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Sums of odd-ly many fractions and the distribution of primes
Multi angle
Authors : Kuperberg, Vivian (Author of the conference)
CIRM (Publisher )
11D68 - Rational numbers as sums of fractions
11D79 - Congruences in many variables
11N05 - Distribution of primes
CIRM (Publisher )
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Abstract :
In this talk, I will discuss new bounds on constrained sets of fractions. Specifically, I will discuss the answer to the following question, which arises in several areas of number theory: For an integer $k\geq2$, consider the set of $k$-tuples of reduced fractions $\frac{a1}{q1} , . . . , \frac{ak}{qk} \in I$, where $I$ is an interval around 0. How many $k$-tuples are there with $\sum_{i} \frac{ai}{qi} \in \mathbb{Z} $? When $k$ is even, the answer is well-known: the main contribution to the number of solutions comes from “diagonal” terms, where the fractions $\frac{ai}{qi}$ cancel in pairs. When $k$ is odd, the answer is much more mysterious! In joint work with Bloom, we prove a near-optimal upper bound on this problem when $k$ is odd. I will also discuss applications of this problem to estimating moments of the distributions of primes and reduced residues.
Keywords : odd moments; relative gcds; Manin's conjecture
MSC Codes : Keywords : odd moments; relative gcds; Manin's conjecture
11D68 - Rational numbers as sums of fractions
11D79 - Congruences in many variables
11N05 - Distribution of primes
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 15/07/2025 Conference Date : 23/06/2025 Subseries : Research talks arXiv category : Number Theory Mathematical Area(s) : Combinatorics ; Number Theory Format : MP4 (.mp4) - HD Video Time : 00:34:19 Targeted Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students Download : https://videos.cirm-math.fr/2025-06-23_Kuperberg.mp4 
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Information on the Event
Event Title : Prime numbers and arithmetic randomness / Nombres premiers et aléa arithmétiqueEvent Organizers : Elsholtz, Christian ; Ostafe, Alina ; Rivat, Joël ; Stoll, Thomas ; Swaenepoel, Cathy Dates : 23/06/2025 - 27/06/2025 Event Year : 2025 Event URL : https://conferences.cirm-math.fr/3213.html
Citation Data
DOI : 10.24350/CIRM.V.20367803Cite this video as: Kuperberg, Vivian (2025). Sums of odd-ly many fractions and the distribution of primes. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20367803 URI : http://dx.doi.org/10.24350/CIRM.V.20367803 |
See Also
- [Multi angle] Primes in arithmetic progressions and bounded gaps / Author of the conference Stadlmann, Julia.
- [Multi angle] Recent progress on some irrationality questions of Erdős / Author of the conference Pratt, Kyle.
- [Multi angle] Improved bounds for the Fourier uniformity conjecture / Author of the conference Pilatte, Cédric.
- [Multi angle] A two dimensional delta symbol method and applications / Author of the conference Li, Junxian.
Bibliography
- BETTIN, Sandro et DESTAGNOL, Kevin. THE POWER-SAVING MANIN–PEYRE CONJECTURE FOR A SENARY CUBIC. Mathematika, 2019, vol. 65, no 4, p. 789-830. - https://doi.org/10.1112/S0025579319000159
- BLOMER, Valentin et BRÜDERN, Jörg. The density of rational points on a certain threefold. In : Contributions in Analytic and Algebraic Number Theory: Festschrift for SJ Patterson. New York, NY : Springer New York, 2011. p. 1-15. - https://doi.org/10.1007/978-1-4614-1219-9_1
- BLOMER, Valentin, BRÜDERN, Jörg. and SALBERGER, Per. On a certain senary cubic form, Proc. Lond. Math. Soc. (3) 108 (2014), no. 4, 911– 964. - https://doi.org/10.1112/plms/pdt043
- BLOMER, Valentin, BRÜDERN, Jörg, et SALBERGER, Per. The Manin–Peyre formula for a certain biprojective threefold. Mathematische Annalen, 2018, vol. 370, p. 491-553. - https://doi.org/10.1007/s00208-017-1621-4
