My research focuses on the properties of the distribution of prime numbers.
Primes are among the oldest and most important concepts in mathematics and have been studied for millennia. Many conjectures concerning their distribution, like Goldbach's conjecture and the Twin Prime conjecture, have held up astonishingly well over the centuries, as far as numerical evidence is concerned. However, despite plenty of active research in the area and several promising results in the past couple of decades, proofs of such conjectures still seem impossibly far out of reach - the disparity between what we believe to be true and what we know to be true remains very large.
As such, I am particularly interested in methods which combine the following two branches of number theory: The complex analytic approach from classical analytic number theory, which counts primes by estimating complex integrals of special functions, known as Dirichlet series, and has led to key results, such as the prime number theorem, but is often limited by a lack of strong enough bounds on Dirichlet series. And the combinatorial approach from sieve theory, sifting out numbers with many prime factors from certain sets, which has provided good bounds on the number of integers with only few prime factors, but struggles with detecting primes themselves.
A combination of these two different approaches can compensate for their individual deficiencies and give some improvements over results obtained via classical methods. Recently, I used one such method, known as Harman’s sieve, to improve an upper bound on the mean square gap between primes.
Merton is where my mathematical journey began, as an undergraduate in 2016. It has been really wonderful both to get taught and to teach here, and I am incredibly grateful to the college for all the support and opportunities it has provided me with.