Sums of Odd-ly Many Fractions and Distribution of Primes

Explore new bounds on constrained sets of fractions in this mathematical conference talk that addresses a fundamental question arising in several areas of number theory. Discover the answer to how many k-tuples of reduced fractions exist within an interval around zero when their sum equals an integer, with particular focus on the mysterious case when k is odd. Learn about the well-understood even k scenario where diagonal terms dominate through fraction pairs that cancel each other out, then delve into the significantly more complex odd k case. Examine the near-optimal upper bound proof developed in joint work with Bloom for odd k values, and understand the sophisticated mathematical techniques required to tackle this challenging problem. Gain insights into the practical applications of this research, including its use in estimating moments of prime number distributions and reduced residues. Follow the mathematical reasoning behind constrained fraction sets and their connections to broader number theory concepts, presented during the thematic meeting on "Prime numbers and arithmetic randomness" at the Centre International de Rencontres Mathématiques in Marseille, France.