- BLOOM, Thomas F. et MAYNARD, James. A new upper bound for sets with no square differences. Compositio Mathematica, 2022, vol. 158, no 8, p. 1777-1798. - https://doi.org/10.1112/S0010437X22007679
- BOURGAIN, Jean, DILWORTH, Stephen J., FORD, Kevin, et al. Breaking the k2 barrier for explicit RIP matrices. In : Proceedings of the forty-third annual ACM symposium on Theory of computing. 2011. p. 637-644. - https://doi.org/10.1145/1993636.1993721
- DE LA BRETÈCHE, Régis. Sur le nombre de matrices aléatoires à coefficients rationnels, Q. J. Math. 68 (2017), no. 3, 935– 955. - https://doi.org/10.1093/qmath/hax006
- DE LA BRETÈCHE, Régis. Réepartition conjointe de trois nombres premiers et applications. arXiv preprint arXiv:2410.02480, 2024. - https://doi.org/10.48550/arXiv.2410.02480
- Dedekind, R., 1930. Gesammelte mathematische Werke / Richard Dedekind. Hrsg. von Robert Fricke; Emmy Noether; Øystein Ore ; Bd. 1. Vieweg, Braunschweig. - https://doi.org/10.24355/dbbs.084-200908190200-5
- DESTAGNOL K. La conjecture de Manin pour une famille de variétés en dimension supérieure. Mathematical Proceedings of the Cambridge Philosophical Society. 2019;166(3):433-486. - https://doi.org/110.1017/S030500411800004X
- GALLAGHER, Patrick X. On the distribution of primes in short intervals. Mathematika, 1976, vol. 23, no 1, p. 4-9. - https://doi.org/10.1112/S0025579300016442
- GORODETSKY, Ofir. High and odd moments in the Erd\H {o} s--Kac theorem. arXiv preprint arXiv:2501.00351, 2024. - https://doi.org/10.48550/arXiv.2501.00351
- GORODETSKY, Ofir, MANGEREL, Alexander, et RODGERS, Brad. Squarefrees are Gaussian in short intervals. Journal für die reine und angewandte Mathematik (Crelles Journal), 2023, vol. 2023, no 795, p. 1-44. - https://doi.org/10.1515/crelle-2022-0066
- HARDY, Godfrey H. et LITTLEWOOD, John E. Some problems of ‘Partitio numerorum'; III: On the expression of a number as a sum of primes. Acta mathematica, 1923, vol. 44, no 1, p. 1-70. - https://doi.org/10.1007/BF02403921
- HEATH-BROWN, D. Roger. The density of rational points on Cayley's cubic surface. arXiv preprint math/0210333, 2002. - https://doi.org/10.48550/arXiv.math/0210333
- KUPERBERG, Vivian. Odd moments in the distribution of primes. Algebra & Number Theory, 2025, vol. 19, no 4, p. 617-666. - https://doi.org/10.2140/ant.2025.19.617
- LEMKE OLIVER, Robert J. et SOUNDARARAJAN, Kannan. Unexpected biases in the distribution of consecutive primes. Proceedings of the National Academy of Sciences, 2016, vol. 113, no 31, p. E4446-E4454. - https://doi.org/10.1073/pnas.1605366113
- MONTGOMERY, Hugh L. et SOUNDARARAJAN, Kannan. Primes in short intervals. Communications in mathematical physics, 2004, vol. 252, no 1, p. 589-617. - https://doi.org/10.1007/s00220-004-1222-4
- MONTGOMERY, Hugh L. et VAUGHAN, Robert C. On the distribution of reduced residues. Annals of Mathematics, 1986, vol. 123, no 2, p. 311-333. - https://doi.org/10.2307/1971274
- MONTGOMERY, Hugh L. et VAUGHAN, Robert C. Multiplicative number theory I: Classical theory. Cambridge university press, 2007. - https://doi.org/10.1017/CBO9780511618314
- SHPARLINSKI, Igor E. Linear equations with rational fractions of bounded height and stochastic matrices. Quarterly Journal of Mathematics, 2018, vol. 69, no 2, p. 487-499. - https://doi.org/10.1093/qmath/hax049
- DEDEKIND, R. Gesammelte mathematische Werke, Band 2, Hrsg. v. Robert Fricke, Emmy Noether u. Oeystein Ore., Uber Zerlegungen von Zahlen durch ihren größten gemeinsamen Teiler,(Festschrift der Universität Braunschweig, 1897). 1931. - https://doi.org/10.24355/dbbs.084-200908190200-5
- ELSHOLTZ, Christian. Sums of K unit fractions. Transactions of the American Mathematical society, 2001, vol. 353, no 8, p. 3209-3227. - https://doi.org/10.1090/S0002-9947-01-02782-9
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