Next Seminar:  June 26, 2026 at TU Wien

Time:  June 26, 13:00 - 18:00

Location: Freihaus TU Wien (Wiedner Hauptstr. 8-10), Turm A, 7th floor, Zeichensaal 3

Schedule:  

13:00-14:00:   Rena Chu

Title: Short character sums evaluated at polynomials 

Abstract: Let p be a prime. Bounding short sums of Dirichlet characters mod p is a classical problem in analytic number theory, and the celebrated work of Burgess provides nontrivial bounds for sums as short as p1/4 + ε for all ε >0. In this talk, we will first survey known bounds in the original and generalized settings. Then we discuss the so-called "Burgess method'' and present a new variation and new results for characters evaluated at polynomials.

14:00-14:30: Tea break

14:30-15:30: Yijie Diao

Title: Square values of odd-degree binary forms 

Abstract: In this talk, I will present a new upper bound for the number of integer pairs for which an odd-degree binary form takes square value. As an application, this improves a recent result of Heath-Brown on sums of three square-full integers. 

15:30-16:30: Break

16:00-17:00:  Julia Stadlmann

Title: The large sieve inequality and additive decompositions of sums of squares 

Abstract: Ostmann’s problem asks if there are sets A1 and A2 with |A1|, |A2| > 1 so that the sumset A1 + A2 differs from the set of primes by only finitely many elements. It is believed that no such A1 and A2 exist, but to date the problem remains open. A major obstacle to the resolution of Ostmann's problem is the treatment of A1 and A2 which both occupy approximately half the residue classes mod p for large primes p, and an example of such a set are the squares. Motivated by this obstacle, we study additive decompositions of sums of squares.

Although the set of sums of two squares can be written as a sumset in uncountably many different ways, any non-trivial sumset decomposition must consist of two sets of roughly equal size: We show that if |A1|, |A2| > 1 and A1+A2 is the set of squares, then sqrt(x)/(log x)^(7/2) << |A1 ∩ [1,x]|, |A2 ∩ [1,x]| << sqrt(x)*(log x)^3. The key ingredient of our proof is a new large sieve bound for sets which are missing various residue classes modulo prime squares. That bound is a significant improvement over the corresponding Johnsen-Selberg sieve inequality for certain interesting residue class configurations modulo prime squares. This is joint work with Christian Elsholtz.

18:00: Dinner