On the Riemann Zeta-Function, Fourier Optimization and Quadratic Forms

Explore the fascinating intersection of analytic number theory and Fourier analysis in this 49-minute lecture by Emily Quesada Herrera from TU Graz, Austria. Delve into the distribution of values and zeros of the Riemann zeta-function, as well as the distribution of integers and primes represented by quadratic forms. Examine Selberg's central limit theorem and its implications for the logarithm of the Riemann zeta-function on the critical line. Discover recent joint works addressing Berry's 1988 conjecture, conditional on the Riemann Hypothesis and a strong version of the pair correlation conjecture. Investigate a Fourier analysis approach to studying integers and primes represented by binary quadratic forms, tracing back to Fermat's classical problem. Cover topics including Fourier transforms, zeros of the Riemann zeta-function, pair correlation, number variance of zeta zeros, and congruence sums in this comprehensive exploration of advanced mathematical concepts.

Intro
Fourier transform
The zeros of the Riemann zeta-function
Pair correlation: the finer vertical distribution of zeros
Selberg's central limit theorem
Number variance of zeta zeros
Berry's conjecture
Fermat, Euler and beyond!
Congruence sums: a variation of the ellipse problem