{"external_id":{"arxiv":["2312.09021"]},"date_updated":"2026-07-14T11:45:18Z","quality_controlled":"1","month":"07","volume":131,"title":"Odd moments and adding fractions","arxiv":1,"article_processing_charge":"No","date_published":"2025-07-01T00:00:00Z","OA_type":"green","publication_identifier":{"issn":["0024-6115"],"eissn":["1460-244X"]},"type":"journal_article","status":"public","oa_version":"Preprint","abstract":[{"text":"We prove near-optimal upper bounds for the oddmoments of the distribution of coprime residues inshort intervals, confirming a conjecture of Montgomeryand Vaughan. As an application, we prove near-optimalupper bounds for the average of the refined singularseries in the Hardy–Littlewood conjectures concerningthe number of prime 𝑘-tuples for 𝑘 odd. The mainnew ingredient is a near-optimal upper bound for thenumber of solutions to ∑1 ⩽𝑖 ⩽𝑘𝑎 𝑖𝑞𝑖∈ ℤ when 𝑘 is odd,with gcd(𝑎𝑖 , 𝑞𝑖 ) = 1 and restrictions on the size of thenumerators and denominators, which is of indepen-dent interest.","lang":"eng"}],"citation":{"ama":"Bloom TF, Kuperberg VZ. Odd moments and adding fractions. <i>Proceedings of the London Mathematical Society</i>. 2025;131(1). doi:<a href=\"https://doi.org/10.1112/plms.70068\">10.1112/plms.70068</a>","ieee":"T. F. Bloom and V. Z. Kuperberg, “Odd moments and adding fractions,” <i>Proceedings of the London Mathematical Society</i>, vol. 131, no. 1. Wiley, 2025.","ista":"Bloom TF, Kuperberg VZ. 2025. Odd moments and adding fractions. Proceedings of the London Mathematical Society. 131(1), e70068.","short":"T.F. Bloom, V.Z. Kuperberg, Proceedings of the London Mathematical Society 131 (2025).","mla":"Bloom, Thomas F., and Vivian Zieve Kuperberg. “Odd Moments and Adding Fractions.” <i>Proceedings of the London Mathematical Society</i>, vol. 131, no. 1, e70068, Wiley, 2025, doi:<a href=\"https://doi.org/10.1112/plms.70068\">10.1112/plms.70068</a>.","chicago":"Bloom, Thomas F., and Vivian Zieve Kuperberg. “Odd Moments and Adding Fractions.” <i>Proceedings of the London Mathematical Society</i>. Wiley, 2025. <a href=\"https://doi.org/10.1112/plms.70068\">https://doi.org/10.1112/plms.70068</a>.","apa":"Bloom, T. F., &#38; Kuperberg, V. Z. (2025). Odd moments and adding fractions. <i>Proceedings of the London Mathematical Society</i>. Wiley. <a href=\"https://doi.org/10.1112/plms.70068\">https://doi.org/10.1112/plms.70068</a>"},"scopus_import":"1","article_number":"e70068","publisher":"Wiley","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2312.09021"}],"author":[{"full_name":"Bloom, Thomas F.","last_name":"Bloom","first_name":"Thomas F."},{"last_name":"Kuperberg","first_name":"Vivian Zieve","id":"c3bac823-112d-11f0-a3f5-c264f852e697","full_name":"Kuperberg, Vivian Zieve"}],"language":[{"iso":"eng"}],"doi":"10.1112/plms.70068","oa":1,"publication_status":"published","extern":"1","mathsc":["11N69","11N05","14G05"],"intvolume":"       131","year":"2025","date_created":"2026-06-29T13:00:46Z","article_type":"original","day":"01","OA_place":"repository","publication":"Proceedings of the London Mathematical Society","issue":"1","_id":"22203","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"}