[{"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2109.03767"}],"publisher":"Mathematical Sciences Publishers","_id":"22191","mathsc":["11N05","11N13"],"doi":"10.2140/ant.2025.19.617","author":[{"first_name":"Vivian Zieve","last_name":"Kuperberg","id":"c3bac823-112d-11f0-a3f5-c264f852e697","full_name":"Kuperberg, Vivian Zieve"}],"date_created":"2026-06-29T12:56:09Z","type":"journal_article","intvolume":"        19","OA_place":"repository","day":"24","external_id":{"unknown":["2109.03767"]},"oa_version":"Preprint","extern":"1","month":"03","oa":1,"year":"2025","volume":19,"date_published":"2025-03-24T00:00:00Z","article_type":"original","publication_status":"published","abstract":[{"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.","lang":"eng"}],"quality_controlled":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"OA_type":"green","publication_identifier":{"issn":["1937-0652"],"eissn":["1944-7833"]},"date_updated":"2026-07-14T10:52:02Z","title":"Odd moments in the distribution of primes","article_processing_charge":"No","issue":"4","status":"public","scopus_import":"1","publication":"Algebra & Number Theory","citation":{"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>","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>","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.","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>.","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>.","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."},"page":"617-666"},{"article_processing_charge":"No","title":"Sums of singular series along arithmetic progressions and with smooth weights","publication_identifier":{"issn":["1793-0421"],"eissn":["1793-7310"]},"date_updated":"2026-07-14T11:04:58Z","OA_type":"green","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"quality_controlled":"1","publication":"International Journal of Number Theory","page":"53-74","citation":{"ieee":"V. Z. Kuperberg, “Sums of singular series along arithmetic progressions and with smooth weights,” <i>International Journal of Number Theory</i>, vol. 21, no. 01. World Scientific Publishing, pp. 53–74, 2025.","ama":"Kuperberg VZ. Sums of singular series along arithmetic progressions and with smooth weights. <i>International Journal of Number Theory</i>. 2025;21(01):53-74. doi:<a href=\"https://doi.org/10.1142/s1793042125500046\">10.1142/s1793042125500046</a>","apa":"Kuperberg, V. Z. (2025). Sums of singular series along arithmetic progressions and with smooth weights. <i>International Journal of Number Theory</i>. World Scientific Publishing. <a href=\"https://doi.org/10.1142/s1793042125500046\">https://doi.org/10.1142/s1793042125500046</a>","short":"V.Z. Kuperberg, International Journal of Number Theory 21 (2025) 53–74.","mla":"Kuperberg, Vivian Zieve. “Sums of Singular Series along Arithmetic Progressions and with Smooth Weights.” <i>International Journal of Number Theory</i>, vol. 21, no. 01, World Scientific Publishing, 2025, pp. 53–74, doi:<a href=\"https://doi.org/10.1142/s1793042125500046\">10.1142/s1793042125500046</a>.","ista":"Kuperberg VZ. 2025. Sums of singular series along arithmetic progressions and with smooth weights. International Journal of Number Theory. 21(01), 53–74.","chicago":"Kuperberg, Vivian Zieve. “Sums of Singular Series along Arithmetic Progressions and with Smooth Weights.” <i>International Journal of Number Theory</i>. World Scientific Publishing, 2025. <a href=\"https://doi.org/10.1142/s1793042125500046\">https://doi.org/10.1142/s1793042125500046</a>."},"scopus_import":"1","status":"public","issue":"01","type":"journal_article","intvolume":"        21","author":[{"last_name":"Kuperberg","first_name":"Vivian Zieve","full_name":"Kuperberg, Vivian Zieve","id":"c3bac823-112d-11f0-a3f5-c264f852e697"}],"date_created":"2026-06-29T12:57:46Z","doi":"10.1142/s1793042125500046","_id":"22195","publisher":"World Scientific Publishing","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2301.06095","open_access":"1"}],"article_type":"original","publication_status":"published","abstract":[{"text":"Sums of the singular series constants that appear in the Hardy–Littlewood k-tuples conjectures have long been studied in connection to the distribution of primes. We study constrained sums of singular series, where the sum is taken over sets whose elements are specified modulo r or weighted by smooth functions. We show that the value of the sum is governed by incidences modulo r of elements of the set in the case of arithmetic progressions and by pairings of the smooth functions in the case of weights. These sums shed light on sums of singular series in other formats. ","lang":"eng"}],"arxiv":1,"year":"2025","oa":1,"volume":21,"date_published":"2025-01-01T00:00:00Z","month":"01","extern":"1","external_id":{"arxiv":["2301.06095"]},"oa_version":"Preprint","OA_place":"repository","day":"01"},{"OA_place":"repository","day":"20","external_id":{"arxiv":["2212.04969"]},"oa_version":"Preprint","extern":"1","month":"03","oa":1,"year":"2025","volume":12,"date_published":"2025-03-20T00:00:00Z","arxiv":1,"article_type":"original","publication_status":"published","abstract":[{"text":"In Kuperberg and Lal´ın [Forum Math. 34 (2022), pp. 711–747],\r\nthe authors studied the mean-square of certain sums of the divisor function\r\ndk(f) over the function field Fq[T] in the limit as q →∞ and related these\r\nsums to integrals over the ensemble of symplectic matrices, along similar lines\r\nas previous work of Keating, Rodgers, Roditty-Gershon and Rudnick [Math. Z.\r\n288 (2018), pp. 167–198] for unitary matrices. We present an analogous problem yielding an integral over the ensemble of orthogonal matrices and pursue\r\na more detailed study of both the symplectic and orthogonal matrix integrals,\r\nrelating them to symmetric function theory. The function field results lead to\r\nconjectures concerning analogous questions over number fields.","lang":"eng"}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2212.04969"}],"publisher":"American Mathematical Society","_id":"22200","mathsc":["11N60","05A15","11M50","11N56"],"doi":"10.1090/btran/186","date_created":"2026-06-29T12:59:46Z","author":[{"full_name":"Kuperberg, Vivian Zieve","id":"c3bac823-112d-11f0-a3f5-c264f852e697","last_name":"Kuperberg","first_name":"Vivian Zieve"},{"full_name":"Lalín, Matilde","first_name":"Matilde","last_name":"Lalín"}],"intvolume":"        12","type":"journal_article","issue":"10","status":"public","scopus_import":"1","publication":"Transactions of the American Mathematical Society, Series B","citation":{"short":"V.Z. Kuperberg, M. Lalín, Transactions of the American Mathematical Society, Series B 12 (2025) 323–370.","mla":"Kuperberg, Vivian Zieve, and Matilde Lalín. “Symplectic Conjectures for Sums of Divisor Functions and Explorations of an Orthogonal Regime.” <i>Transactions of the American Mathematical Society, Series B</i>, vol. 12, no. 10, American Mathematical Society, 2025, pp. 323–70, doi:<a href=\"https://doi.org/10.1090/btran/186\">10.1090/btran/186</a>.","ista":"Kuperberg VZ, Lalín M. 2025. Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime. Transactions of the American Mathematical Society, Series B. 12(10), 323–370.","chicago":"Kuperberg, Vivian Zieve, and Matilde Lalín. “Symplectic Conjectures for Sums of Divisor Functions and Explorations of an Orthogonal Regime.” <i>Transactions of the American Mathematical Society, Series B</i>. American Mathematical Society, 2025. <a href=\"https://doi.org/10.1090/btran/186\">https://doi.org/10.1090/btran/186</a>.","ieee":"V. Z. Kuperberg and M. Lalín, “Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime,” <i>Transactions of the American Mathematical Society, Series B</i>, vol. 12, no. 10. American Mathematical Society, pp. 323–370, 2025.","ama":"Kuperberg VZ, Lalín M. Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime. <i>Transactions of the American Mathematical Society, Series B</i>. 2025;12(10):323-370. doi:<a href=\"https://doi.org/10.1090/btran/186\">10.1090/btran/186</a>","apa":"Kuperberg, V. Z., &#38; Lalín, M. (2025). Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime. <i>Transactions of the American Mathematical Society, Series B</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/btran/186\">https://doi.org/10.1090/btran/186</a>"},"page":"323-370","quality_controlled":"1","language":[{"iso":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","OA_type":"green","publication_identifier":{"issn":["2330-0000"]},"date_updated":"2026-07-14T11:24:55Z","title":"Symplectic conjectures for sums of divisor functions and explorations of an orthogonal regime","article_processing_charge":"No"},{"extern":"1","month":"05","OA_place":"repository","day":"01","external_id":{"arxiv":["2410.17939"]},"oa_version":"Preprint","article_type":"original","publication_status":"published","abstract":[{"text":"Kuperberg and Lalín stated some conjectures on the variance of certain sums of the divisor function dk(n) over number fields, which were inspired by analogous results over function fields proven by the authors. These problems are related to certain symplectic matrix integrals. While the function field results can be directly related to the random matrix integrals, the connection between the random matrix integrals and the number field results is less direct and involves arithmetic factors. The goal of this article is to give heuristic arguments for the formulas of these arithmetic factors.","lang":"eng"}],"year":"2025","oa":1,"date_published":"2025-05-01T00:00:00Z","volume":71,"arxiv":1,"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2410.17939","open_access":"1"}],"publisher":"Wiley","_id":"22199","mathsc":["11N60","05A15","11M50","11N56"],"type":"journal_article","intvolume":"        71","doi":"10.1112/mtk.70029","date_created":"2026-06-29T12:59:16Z","author":[{"first_name":"Vivian Zieve","last_name":"Kuperberg","full_name":"Kuperberg, Vivian Zieve","id":"c3bac823-112d-11f0-a3f5-c264f852e697"},{"full_name":"Lalín, Matilde","last_name":"Lalín","first_name":"Matilde"}],"issue":"3","scopus_import":"1","publication":"Mathematika","citation":{"apa":"Kuperberg, V. Z., &#38; Lalín, M. (2025). Arithmetic constants for symplectic variances of the divisor function. <i>Mathematika</i>. Wiley. <a href=\"https://doi.org/10.1112/mtk.70029\">https://doi.org/10.1112/mtk.70029</a>","ama":"Kuperberg VZ, Lalín M. Arithmetic constants for symplectic variances of the divisor function. <i>Mathematika</i>. 2025;71(3). doi:<a href=\"https://doi.org/10.1112/mtk.70029\">10.1112/mtk.70029</a>","ieee":"V. Z. Kuperberg and M. Lalín, “Arithmetic constants for symplectic variances of the divisor function,” <i>Mathematika</i>, vol. 71, no. 3. Wiley, 2025.","chicago":"Kuperberg, Vivian Zieve, and Matilde Lalín. “Arithmetic Constants for Symplectic Variances of the Divisor Function.” <i>Mathematika</i>. Wiley, 2025. <a href=\"https://doi.org/10.1112/mtk.70029\">https://doi.org/10.1112/mtk.70029</a>.","mla":"Kuperberg, Vivian Zieve, and Matilde Lalín. “Arithmetic Constants for Symplectic Variances of the Divisor Function.” <i>Mathematika</i>, vol. 71, no. 3, e70029, Wiley, 2025, doi:<a href=\"https://doi.org/10.1112/mtk.70029\">10.1112/mtk.70029</a>.","ista":"Kuperberg VZ, Lalín M. 2025. Arithmetic constants for symplectic variances of the divisor function. Mathematika. 71(3), e70029.","short":"V.Z. Kuperberg, M. Lalín, Mathematika 71 (2025)."},"status":"public","OA_type":"green","quality_controlled":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"article_number":"e70029","article_processing_charge":"No","publication_identifier":{"issn":["0025-5793","2041-7942"]},"date_updated":"2026-07-14T11:20:10Z","title":"Arithmetic constants for symplectic variances of the divisor function"},{"publisher":"Springer Nature","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2305.02662","open_access":"1"}],"_id":"22201","type":"journal_article","intvolume":"       267","doi":"10.1007/s11856-024-2711-0","date_created":"2026-06-29T13:00:06Z","author":[{"full_name":"de la Bretèche, Régis","last_name":"de la Bretèche","first_name":"Régis"},{"first_name":"Vivian Zieve","last_name":"Kuperberg","id":"c3bac823-112d-11f0-a3f5-c264f852e697","full_name":"Kuperberg, Vivian Zieve"}],"month":"06","extern":"1","OA_place":"repository","day":"01","external_id":{"arxiv":["2305.02662"]},"oa_version":"Preprint","article_type":"original","publication_status":"published","abstract":[{"lang":"eng","text":"We consider an analog of a conjecture of Montgomery and Soundararajan on the moments of primes in short intervals in number fields; this analog was discussed and heuristically derived in a paper of the second author, Rodgers, and Roditty-Gershon. Adapting work of the first author and Fiorilli in the integer case, we establish lower bounds on a weighted version of these moments which agree with the conjectured values."