\/mml:mn>\n r<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n \n \n \u03c0\n \n <\/mml:mi>\n \n 2<\/mml:mn>\n r<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n 2<\/mml:mn>\n \n C<\/mml:mi>\n \n 2<\/mml:mn>\n r<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n \n \n l<\/mml:mi>\n i<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msub>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\omega _{2r}(x)=\\pi _{2r}(x)- 2C_{2r}\\,\\mathrm {li}_2(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n that corresponds to Riemann\u2019s formula for\n \n \n \n \n \n \u03c0\n \n <\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n \n \u2212\n \n <\/mml:mo>\n \n l<\/mml:mi>\n i<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\pi (x)-\\mathrm {li}(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . However, there is a heuristic approximate formula for averages of the remainders\n \n \n \n \n \n \n \u03c9\n \n <\/mml:mi>\n \n 2<\/mml:mn>\n r<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\omega _{2r}(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n which is supported by numerical results.\n <\/p>","DOI":"10.1090\/s0025-5718-09-02312-6","type":"journal-article","created":{"date-parts":[[2010,2,4]],"date-time":"2010-02-04T11:19:48Z","timestamp":1265282388000},"page":"1209-1229","source":"Crossref","is-referenced-by-count":2,"title":["Average prime-pair counting formula"],"prefix":"10.1090","volume":"79","author":[{"given":"Jaap","family":"Korevaar","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Herman","family":"te Riele","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2009,9,25]]},"reference":[{"key":"1","unstructured":"R.F. Arenstorf, There are infinitely many prime twins. Available on the internet, at http:\/\/arxiv.org\/abs\/math\/0405509v1. Article posted May 26, 2004; withdrawn June 9, 2004."},{"key":"2","doi-asserted-by":"publisher","first-page":"363","DOI":"10.2307\/2004056","article-title":"A heuristic asymptotic formula concerning the distribution of prime numbers","volume":"16","author":"Bateman, Paul T.","year":"1962","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"3","first-page":"119","article-title":"Primes represented by irreducible polynomials in one variable","author":"Bateman, Paul T.","year":"1965"},{"issue":"231","key":"4","doi-asserted-by":"publisher","first-page":"1285","DOI":"10.1090\/S0025-5718-99-01104-7","article-title":"A new bound for the smallest \ud835\udc65 with \ud835\udf0b(\ud835\udc65)>\ud835\udc59\ud835\udc56(\ud835\udc65)","volume":"69","author":"Bays, Carter","year":"2000","journal-title":"Math. 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Private communication including spreadsheet attachment tot10-tot.csv, February 2007."},{"key":"8","series-title":"Graduate Texts in Mathematics","isbn-type":"print","volume-title":"Multiplicative number theory","volume":"74","author":"Davenport, Harold","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/0387950974","edition":"3"},{"key":"9","isbn-type":"print","volume-title":"Riemann's zeta function","author":"Edwards, H. M.","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0486417409"},{"issue":"1","key":"10","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1215\/ijm\/1255986635","article-title":"Some singular series averages and the distribution of Goldbach numbers in short intervals","volume":"39","author":"Friedlander, J. B.","year":"1995","journal-title":"Illinois J. 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E.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0521397898"},{"key":"14","series-title":"A Wiley-Interscience Publication","isbn-type":"print","volume-title":"The Riemann zeta-function","author":"Ivi\u0107, Aleksandar","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/047180634X"},{"issue":"1","key":"15","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1016\/S0019-3577(05)80013-8","article-title":"Distributional Wiener-Ikehara theorem and twin primes","volume":"16","author":"Korevaar, Jacob","year":"2005","journal-title":"Indag. Math. (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0019-3577","issn-type":"print"},{"key":"16","doi-asserted-by":"crossref","unstructured":"J. Korevaar, Prime pairs and the zeta function. J. Approx. 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L.","year":"1973"},{"key":"21","unstructured":"T. R. Nicely, Enumeration of the twin-prime pairs to 10\u00b9\u2076. See the internet, http:\/\/www.trnicely.net, September 2008."},{"issue":"177","key":"22","doi-asserted-by":"publisher","first-page":"323","DOI":"10.2307\/2007893","article-title":"On the sign of the difference \ud835\udf0b(\ud835\udc65)-\ud835\udc59\ud835\udc56(\ud835\udc65)","volume":"48","author":"te Riele, Herman J. J.","year":"1987","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"23","isbn-type":"print","volume-title":"The theory of the Riemann zeta-function","author":"Titchmarsh, E. C.","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/0198533691","edition":"2"},{"key":"24","series-title":"Cambridge Mathematical Library","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511608759","volume-title":"A course of modern analysis","author":"Whittaker, E. 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Undated attachment to private communication, July 2008."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2010-79-270\/S0025-5718-09-02312-6\/S0025-5718-09-02312-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2010-79-270\/S0025-5718-09-02312-6\/S0025-5718-09-02312-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:24:46Z","timestamp":1776788686000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2010-79-270\/S0025-5718-09-02312-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,9,25]]},"references-count":25,"journal-issue":{"issue":"270","published-print":{"date-parts":[[2010,4]]}},"alternative-id":["S0025-5718-09-02312-6"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-09-02312-6","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2009,9,25]]}}}