[{"OA_type":"green","OA_place":"repository","language":[{"iso":"eng"}],"publication":"Algebra & Number Theory","scopus_import":"1","issue":"4","publication_identifier":{"eissn":["1944-7833"],"issn":["1937-0652"]},"page":"617-666","citation":{"chicago":"Kuperberg, Vivian Zieve. “Odd Moments in the Distribution of Primes.” <i>Algebra &#38; Number Theory</i>. Mathematical Sciences Publishers, 2025. <a href=\"https://doi.org/10.2140/ant.2025.19.617\">https://doi.org/10.2140/ant.2025.19.617</a>.","apa":"Kuperberg, V. Z. (2025). Odd moments in the distribution of primes. <i>Algebra &#38; Number Theory</i>. Mathematical Sciences Publishers. <a href=\"https://doi.org/10.2140/ant.2025.19.617\">https://doi.org/10.2140/ant.2025.19.617</a>","mla":"Kuperberg, Vivian Zieve. “Odd Moments in the Distribution of Primes.” <i>Algebra &#38; Number Theory</i>, vol. 19, no. 4, Mathematical Sciences Publishers, 2025, pp. 617–66, doi:<a href=\"https://doi.org/10.2140/ant.2025.19.617\">10.2140/ant.2025.19.617</a>.","ama":"Kuperberg VZ. Odd moments in the distribution of primes. <i>Algebra &#38; Number Theory</i>. 2025;19(4):617-666. doi:<a href=\"https://doi.org/10.2140/ant.2025.19.617\">10.2140/ant.2025.19.617</a>","ista":"Kuperberg VZ. 2025. Odd moments in the distribution of primes. Algebra &#38; Number Theory. 19(4), 617–666.","short":"V.Z. Kuperberg, Algebra &#38; Number Theory 19 (2025) 617–666.","ieee":"V. Z. Kuperberg, “Odd moments in the distribution of primes,” <i>Algebra &#38; Number Theory</i>, vol. 19, no. 4. Mathematical Sciences Publishers, pp. 617–666, 2025."},"date_updated":"2026-07-14T10:52:02Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"Odd moments in the distribution of primes","_id":"22191","status":"public","date_created":"2026-06-29T12:56:09Z","author":[{"full_name":"Kuperberg, Vivian Zieve","last_name":"Kuperberg","first_name":"Vivian Zieve","id":"c3bac823-112d-11f0-a3f5-c264f852e697"}],"oa_version":"Preprint","type":"journal_article","article_type":"original","abstract":[{"lang":"eng","text":"Montgomery and Soundararajan showed that the distribution of ψ(x+H)−ψ(x), for 0≤ x ≤ N, is approximately normal with mean ∼ H and variance ∼ H log(N/H), when N\r\nδ ≤ H ≤ N\r\n1−δ\r\n. Their work depends\r\non showing that sums Rk (h) of k-term singular series are µk (−h log h + Ah)\r\nk/2 + Ok (h\r\nk/2−1/(7k)+ε\r\n),\r\nwhere A is a constant and µk are the Gaussian moment constants. We study lower-order terms in the size\r\nof these moments. We conjecture that when k is odd, Rk (h) ≍ h\r\n(k−1)/2\r\n(log h)\r\n(k+1)/2\r\n. We prove an upper\r\nbound with the correct power of h when k = 3, and prove analogous upper bounds in the function field\r\nsetting when k = 3 and k = 5. We provide further evidence for this conjecture in the form of numerical\r\ncomputations."}],"extern":"1","volume":19,"doi":"10.2140/ant.2025.19.617","day":"24","oa":1,"external_id":{"unknown":["2109.03767"]},"publisher":"Mathematical Sciences Publishers","month":"03","mathsc":["11N05","11N13"],"date_published":"2025-03-24T00:00:00Z","article_processing_charge":"No","fulldoi":"https://doi.org/10.2140/ant.2025.19.617","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2109.03767","open_access":"1"}],"publication_status":"published","quality_controlled":"1","year":"2025","intvolume":"        19"}]