}],"oa":1,"year":"2025","date_published":"2025-06-01T00:00:00Z","volume":267,"arxiv":1,"OA_type":"green","quality_controlled":"1","language":[{"iso":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","publication_identifier":{"issn":["0021-2172"],"eissn":["1565-8511"]},"date_updated":"2026-07-14T11:28:49Z","title":"Lower bounds on weighted moments of primes in short intervals in number fields","issue":"1","scopus_import":"1","publication":"Israel Journal of Mathematics","citation":{"short":"R. de la Bretèche, V.Z. Kuperberg, Israel Journal of Mathematics 267 (2025) 437–461.","chicago":"Bretèche, Régis de la, and Vivian Zieve Kuperberg. “Lower Bounds on Weighted Moments of Primes in Short Intervals in Number Fields.” <i>Israel Journal of Mathematics</i>. Springer Nature, 2025. <a href=\"https://doi.org/10.1007/s11856-024-2711-0\">https://doi.org/10.1007/s11856-024-2711-0</a>.","ista":"de la Bretèche R, Kuperberg VZ. 2025. Lower bounds on weighted moments of primes in short intervals in number fields. Israel Journal of Mathematics. 267(1), 437–461.","mla":"de la Bretèche, Régis, and Vivian Zieve Kuperberg. “Lower Bounds on Weighted Moments of Primes in Short Intervals in Number Fields.” <i>Israel Journal of Mathematics</i>, vol. 267, no. 1, Springer Nature, 2025, pp. 437–61, doi:<a href=\"https://doi.org/10.1007/s11856-024-2711-0\">10.1007/s11856-024-2711-0</a>.","ama":"de la Bretèche R, Kuperberg VZ. Lower bounds on weighted moments of primes in short intervals in number fields. <i>Israel Journal of Mathematics</i>. 2025;267(1):437-461. doi:<a href=\"https://doi.org/10.1007/s11856-024-2711-0\">10.1007/s11856-024-2711-0</a>","apa":"de la Bretèche, R., &#38; Kuperberg, V. Z. (2025). Lower bounds on weighted moments of primes in short intervals in number fields. <i>Israel Journal of Mathematics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s11856-024-2711-0\">https://doi.org/10.1007/s11856-024-2711-0</a>","ieee":"R. de la Bretèche and V. Z. Kuperberg, “Lower bounds on weighted moments of primes in short intervals in number fields,” <i>Israel Journal of Mathematics</i>, vol. 267, no. 1. Springer Nature, pp. 437–461, 2025."},"page":"437-461","status":"public"},{"mathsc":["11N69","11N05","14G05"],"_id":"22203","publisher":"Wiley","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2312.09021"}],"intvolume":"       131","type":"journal_article","date_created":"2026-06-29T13:00:46Z","author":[{"full_name":"Bloom, Thomas F.","last_name":"Bloom","first_name":"Thomas F."},{"id":"c3bac823-112d-11f0-a3f5-c264f852e697","full_name":"Kuperberg, Vivian Zieve","first_name":"Vivian Zieve","last_name":"Kuperberg"}],"doi":"10.1112/plms.70068","extern":"1","month":"07","oa_version":"Preprint","external_id":{"arxiv":["2312.09021"]},"day":"01","OA_place":"repository","abstract":[{"lang":"eng","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."}],"article_type":"original","publication_status":"published","arxiv":1,"date_published":"2025-07-01T00:00:00Z","volume":131,"oa":1,"year":"2025","OA_type":"green","language":[{"iso":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","article_processing_charge":"No","article_number":"e70068","title":"Odd moments and adding fractions","date_updated":"2026-07-14T11:45:18Z","publication_identifier":{"eissn":["1460-244X"],"issn":["0024-6115"]},"issue":"1","citation":{"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>.","ista":"Bloom TF, Kuperberg VZ. 2025. Odd moments and adding fractions. Proceedings of the London Mathematical Society. 131(1), e70068.","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>.","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.","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>","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>"},"publication":"Proceedings of the London Mathematical Society","scopus_import":"1","status":"public"},{"article_processing_charge":"No","publication_identifier":{"issn":["1474-7480"],"eissn":["1475-3030"]},"date_updated":"2026-07-14T11:50:36Z","title":"Positive density for consecutive runs of sums of two squares","OA_type":"green","quality_controlled":"1","language":[{"iso":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","scopus_import":"1","publication":"Journal of the Institute of Mathematics of Jussieu","page":"1995-2046","citation":{"chicago":"Kimmel, Noam, and Vivian Zieve Kuperberg. “Positive Density for Consecutive Runs of Sums of Two Squares.” <i>Journal of the Institute of Mathematics of Jussieu</i>. Cambridge University Press, 2025. <a href=\"https://doi.org/10.1017/s1474748025000131\">https://doi.org/10.1017/s1474748025000131</a>.","mla":"Kimmel, Noam, and Vivian Zieve Kuperberg. “Positive Density for Consecutive Runs of Sums of Two Squares.” <i>Journal of the Institute of Mathematics of Jussieu</i>, vol. 24, no. 5, Cambridge University Press, 2025, pp. 1995–2046, doi:<a href=\"https://doi.org/10.1017/s1474748025000131\">10.1017/s1474748025000131</a>.","ista":"Kimmel N, Kuperberg VZ. 2025. Positive density for consecutive runs of sums of two squares. Journal of the Institute of Mathematics of Jussieu. 24(5), 1995–2046.","short":"N. Kimmel, V.Z. Kuperberg, Journal of the Institute of Mathematics of Jussieu 24 (2025) 1995–2046.","apa":"Kimmel, N., &#38; Kuperberg, V. Z. (2025). Positive density for consecutive runs of sums of two squares. <i>Journal of the Institute of Mathematics of Jussieu</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/s1474748025000131\">https://doi.org/10.1017/s1474748025000131</a>","ama":"Kimmel N, Kuperberg VZ. Positive density for consecutive runs of sums of two squares. <i>Journal of the Institute of Mathematics of Jussieu</i>. 2025;24(5):1995-2046. doi:<a href=\"https://doi.org/10.1017/s1474748025000131\">10.1017/s1474748025000131</a>","ieee":"N. Kimmel and V. Z. Kuperberg, “Positive density for consecutive runs of sums of two squares,” <i>Journal of the Institute of Mathematics of Jussieu</i>, vol. 24, no. 5. Cambridge University Press, pp. 1995–2046, 2025."