Student Analytic Number Theory

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…

Abstract

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

Abstract: In 1970, Erdos and Sarkozy wrote a joint paper studying sequences of integers a1 < a2 < . . . having what they called property P, meaning that no a_i divides the sum of two larger a_j , a_k. In the paper, it was stated that the authors believed that a subset A ⊂ [n]…

The functional equation of the Estermann function (the additive twist of zeta(s)^2) is morally equivalent to the Voronoi summation formula. This can be used, among other things, to study the correlations of the divisor counting function d(n). Motivated by the divisor correlation problem in the…

In this work, we establish a clear-cut criterion for determining when an affine subspace of R^n is extremal. Specifically, we investigate the behavior of the diophantine exponent of an affine subspace and determine when it is minimal (equal to the Dirichlet exponent 1/n). Our…