Number Theory Seminar | Sourabhashis Das | Prime Gaps in Arithmetic Progressions Are Graphic

A sequence of positive integers is called graphic if it arises as the degree sequence of a simple graph. In this talk, we explore a surprising connection between this classical graph-theoretic notion and the distribution of primes.

Let us call a simple graph on \(n \geq 2\) vertices a prime gap graph if its vertex degrees are \(1\) and the first \(n-1\) prime gaps. Growing out of an empirical observation by Toroczkai, he and his collaborators established that a prime gap graph exists for every sufficiently large \(n\), and for every \(n \geq 2\) under the Riemann Hypothesis, and that an infinite sequence of such prime gap graphs can be generated via the degree-preserving growth process---the first naturally occurring infinite sequence of positive integers identified as graphic.

In this talk, we extend these results to prime gaps in arithmetic progressions. Specifically, we show that the gap sequences formed by consecutive primes in a fixed progression \(a \pmod{q}\), with \(\gcd(a,q)=1\), are likewise graphic. The analytical tools required stem from Heath-Brown's work on bounding moments of large prime gaps. We demonstrate these tools in the context of primes in arithmetic progressions and explain how they drive our proof.

This is joint work with Keshav Aggarwal and is a work in progress.

MC 5501