},"status":"public","issue":"5","type":"journal_article","intvolume":"        24","doi":"10.1017/s1474748025000131","date_created":"2026-06-29T13:01:08Z","author":[{"last_name":"Kimmel","first_name":"Noam","full_name":"Kimmel, Noam"},{"id":"c3bac823-112d-11f0-a3f5-c264f852e697","full_name":"Kuperberg, Vivian Zieve","last_name":"Kuperberg","first_name":"Vivian Zieve"}],"main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2406.04174","open_access":"1"}],"publisher":"Cambridge University Press","_id":"22204","publication_status":"published","article_type":"original","abstract":[{"text":"We study the distribution of consecutive sums of two squares in arithmetic progressions. We\r\nshow that for any odd squarefree modulus q, any two reduced congruence classes a1 and a2 mod q,\r\nand any r1,r2 ≥ 1, a positive density of sums of two squares begin a chain of r1 consecutive sums of\r\ntwo squares, all of which are a1 mod q, followed immediately by a chain of r2 consecutive sums of two\r\nsquares, all of which are a2 mod q. This is an analog of the result of Maynard for the sequence of primes,\r\nshowing that for any reduced congruence class a mod q and for any r ≥ 1, a positive density of primes\r\nbegin a sequence of r consecutive primes, all of which are a mod q","lang":"eng"}],"oa":1,"year":"2025","volume":24,"date_published":"2025-09-01T00:00:00Z","arxiv":1,"extern":"1","month":"09","OA_place":"repository","day":"01","external_id":{"arxiv":["2406.04174"]},"oa_version":"Preprint"},{"issue":"4","status":"public","citation":{"ieee":"M. Afifurrahman, V. Z. Kuperberg, A. Ostafe, and I. E. Shparlinski, “Statistics of ranks, determinants and characteristic polynomials of rational matrices,” <i>Forum Mathematicum</i>, vol. 37, no. 4. De Gruyter, 2024.","ama":"Afifurrahman M, Kuperberg VZ, Ostafe A, Shparlinski IE. Statistics of ranks, determinants and characteristic polynomials of rational matrices. <i>Forum Mathematicum</i>. 2024;37(4). doi:<a href=\"https://doi.org/10.1515/forum-2024-0114\">10.1515/forum-2024-0114</a>","apa":"Afifurrahman, M., Kuperberg, V. Z., Ostafe, A., &#38; Shparlinski, I. E. (2024). Statistics of ranks, determinants and characteristic polynomials of rational matrices. <i>Forum Mathematicum</i>. De Gruyter. <a href=\"https://doi.org/10.1515/forum-2024-0114\">https://doi.org/10.1515/forum-2024-0114</a>","short":"M. Afifurrahman, V.Z. Kuperberg, A. Ostafe, I.E. Shparlinski, Forum Mathematicum 37 (2024).","ista":"Afifurrahman M, Kuperberg VZ, Ostafe A, Shparlinski IE. 2024. Statistics of ranks, determinants and characteristic polynomials of rational matrices. Forum Mathematicum. 37(4).","mla":"Afifurrahman, Muhammad, et al. “Statistics of Ranks, Determinants and Characteristic Polynomials of Rational Matrices.” <i>Forum Mathematicum</i>, vol. 37, no. 4, De Gruyter, 2024, doi:<a href=\"https://doi.org/10.1515/forum-2024-0114\">10.1515/forum-2024-0114</a>.","chicago":"Afifurrahman, Muhammad, Vivian Zieve Kuperberg, Alina Ostafe, and Igor E. Shparlinski. “Statistics of Ranks, Determinants and Characteristic Polynomials of Rational Matrices.” <i>Forum Mathematicum</i>. De Gruyter, 2024. <a href=\"https://doi.org/10.1515/forum-2024-0114\">https://doi.org/10.1515/forum-2024-0114</a>."},"publication":"Forum Mathematicum","scopus_import":"1","language":[{"iso":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","OA_type":"green","title":"Statistics of ranks, determinants and characteristic polynomials of rational matrices","date_updated":"2026-07-14T10:48:41Z","publication_identifier":{"issn":["0933-7741"],"eissn":["1435-5337"]},"article_processing_charge":"No","oa_version":"Preprint","external_id":{"arxiv":["2401.10086"]},"OA_place":"repository","month":"01","extern":"1","arxiv":1,"volume":37,"date_published":"2024-01-01T00:00:00Z","year":"2024","oa":1,"abstract":[{"lang":"eng","text":"We consider the set of 𝑚×𝑛 matrices with rational entries having numerator and denominator of size at most H and obtain various upper bounds on the number of such matrices of a given rank, or with a given determinant, or a given characteristic polynomial. We also consider similar questions for matrices whose entries are Egyptian fractions."}],"publication_status":"published","article_type":"original","mathsc":["11C20","15B36","15B52"],"_id":"22190","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2401.10086","open_access":"1"}],"publisher":"De Gruyter","date_created":"2026-06-29T12:55:47Z","author":[{"full_name":"Afifurrahman, Muhammad","first_name":"Muhammad","last_name":"Afifurrahman"},{"last_name":"Kuperberg","first_name":"Vivian Zieve","full_name":"Kuperberg, Vivian Zieve","id":"c3bac823-112d-11f0-a3f5-c264f852e697"},{"first_name":"Alina","last_name":"Ostafe","full_name":"Ostafe, Alina"},{"full_name":"Shparlinski, Igor E.","first_name":"Igor E.","last_name":"Shparlinski"}],"doi":"10.1515/forum-2024-0114","type":"journal_article","intvolume":"        37"},{"author":[{"full_name":"Kimmel, Noam","last_name":"Kimmel","first_name":"Noam"},{"first_name":"Vivian Zieve","last_name":"Kuperberg","id":"c3bac823-112d-11f0-a3f5-c264f852e697","full_name":"Kuperberg, Vivian Zieve"}],"date_created":"2026-06-29T12:58:28Z","doi":"10.1016/j.jnt.2024.05.003","type":"journal_article","intvolume":"       264","_id":"22197","publisher":"Elsevier","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2306.12855"}],"arxiv":1,"year":"2024","date_published":"2024-11-01T00:00:00Z","volume":264,"publication_status":"published","article_type":"original","abstract":[{"lang":"eng","text":"We study the distribution of consecutive sums of two squares\r\nin arithmetic progressions. If {En}n∈N is the sequence of\r\nsums of two squares in increasing order, we show that for\r\nany modulus q and any congruence classes a1, a2, a3 mod q\r\nwhich are admissible in the sense that there are solutions\r\nto x2 + y2 ≡ ai mod q, there exist infinitely many n with\r\nEn+i−1 ≡ ai mod q, for i =1, 2, 3. We also show that for\r\nany r1, r2 ≥ 1, there exist infinitely many n with En+i−1 ≡\r\na1 mod q for 1 ≤ i ≤ r1 and En+i−1 ≡ a2 mod q for\r\nr1 +1 ≤ i ≤ r1 + r2"}],"external_id":{"arxiv":["2306.12855"]},"oa_version":"Preprint","OA_place":"repository","day":"01","month":"11","extern":"1","title":"Consecutive runs of sums of two squares","publication_identifier":{"issn":["0022-314X"]},"date_updated":"2026-07-14T11:10:45Z","article_processing_charge":"No","language":[{"iso":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","OA_type":"green","status":"public","publication":"Journal of Number Theory","page":"135-147","citation":{"ieee":"N. Kimmel and V. Z. Kuperberg, “Consecutive runs of sums of two squares,” <i>Journal of Number Theory</i>, vol. 264. Elsevier, pp. 135–147, 2024.","ama":"Kimmel N, Kuperberg VZ. Consecutive runs of sums of two squares. <i>Journal of Number Theory</i>. 