[{"publication":"The Ramanujan Journal","status":"public","date_updated":"2026-07-14T10:59:34Z","external_id":{"arxiv":["2001.09513"]},"date_published":"2022-06-01T00:00:00Z","intvolume":"        58","day":"01","abstract":[{"lang":"eng","text":"Gross and Smith have put forward generalizations of Hardy–Littlewood twin prime\r\nconjectures for algebraic number fields. We estimate the behaviour of sums of a singular series that arises in these conjectures, up to lower-order terms. More exactly,\r\nwhere S(η) is the singular series, we find asymptotic formulas for smoothed sums of\r\nS(η)−1. Based upon Gross and Smith’s conjectures, we use our result to suggest that\r\nfor large enough ‘short intervals’ in an algebraic number field K, the variance of counts\r\nof prime elements in a random short interval deviates from a Cramér model prediction\r\nby a universal factor, independent of K. The conjecture over number fields generalizes\r\na classical conjecture of Goldston and Montgomery over the integers. Numerical data\r\nare provided supporting the conjecture."}],"author":[{"id":"c3bac823-112d-11f0-a3f5-c264f852e697","last_name":"Kuperberg","full_name":"Kuperberg, Vivian Zieve","first_name":"Vivian Zieve"},{"last_name":"Rodgers","full_name":"Rodgers, Brad","first_name":"Brad"},{"last_name":"Roditty-Gershon","full_name":"Roditty-Gershon, Edva","first_name":"Edva"}],"publisher":"Springer Nature","OA_type":"green","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2001.09513","open_access":"1"}],"page":"291-317","year":"2022","OA_place":"repository","_id":"22193","extern":"1","publication_identifier":{"issn":["1382-4090"],"eissn":["1572-9303"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","type":"journal_article","article_type":"original","arxiv":1,"volume":58,"keyword":["Primes","Short intervals","Ramanujan sums","Singular series","Number fields"],"mathsc":["11N05","11R47"],"date_created":"2026-06-29T12:57:02Z","doi":"10.1007/s11139-022-00561-9","citation":{"ama":"Kuperberg VZ, Rodgers B, Roditty-Gershon E. Sums of singular series and primes in short intervals in algebraic number fields. <i>The Ramanujan Journal</i>. 2022;58(2):291-317. doi:<a href=\"https://doi.org/10.1007/s11139-022-00561-9\">10.1007/s11139-022-00561-9</a>","short":"V.Z. Kuperberg, B. Rodgers, E. Roditty-Gershon, The Ramanujan Journal 58 (2022) 291–317.","apa":"Kuperberg, V. Z., Rodgers, B., &#38; Roditty-Gershon, E. (2022). Sums of singular series and primes in short intervals in algebraic number fields. <i>The Ramanujan Journal</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s11139-022-00561-9\">https://doi.org/10.1007/s11139-022-00561-9</a>","ieee":"V. Z. Kuperberg, B. Rodgers, and E. Roditty-Gershon, “Sums of singular series and primes in short intervals in algebraic number fields,” <i>The Ramanujan Journal</i>, vol. 58, no. 2. Springer Nature, pp. 291–317, 2022.","ista":"Kuperberg VZ, Rodgers B, Roditty-Gershon E. 2022. Sums of singular series and primes in short intervals in algebraic number fields. The Ramanujan Journal. 58(2), 291–317.","chicago":"Kuperberg, Vivian Zieve, Brad Rodgers, and Edva Roditty-Gershon. “Sums of Singular Series and Primes in Short Intervals in Algebraic Number Fields.” <i>The Ramanujan Journal</i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1007/s11139-022-00561-9\">https://doi.org/10.1007/s11139-022-00561-9</a>.","mla":"Kuperberg, Vivian Zieve, et al. “Sums of Singular Series and Primes in Short Intervals in Algebraic Number Fields.” <i>The Ramanujan Journal</i>, vol. 58, no. 2, Springer Nature, 2022, pp. 291–317, doi:<a href=\"https://doi.org/10.1007/s11139-022-00561-9\">10.1007/s11139-022-00561-9</a>."},"quality_controlled":"1","article_processing_charge":"No","publication_status":"published","language":[{"iso":"eng"}],"title":"Sums of singular series and primes in short intervals in algebraic number fields","month":"06","oa":1,"issue":"2","oa_version":"Preprint"}]