Student Analytic Number Theory

We will discuss a paper of Tao and Teräväinen.

Abstract: We investigate K-multimagic squares of order N, these are N × N magic squares which remain magic after raising each element to the kth power for all 2 ⩽ k ⩽ K. Given K ⩾ 2, we consider the problem of establishing the smallest integer N(K) for which there exists non-trivial…

We improve 1987 estimates of Patterson for sums of quartic Gauss sums over primes. Our Type-I and Type-II estimates feature new ideas, including use of the quadratic large sieve over the Gaussian quadratic field, and Suzuki's evaluation of the Fourier-Whittaker coefficients of quartic theta…

We will continue discussing the paper of Walsh, Local uniformity through larger scales.

Abstract: Let f be a Steinhaus or Rademacher random multiplicative function. Even though a lot is known about partial sums of f over positive integers due to work of Harper, the exact distribution is still elusive. However, partial sums of f over some restricted sets of positive integers are…

Abstract: We will discuss cancellation of the Liouville function in almost all short intervals, and the Fourier uniformity conjecture.

Abstract: Over the course of his long career, Paul Erdős stated many problems, and proved many results, about the irrationality of series. We survey some of his problems and proofs in this area, and describe recent progress.

We will discuss a couple of recent papers of Walsh.

Note alternate location.

The dispersion method has found an impressive range of applications in analytic number theory, from bounded gaps between primes to the greatest prime factors of quadratic polynomials. The method requires bounding certain sums of Kloosterman sums, using deep inputs from algebraic geometry and the…

I will discuss joint work in progress with N. Arala, J. R. Getz, J. Hou, C.-H. Hsu, and H. Li, concerning a new, nonabelian circle method and its applications to counting problems of a classical flavor.