2024;264:135-147. doi:<a href=\"https://doi.org/10.1016/j.jnt.2024.05.003\">10.1016/j.jnt.2024.05.003</a>","apa":"Kimmel, N., &#38; Kuperberg, V. Z. (2024). Consecutive runs of sums of two squares. <i>Journal of Number Theory</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jnt.2024.05.003\">https://doi.org/10.1016/j.jnt.2024.05.003</a>","short":"N. Kimmel, V.Z. Kuperberg, Journal of Number Theory 264 (2024) 135–147.","ista":"Kimmel N, Kuperberg VZ. 2024. Consecutive runs of sums of two squares. Journal of Number Theory. 264, 135–147.","mla":"Kimmel, Noam, and Vivian Zieve Kuperberg. “Consecutive Runs of Sums of Two Squares.” <i>Journal of Number Theory</i>, vol. 264, Elsevier, 2024, pp. 135–47, doi:<a href=\"https://doi.org/10.1016/j.jnt.2024.05.003\">10.1016/j.jnt.2024.05.003</a>.","chicago":"Kimmel, Noam, and Vivian Zieve Kuperberg. “Consecutive Runs of Sums of Two Squares.” <i>Journal of Number Theory</i>. Elsevier, 2024. <a href=\"https://doi.org/10.1016/j.jnt.2024.05.003\">https://doi.org/10.1016/j.jnt.2024.05.003</a>."},"scopus_import":"1"},{"_id":"22192","publisher":"Oxford University Press","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2210.09775"}],"intvolume":"        74","type":"journal_article","author":[{"last_name":"Kuperberg","first_name":"Vivian Zieve","id":"c3bac823-112d-11f0-a3f5-c264f852e697","full_name":"Kuperberg, Vivian Zieve"}],"date_created":"2026-06-29T12:56:30Z","doi":"10.1093/qmath/haad030","extern":"1","month":"12","oa_version":"Preprint","external_id":{"arxiv":["2210.09775"]},"day":"01","OA_place":"repository","abstract":[{"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)).","lang":"eng"}],"publication_status":"published","article_type":"original","arxiv":1,"volume":74,"date_published":"2023-12-01T00:00:00Z","oa":1,"year":"2023","OA_type":"green","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"quality_controlled":"1","article_processing_charge":"No","title":"Sums of singular series with large sets and the tail of the distribution of primes","date_updated":"2026-07-14T10:55:52Z","publication_identifier":{"issn":["0033-5606"],"eissn":["1464-3847"]},"issue":"4","page":"1457-1479","citation":{"short":"V.Z. Kuperberg, The Quarterly Journal of Mathematics 74 (2023) 1457–1479.","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>.","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>.","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.","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>","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>","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."},"publication":"The Quarterly Journal of Mathematics","scopus_import":"1","status":"public"},{"issue":"2","status":"public","keyword":["Primes","Short intervals","Ramanujan sums","Singular series","Number fields"],"page":"291-317","citation":{"short":"V.Z. Kuperberg, B. Rodgers, E. Roditty-Gershon, The Ramanujan Journal 58 (2022) 291–317.","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.","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>.","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>.","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.","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>","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>"},"publication":"The Ramanujan Journal","quality_controlled":"1","language":[{"iso":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","OA_type":"green","date_updated":"2026-07-14T10:59:34Z","publication_identifier":{"eissn":["1572-9303"],"issn":["1382-4090"]},"title":"Sums of singular series and primes in short intervals in algebraic number fields","article_processing_charge":"No","day":"01","OA_place":"repository","oa_version":"Preprint","external_id":{"arxiv":["2001.09513"]},"month":"06","extern":"1","volume":58,"date_published":"2022-06-01T00:00:00Z","oa":1,"year":"2022","arxiv":1,"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."}],"article_type":"original","publication_status":"published","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2001.09513"}],"publisher":"Springer Nature","mathsc":["11N05","11R47"],"_id":"22193","doi":"10.1007/s11139-022-00561-9","author":[{"id":"c3bac823-112d-11f0-a3f5-c264f852e697","full_name":"Kuperberg, Vivian Zieve","last_name":"Kuperberg","first_name":"Vivian Zieve"},{"last_name":"Rodgers","first_name":"Brad","full_name":"Rodgers, Brad"},{"last_name":"Roditty-Gershon","first_name":"Edva","full_name":"Roditty-Gershon, Edva"}],"date_created":"2026-06-29T12:57:02Z","type":"journal_article","intvolume":"        58"},{"_id":"22194","publisher":"De Gruyter","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2107.01437","open_access":"1"}],"type":"journal_article","intvolume":"        34","author":[{"id":"c3bac823-112d-11f0-a3f5-c264f852e697","full_name":"Kuperberg, Vivian Zieve","last_name":"Kuperberg","first_name":"Vivian Zieve"},{"first_name":"Matilde","last_name":"Lalín","full_name":"Lalín, Matilde"}],"date_created":"2026-06-29T12:57:26Z","doi":"10.1515/forum-2021-0171","month":"03","extern":"1","external_id":{"arxiv":["2107.01437"]},"oa_version":"Preprint","OA_place":"repository","day":"26","publication_status":"published","article_type":"original","abstract":[{"lang":"eng","text":"In [J. P. Keating, B. Rodgers, E. Roditty-Gershon and Z. Rudnick, Sums of divisor functions in 𝔽𝑞⁢[𝑡] and matrix integrals, Math. Z. 288 2018, 1–2, 167–198], the authors established relationships of the mean-square of sums of the divisor function 𝑑𝑘⁢(𝑓) over short intervals and over arithmetic progressions for the function field 𝔽𝑞⁢[𝑇] to certain integrals over the ensemble of unitary matrices. We consider similar problems leading to distributions over the ensemble of symplectic matrices. We also consider analogous questions involving convolutions of the von Mangoldt function."}],"arxiv":1,"year":"2022","oa":1,"volume":34,"date_published":"2022-03-26T00:00:00Z","OA_type":"green","language":[{"iso":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","article_processing_charge":"No","title":"Sums of divisor functions and von Mangoldt convolutions in 𝔽q[T] leading to symplectic distributions","publication_identifier":{"eissn":["1435-5337"],"issn":["0933-7741"]},"date_updated":"2026-07-14T11:02:45Z","issue":"3","publication":"Forum Mathematicum","page":"711-747","citation":{"short":"V.Z. Kuperberg, M. Lalín, Forum Mathematicum 34 (2022) 711–747.","chicago":"Kuperberg, Vivian Zieve, and Matilde Lalín. “Sums of Divisor Functions and von Mangoldt Convolutions in 𝔽q[T] Leading to Symplectic Distributions.” <i>Forum Mathematicum</i>. De Gruyter, 2022. <a href=\"https://doi.org/10.1515/forum-2021-0171\">https://doi.org/10.1515/forum-2021-0171</a>.","ista":"Kuperberg VZ, Lalín M. 2022. Sums of divisor functions and von Mangoldt convolutions in 𝔽q[T] leading to symplectic distributions. Forum Mathematicum. 