Student Analytic Number Theory

Suppose A is a subset of the natural numbers with positive density. A classical result in additive combinatorics, Szemerédi’s theorem, states that for each positive integer k, A must have an arithmetic progression of nonzero common difference of length k.In this talk, we shall discuss various…

Quantum unique ergodicity (QUE) describes the equidistribution of the L2-mass of eigenfunctions of the Laplacian as their eigenvalues approach infinity. My focus lies on a specific variant known as holomorphic QUE, which concerns the distribution of the L2-mass of normalized…

Let r_0(n) be the number of representations of n as a sum of two squares, and r_1(n) count the number of representations of n as a sum of an integer square and a prime square. The asymptotic formulas for the moments of r_0(n), with k greater than 1 summed over n up to x are well-known via…

The spectral theory of automorphic forms finds remarkable applications in analytic number theory. Notably, it is utilised in results concerning the distribution of primes in large arithmetic progressions and in questions on variants of the fourth moment of the zeta function. Traditionally, these…

It was asked by E. Szemer\'edi if, for a finite set $A\subset \mathbf{Z}$, one can improve estimates for $\max\{|A+A|,|A\cdot A|\}$, under the constraint that all integers involved have a bounded number of prime factors -- that is, each $a\in A$ satisfies $\omega(a)\leq k$. In our paper we show…

The study of exponential sums with multiplicative coefficients is classical in analytic number theory, yet our understanding of them is far from complete. This is unsurprising, seeing as multiplicative functions alone are often difficult objects to grasp. However, in recent years, our…

We count squarefree numbers in short intervals [X, X+H] for H > X^{1/5 - \delta}, where \delta > 0 is some absolute constant. This improves on the exponent 1/5 shown by Filaseta and Trifonov in 1992. 

In improving bounds on the number of integers in a short interval divisible by…

Abstract: Inspired by a recent breakthrough work of Gorodetsky, Matomaki, Radziwill and Rodgers on variance of squarefree numbers in short intervals, a similar study for variance of squarefull numbers in short intervals was carried out. In this talk, I will highlight some of the journeys in this…

In this talk we will discuss the behaviour of the Riemann zeta on the critical line, and in particular, its correlations in various ranges. We will prove a new result for correlations of squares, where shifts may be up to size $T^{3/2-\varepsilon}$. We will also explain how this result relates…

Abstract