{"scopus_import":"1","day":"01","OA_type":"green","publisher":"Oxford University Press","doi":"10.1093/qmath/haad030","arxiv":1,"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2210.09775","open_access":"1"}],"year":"2023","intvolume":"        74","OA_place":"repository","external_id":{"arxiv":["2210.09775"]},"language":[{"iso":"eng"}],"oa":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2026-06-29T12:56:30Z","oa_version":"Preprint","issue":"4","date_updated":"2026-07-14T10:55:52Z","month":"12","abstract":[{"lang":"eng","text":"In 1976, Gallagher showed that the Hardy–Littlewood conjectures on prime k-tuples imply that the\r\ndistribution of primes in log-size intervals is Poissonian. He did so by computing average values\r\nof the singular series constants over different sets of a fixed size k contained in an interval [1,h]\r\nas h → ∞, and then using this average to compute moments of the distribution of primes. In this\r\npaper, we study averages where k is relatively large with respect to h. We then apply these averages\r\nto the tail of the distribution. For example, we show, assuming appropriate Hardy–Littlewood\r\nconjectures and in certain ranges of the parameters, the number of intervals [n,n + λlogx] with\r\nn ≤ x containing at least k primes is ≪ x exp(−k/(λe))."}],"volume":74,"article_type":"original","_id":"22192","status":"public","citation":{"mla":"Kuperberg, Vivian Zieve. “Sums of Singular Series with Large Sets and the Tail of the Distribution of Primes.” <i>The Quarterly Journal of Mathematics</i>, vol. 74, no. 4, Oxford University Press, 2023, pp. 1457–79, doi:<a href=\"https://doi.org/10.1093/qmath/haad030\">10.1093/qmath/haad030</a>.","ieee":"V. Z. Kuperberg, “Sums of singular series with large sets and the tail of the distribution of primes,” <i>The Quarterly Journal of Mathematics</i>, vol. 74, no. 4. Oxford University Press, pp. 1457–1479, 2023.","chicago":"Kuperberg, Vivian Zieve. “Sums of Singular Series with Large Sets and the Tail of the Distribution of Primes.” <i>The Quarterly Journal of Mathematics</i>. Oxford University Press, 2023. <a href=\"https://doi.org/10.1093/qmath/haad030\">https://doi.org/10.1093/qmath/haad030</a>.","ista":"Kuperberg VZ. 2023. Sums of singular series with large sets and the tail of the distribution of primes. The Quarterly Journal of Mathematics. 74(4), 1457–1479.","short":"V.Z. Kuperberg, The Quarterly Journal of Mathematics 74 (2023) 1457–1479.","apa":"Kuperberg, V. Z. (2023). Sums of singular series with large sets and the tail of the distribution of primes. <i>The Quarterly Journal of Mathematics</i>. Oxford University Press. <a href=\"https://doi.org/10.1093/qmath/haad030\">https://doi.org/10.1093/qmath/haad030</a>","ama":"Kuperberg VZ. Sums of singular series with large sets and the tail of the distribution of primes. <i>The Quarterly Journal of Mathematics</i>. 2023;74(4):1457-1479. doi:<a href=\"https://doi.org/10.1093/qmath/haad030\">10.1093/qmath/haad030</a>"},"article_processing_charge":"No","publication_identifier":{"eissn":["1464-3847"],"issn":["0033-5606"]},"type":"journal_article","extern":"1","publication":"The Quarterly Journal of Mathematics","quality_controlled":"1","page":"1457-1479","publication_status":"published","author":[{"id":"c3bac823-112d-11f0-a3f5-c264f852e697","first_name":"Vivian Zieve","last_name":"Kuperberg","full_name":"Kuperberg, Vivian Zieve"}],"date_published":"2023-12-01T00:00:00Z","title":"Sums of singular series with large sets and the tail of the distribution of primes"}