34(3), 711–747.","mla":"Kuperberg, Vivian Zieve, and Matilde Lalín. “Sums of Divisor Functions and von Mangoldt Convolutions in 𝔽q[T] Leading to Symplectic Distributions.” <i>Forum Mathematicum</i>, vol. 34, no. 3, De Gruyter, 2022, pp. 711–47, doi:<a href=\"https://doi.org/10.1515/forum-2021-0171\">10.1515/forum-2021-0171</a>.","apa":"Kuperberg, V. Z., &#38; Lalín, M. (2022). Sums of divisor functions and von Mangoldt convolutions in 𝔽q[T] leading to symplectic distributions. <i>Forum Mathematicum</i>. De Gruyter. <a href=\"https://doi.org/10.1515/forum-2021-0171\">https://doi.org/10.1515/forum-2021-0171</a>","ama":"Kuperberg VZ, Lalín M. Sums of divisor functions and von Mangoldt convolutions in 𝔽q[T] leading to symplectic distributions. <i>Forum Mathematicum</i>. 2022;34(3):711-747. doi:<a href=\"https://doi.org/10.1515/forum-2021-0171\">10.1515/forum-2021-0171</a>","ieee":"V. Z. Kuperberg and M. Lalín, “Sums of divisor functions and von Mangoldt convolutions in 𝔽q[T] leading to symplectic distributions,” <i>Forum Mathematicum</i>, vol. 34, no. 3. De Gruyter, pp. 711–747, 2022."},"scopus_import":"1","status":"public"},{"OA_type":"green","quality_controlled":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"article_processing_charge":"No","publication_identifier":{"issn":["0022-314X"]},"date_updated":"2026-07-14T11:08:14Z","title":"On pseudo-polynomials divisible only by a sparse set of primes and α-primary pseudo-polynomials","scopus_import":"1","publication":"Journal of Number Theory","page":"531-541","citation":{"ista":"Kuperberg VZ. 2022. On pseudo-polynomials divisible only by a sparse set of primes and α-primary pseudo-polynomials. Journal of Number Theory. 241, 531–541.","mla":"Kuperberg, Vivian Zieve. “On Pseudo-Polynomials Divisible Only by a Sparse Set of Primes and α-Primary Pseudo-Polynomials.” <i>Journal of Number Theory</i>, vol. 241, Elsevier, 2022, pp. 531–41, doi:<a href=\"https://doi.org/10.1016/j.jnt.2022.04.006\">10.1016/j.jnt.2022.04.006</a>.","chicago":"Kuperberg, Vivian Zieve. “On Pseudo-Polynomials Divisible Only by a Sparse Set of Primes and α-Primary Pseudo-Polynomials.” <i>Journal of Number Theory</i>. Elsevier, 2022. <a href=\"https://doi.org/10.1016/j.jnt.2022.04.006\">https://doi.org/10.1016/j.jnt.2022.04.006</a>.","short":"V.Z. Kuperberg, Journal of Number Theory 241 (2022) 531–541.","ieee":"V. Z. Kuperberg, “On pseudo-polynomials divisible only by a sparse set of primes and α-primary pseudo-polynomials,” <i>Journal of Number Theory</i>, vol. 241. Elsevier, pp. 531–541, 2022.","ama":"Kuperberg VZ. On pseudo-polynomials divisible only by a sparse set of primes and α-primary pseudo-polynomials. <i>Journal of Number Theory</i>. 2022;241:531-541. doi:<a href=\"https://doi.org/10.1016/j.jnt.2022.04.006\">10.1016/j.jnt.2022.04.006</a>","apa":"Kuperberg, V. Z. (2022). On pseudo-polynomials divisible only by a sparse set of primes and α-primary pseudo-polynomials. <i>Journal of Number Theory</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jnt.2022.04.006\">https://doi.org/10.1016/j.jnt.2022.04.006</a>"},"keyword":["Pseudo-polynomials","Chinese remainder theorem","Ruzsa’s conjecture"],"status":"public","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2006.02527","open_access":"1"}],"publisher":"Elsevier","_id":"22196","intvolume":"       241","type":"journal_article","doi":"10.1016/j.jnt.2022.04.006","author":[{"id":"c3bac823-112d-11f0-a3f5-c264f852e697","full_name":"Kuperberg, Vivian Zieve","last_name":"Kuperberg","first_name":"Vivian Zieve"}],"date_created":"2026-06-29T12:58:07Z","extern":"1","month":"05","OA_place":"repository","day":"18","external_id":{"arxiv":["2006.02527"]},"oa_version":"Preprint","article_type":"original","publication_status":"published","abstract":[{"lang":"eng","text":"We explore two questions about pseudo-polynomials, which\r\nare functions f : N → Z such that k divides f(n + k) −\r\nf(n) for all n, k. First, for certain arbitrarily sparse sets R, we\r\nconstruct pseudo-polynomials f with p|f(n) for some n only if\r\np ∈ R. This implies that not all pseudo-polynomials satisfy an\r\nassumption of a recent paper of Kowalski and Soundararajan.\r\nWe also consider α-primary pseudo-polynomials, where the\r\npseudo-polynomial condition is only required for k lying in\r\na set of primes of density α. We show that if an α-primary\r\npseudo-polynomial is O(e(β−)n), where β = √7\r\n3 − 1\r\n6 ≈ 0.715,\r\nthen it is a polynomial."}],"oa":1,"year":"2022","volume":241,"date_published":"2022-05-18T00:00:00Z","arxiv":1},{"issue":"1","status":"public","citation":{"short":"M. Kassabov, V.Z. Kuperberg, T.R. Riley, Journal of Combinatorial Algebra 3 (2019) 41–70.","ista":"Kassabov M, Kuperberg VZ, Riley TR. 2019. Soficity and variations on Higman’s group. Journal of Combinatorial Algebra. 3(1), 41–70.","mla":"Kassabov, Martin, et al. “Soficity and Variations on Higman’s Group.” <i>Journal of Combinatorial Algebra</i>, vol. 3, no. 1, European Mathematical Society, 2019, pp. 41–70, doi:<a href=\"https://doi.org/10.4171/jca/26\">10.4171/jca/26</a>.","chicago":"Kassabov, Martin, Vivian Zieve Kuperberg, and Timothy R. Riley. “Soficity and Variations on Higman’s Group.” <i>Journal of Combinatorial Algebra</i>. European Mathematical Society, 2019. <a href=\"https://doi.org/10.4171/jca/26\">https://doi.org/10.4171/jca/26</a>.","ieee":"M. Kassabov, V. Z. Kuperberg, and T. R. Riley, “Soficity and variations on Higman’s group,” <i>Journal of Combinatorial Algebra</i>, vol. 3, no. 1. European Mathematical Society, pp. 41–70, 2019.","apa":"Kassabov, M., Kuperberg, V. Z., &#38; Riley, T. R. (2019). Soficity and variations on Higman’s group. <i>Journal of Combinatorial Algebra</i>. European Mathematical Society. <a href=\"https://doi.org/10.4171/jca/26\">https://doi.org/10.4171/jca/26</a>","ama":"Kassabov M, Kuperberg VZ, Riley TR. Soficity and variations on Higman’s group. <i>Journal of Combinatorial Algebra</i>. 