University of Illinois, Urbana-Champaign
Number Theory Seminar
Fall 2026

This is the homepage for the UIUC Number Theory Seminar.

We will meet in Sidney Lu MEB 3101.

Seminars from 2012-2022 are archived here.

September 15: Julia Stadlmann (UIUC)
Title: Bounded gaps between primes
Abstract: The Polymath project proved that there are infinitely many prime gaps which are no larger than 246. In this talk I will discuss how the Bombieri-Vinogradov theorem can be combined with newer equidistribution estimates for smooth moduli to obtain an improvement on this result. A key ingredient is a new choice of support for the functions appearing in the corresponding optimization problem for the GPY sieve. To choose this support, we produce various equidistribution conditions for moduli with a large smooth factor, and reduce the calculation of relevant integrals to repeated matrix multiplication.

September 22: Alisa Sedunova (Purdue University)
Title: On the hyperbolic prime number theorem
Abstract: Friedlander and Iwaniec proved that the number of points of the orbit $\{\gamma i \colon \gamma\in\SL_2(\Z)\}$ lying at a distance $p-2$, $p\le x$ prime, from the origin $i$ of the upper half-plane has the order of magnitude $x/\log x$: the upper bound being unconditional, while the lower one is under a strong hypothesis on the distribution of primes in arithmetic progressions. We consider instead square-free distances and prove unconditionally an asymptotic formula with order of magnitude $x$. Employing linear sieve, we also show unconditionally that square-free distances $n$ with at most $7$ prime factors contribute $\gg x/\log x$. Finally, under a hypothesis on the level of distribution $x^{\theta}$ of the sequence $r(n-2)r(n+2)$ in arithmetic progressions (here $r(n)$ is the number of ways to write $n$ as a sum of two squares), the number $7$ can be replaced by $\lfloor2/\theta\rfloor$, which gives $4$ for the analogue of the Bombieri--Vinogradov theorem and $2$ for any $\theta > 2/3$.

September 29: Abhishek Jha (UIUC)
Title: The Poisson Tail Conjecture for primes in short intervals
Abstract: Fifty years ago, Gallagher showed that prime counts in intervals of length λ log(x) are asymptotically Poisson for fixed λ, assuming the Hardy-Littlewood conjectures. In this talk, I will discuss what happens when λ is allowed to grow with x. Under suitable uniform Hardy--Littlewood hypotheses, we obtain a growing range in which the Poisson prediction remains valid and identify a phase transition beyond which it breaks down. The talk will focus on the probabilistic and sieve-theoretic ideas underlying these results.

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University of Illinois, Urbana-ChampaignNumber Theory SeminarFall 2026