English
Sums of odd-ly many fractions and the distribution of primes
Multi angle
Auteurs : Kuperberg, Vivian (Auteur de la Conférence)
CIRM (Editeur )
11D68 - Rational numbers as sums of fractions
11D79 - Congruences in many variables
11N05 - Distribution of primes
CIRM (Editeur )
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Résumé :
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
Codes MSC : 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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Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de publication : 15/07/2025 Date de captation : 23/06/2025 Sous collection : Research talks arXiv category : Number Theory Domaine : Combinatorics ; Number Theory Format : MP4 (.mp4) - HD Durée : 00:34:19 Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students Download : https://videos.cirm-math.fr/2025-06-23_Kuperberg.mp4 
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Informations sur la Rencontre
Nom de la rencontre : Prime numbers and arithmetic randomness / Nombres premiers et aléa arithmétiqueOrganisateurs de la rencontre : Elsholtz, Christian ; Ostafe, Alina ; Rivat, Joël ; Stoll, Thomas ; Swaenepoel, Cathy Dates : 23/06/2025 - 27/06/2025 Année de la rencontre : 2025 URL Congrès : https://conferences.cirm-math.fr/3213.html
Données de citation
DOI : 10.24350/CIRM.V.20367803Citer cette vidéo: 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 |
Voir aussi
- [Multi angle] Primes in arithmetic progressions and bounded gaps / Auteur de la Conférence Stadlmann, Julia.
- [Multi angle] Recent progress on some irrationality questions of Erdős / Auteur de la Conférence Pratt, Kyle.
- [Multi angle] Improved bounds for the Fourier uniformity conjecture / Auteur de la Conférence Pilatte, Cédric.
- [Multi angle] A two dimensional delta symbol method and applications / Auteur de la Conférence Li, Junxian.
Bibliographie
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