2019;3(1):41-70. doi:<a href=\"https://doi.org/10.4171/jca/26\">10.4171/jca/26</a>"},"page":"41-70","publication":"Journal of Combinatorial Algebra","scopus_import":"1","language":[{"iso":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","OA_type":"green","title":"Soficity and variations on Higman’s group","date_updated":"2026-07-14T11:54:52Z","publication_identifier":{"eissn":["2415-6302"]},"article_processing_charge":"No","oa_version":"Preprint","external_id":{"arxiv":["2406.04174"]},"day":"01","OA_place":"repository","month":"02","extern":"1","arxiv":1,"volume":3,"date_published":"2019-02-01T00:00:00Z","year":"2019","oa":1,"abstract":[{"text":"A group is sofic when every finite subset can be well approximated in a finite symmetric group. No example of a non-sofic group is known. Higman's group, which is a circular amalgamation of four copies of the Baumslag–Solitar group, is a candidate. Here we contribute to the discussion of the problem of its soficity in two ways.\r\nWe construct variations on Higman's group replacing the Baumslag–Solitar group by other groups G. We give an elementary condition on G enjoyed for example by Z≀Z and the integral Heisenberg group, under which the resulting group is sofic.\r\n\r\nWe then use soficity to deduce that there exist permutations of Z/nZ that are seemingly pathological in that they have order dividing four and yet locally they behave like exponential functions over most of their domains. Our approach is based on that of Helfgott and Juschenko, who recently showed the soficity of Higman's group would imply some the existence of some similarly pathological functions. Our results call into question their suggestion that this might be a step towards proving the existence of a non-sofic group.","lang":"eng"}],"publication_status":"published","article_type":"original","_id":"22205","publisher":"European Mathematical Society","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2406.04174","open_access":"1"}],"author":[{"last_name":"Kassabov","first_name":"Martin","full_name":"Kassabov, Martin"},{"id":"c3bac823-112d-11f0-a3f5-c264f852e697","full_name":"Kuperberg, Vivian Zieve","last_name":"Kuperberg","first_name":"Vivian Zieve"},{"first_name":"Timothy R.","last_name":"Riley","full_name":"Riley, Timothy R."}],"date_created":"2026-06-29T13:01:27Z","doi":"10.4171/jca/26","intvolume":"         3","type":"journal_article"},{"article_processing_charge":"No","article_number":"P1.19","title":"Chip-firing on trees of loops","date_updated":"2026-07-14T13:09:11Z","publication_identifier":{"eissn":["1077-8926"]},"OA_type":"green","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"quality_controlled":"1","citation":{"ieee":"S. Kailasa, V. Z. Kuperberg, and N. Wawrykow, “Chip-firing on trees of loops,” <i>The Electronic Journal of Combinatorics</i>, vol. 25, no. 1. The Electronic Journal of Combinatorics, 2018.","apa":"Kailasa, S., Kuperberg, V. Z., &#38; Wawrykow, N. (2018). Chip-firing on trees of loops. <i>The Electronic Journal of Combinatorics</i>. The Electronic Journal of Combinatorics. <a href=\"https://doi.org/10.37236/7244\">https://doi.org/10.37236/7244</a>","ama":"Kailasa S, Kuperberg VZ, Wawrykow N. Chip-firing on trees of loops. <i>The Electronic Journal of Combinatorics</i>. 2018;25(1). doi:<a href=\"https://doi.org/10.37236/7244\">10.37236/7244</a>","short":"S. Kailasa, V.Z. Kuperberg, N. Wawrykow, The Electronic Journal of Combinatorics 25 (2018).","ista":"Kailasa S, Kuperberg VZ, Wawrykow N. 2018. Chip-firing on trees of loops. The Electronic Journal of Combinatorics. 25(1), P1.19.","mla":"Kailasa, Sameer, et al. “Chip-Firing on Trees of Loops.” <i>The Electronic Journal of Combinatorics</i>, vol. 25, no. 1, P1.19, The Electronic Journal of Combinatorics, 2018, doi:<a href=\"https://doi.org/10.37236/7244\">10.37236/7244</a>.","chicago":"Kailasa, Sameer, Vivian Zieve Kuperberg, and Nicholas Wawrykow. “Chip-Firing on Trees of Loops.” <i>The Electronic Journal of Combinatorics</i>. The Electronic Journal of Combinatorics, 2018. <a href=\"https://doi.org/10.37236/7244\">https://doi.org/10.37236/7244</a>."},"publication":"The Electronic Journal of Combinatorics","scopus_import":"1","status":"public","keyword":["Chip-firing","Metric graphs","Brill-Noether generality"],"issue":"1","intvolume":"        25","type":"journal_article","date_created":"2026-06-29T13:01:48Z","author":[{"full_name":"Kailasa, Sameer","first_name":"Sameer","last_name":"Kailasa"},{"last_name":"Kuperberg","first_name":"Vivian Zieve","full_name":"Kuperberg, Vivian Zieve","id":"c3bac823-112d-11f0-a3f5-c264f852e697"},{"last_name":"Wawrykow","first_name":"Nicholas","full_name":"Wawrykow, Nicholas"}],"doi":"10.37236/7244","_id":"22206","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1706.04164"}],"publisher":"The Electronic Journal of Combinatorics","abstract":[{"text":"Cools, Draisma, Payne, and Robeva proved that generic metric graphs that are \"paths of loops\" are Brill-Noether general. We show that Brill-Noether generality does not hold for \"trees of loops\": the only trees of loops that are Brill-Noether general are paths of loops. We study various notions of generality and examine which of these graphs satisfy them.","lang":"eng"}],"article_type":"original","publication_status":"published","arxiv":1,"volume":25,"date_published":"2018-01-25T00:00:00Z","year":"2018","oa":1,"month":"01","extern":"1","oa_version":"Preprint","external_id":{"arxiv":["1706.04164"]},"day":"25","OA_place":"repository"},{"_id":"22198","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1512.08762"}],"publisher":"Springer Nature","type":"journal_article","intvolume":"        58","author":[{"full_name":"Connelly, Robert","first_name":"Robert","last_name":"Connelly"},{"first_name":"Matthew","last_name":"Funkhouser","full_name":"Funkhouser, Matthew"},{"last_name":"Kuperberg","first_name":"Vivian Zieve","id":"c3bac823-112d-11f0-a3f5-c264f852e697","full_name":"Kuperberg, Vivian Zieve"},{"full_name":"Solomonides, Evan","first_name":"Evan","last_name":"Solomonides"}],"date_created":"2026-06-29T12:58:50Z","doi":"10.1007/s00454-016-9843-x","extern":"1","month":"01","oa_version":"Preprint","external_id":{"arxiv":["1512.08762"]},"day":"09","OA_place":"repository","abstract":[{"lang":"eng","text":"Packings of equal disks in the plane are known to have density at most\r\nπ/\r\n√\r\n12, although this density is never achieved in the square torus, which is what we\r\ncall the plane modulo the square lattice. We find packings of disks in a square torus\r\nthat we conjecture to be the most dense for certain numbers of packing disks, using\r\ncontinued fractions to approximate 1/\r\n√\r\n3 and 2 −\r\n√\r\n3. We also define a constant to\r\nmeasure the efficiency of a packing motived by a related constant due to Markov for\r\ncontinued fractions. One idea is to use the unique factorization property of Gaussian\r\nintegers to prove that there is an upper bound for the Markov constant for grid-like\r\npackings. By way of contrast, we show that an upper bound by Gruber [In many cases\r\noptimal configurations are almost regular hexagonal, vol. 65, pp. 121–145, 1999;Geom\r\nDedicata 84(1–3):271–320, 2001] for the error for the limiting density of a packing\r\nof equal disks in a planar square, which is on the order of 1/\r\n√\r\nN, is the best possible,\r\nwhereas for our examples for the square torus, the error for the limiting density is on\r\nthe order of 1/N, where N is the number of packing disks."}],"article_type":"original","publication_status":"published","arxiv":1,"date_published":"2017-01-09T00:00:00Z","volume":58,"year":"2017","OA_type":"green","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"quality_controlled":"1","article_processing_charge":"No","title":"Packings of equal disks in a square torus","date_updated":"2026-07-14T11:14:51Z","publication_identifier":{"issn":["0179-5376"],"eissn":["1432-0444"]},"issue":"3","citation":{"short":"R. Connelly, M. Funkhouser, V.Z. Kuperberg, E. Solomonides, Discrete &#38; Computational Geometry 58 (2017) 614–642.","chicago":"Connelly, Robert, Matthew Funkhouser, Vivian Zieve Kuperberg, and Evan Solomonides. “Packings of Equal Disks in a Square Torus.” <i>Discrete &#38; Computational Geometry</i>. Springer Nature, 2017. <a href=\"https://doi.org/10.1007/s00454-016-9843-x\">https://doi.org/10.1007/s00454-016-9843-x</a>.","mla":"Connelly, Robert, et al. “Packings of Equal Disks in a Square Torus.” <i>Discrete &#38; Computational Geometry</i>, vol. 58, no. 3, Springer Nature, 2017, pp. 614–42, doi:<a href=\"https://doi.org/10.1007/s00454-016-9843-x\">10.1007/s00454-016-9843-x</a>.","ista":"Connelly R, Funkhouser M, Kuperberg VZ, Solomonides E. 2017. Packings of equal disks in a square torus. Discrete &#38; Computational Geometry. 58(3), 614–642.","ama":"Connelly R, Funkhouser M, Kuperberg VZ, Solomonides E. Packings of equal disks in a square torus. <i>Discrete &#38; Computational Geometry</i>. 2017;58(3):614-642. doi:<a href=\"https://doi.org/10.1007/s00454-016-9843-x\">10.1007/s00454-016-9843-x</a>","apa":"Connelly, R., Funkhouser, M., Kuperberg, V. Z., &#38; Solomonides, E. (2017). Packings of equal disks in a square torus. <i>Discrete &#38; Computational Geometry</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s00454-016-9843-x\">https://doi.org/10.1007/s00454-016-9843-x</a>","ieee":"R. Connelly, M. Funkhouser, V. Z. Kuperberg, and E. Solomonides, “Packings of equal disks in a square torus,” <i>Discrete &#38; Computational Geometry</i>, vol. 58, no. 3. Springer Nature, pp. 614–642, 2017."},"page":"614-642","publication":"Discrete & Computational Geometry","scopus_import":"1","status":"public"},{"keyword":["modular hadamard matrices","modular symmetric designs"],"status":"public","publication":"Journal of Combinatorial Designs","page":"393-405","citation":{"chicago":"Kuperberg, Vivian Zieve. “Hadamard Matrices modulo p and Small Modular Hadamard Matrices.” <i>Journal of Combinatorial Designs</i>. Wiley, 2016. <a href=\"https://doi.org/10.1002/jcd.21522\">https://doi.org/10.1002/jcd.21522</a>.","ista":"Kuperberg VZ. 2016. Hadamard matrices modulo p and small modular Hadamard matrices. Journal of Combinatorial Designs. 24(9), 393–405.","mla":"Kuperberg, Vivian Zieve. “Hadamard Matrices modulo p and Small Modular Hadamard Matrices.” <i>Journal of Combinatorial Designs</i>, vol. 24, no. 9, Wiley, 2016, pp. 393–405, doi:<a href=\"https://doi.org/10.1002/jcd.21522\">10.1002/jcd.21522</a>.","short":"V.Z. Kuperberg, Journal of Combinatorial Designs 24 (2016) 393–405.","ama":"Kuperberg VZ. Hadamard matrices modulo p and small modular Hadamard matrices. <i>Journal of Combinatorial Designs</i>. 2016;24(9):393-405. doi:<a href=\"https://doi.org/10.1002/jcd.21522\">10.1002/jcd.21522</a>","apa":"Kuperberg, V. Z. (2016). Hadamard matrices modulo p and small modular Hadamard matrices. <i>Journal of Combinatorial Designs</i>. Wiley. <a href=\"https://doi.org/10.1002/jcd.21522\">https://doi.org/10.1002/jcd.21522</a>","ieee":"V. Z. Kuperberg, “Hadamard matrices modulo p and small modular Hadamard matrices,” <i>Journal of Combinatorial Designs</i>, vol. 24, no. 9. Wiley, pp. 393–405, 2016."},"scopus_import":"1","issue":"9","title":"Hadamard matrices modulo p and small modular Hadamard matrices","publication_identifier":{"eissn":["1520-6610"],"issn":["1063-8539"]},"date_updated":"2026-07-14T11:35:58Z","article_processing_charge":"No","language":[{"iso":"eng"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","OA_type":"green","arxiv":1,"year":"2016","oa":1,"volume":24,"date_published":"2016-09-01T00:00:00Z","publication_status":"published","article_type":"original","abstract":[{"lang":"eng","text":"We use modular symmetric designs to study the existence of Hadamard matrices modulo certain primes. We solve the 7-modular and 11-modular versions of the Hadamard conjecture for all but a finite number of cases. In doing so, we state a conjectural sufficient condition for the existence of a p-modular Hadamard matrix for all but finitely many cases. When 2 is a primitive root of a prime p, we conditionally solve this conjecture and therefore the p-modular version of the Hadamard conjecture for all but finitely many cases when p ≡ 3(mod 4), and prove a weaker result for p ≡ 1 (mod 4). Finally, we look at constraints on the existence of m-modular Hadamard matrices when the size of the matrix is small compared to m."}],"external_id":{"arxiv":["1409.0148"]},"oa_version":"Preprint","OA_place":"repository","day":"01","month":"09","extern":"1","date_created":"2026-06-29T13:00:27Z","author":[{"full_name":"Kuperberg, Vivian Zieve","id":"c3bac823-112d-11f0-a3f5-c264f852e697","last_name":"Kuperberg","first_name":"Vivian Zieve"}],"doi":"10.1002/jcd.21522","intvolume":"        24","type":"journal_article","_id":"22202","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1409.0148","open_access":"1"}],"publisher":"Wiley"}]