The PIMS Postdoctoral Fellow Seminar: Emily Quesada-Herrera

In 1859, Bernhard Riemann published his seminal (and only) article in analytic number theory, where he showed that the distribution of prime numbers is connected to the distribution of the complex zeros of the zeta function that bears his name. Studying it as a meromorphic function, he made his famous, unsolved Hypothesis regarding these zeros.

We will discuss some of these topics and their background. An important tool is the Approximate Functional Equation, which approximates zeta in a crucial part of the complex plane - where the interesting zeros are. We’ll see how, after a few intermediate results, this leads to information about both zeros and primes. Important, yet technical tools involve understanding how certain exponential sums can be controlled.

Explicit estimates are those where all constants - usually left implicit in error terms - are given as precisely as possible. Based on joint work with Natasha Dhiman and Habiba Kadiri (University of Lethbridge), we give a new, explicit version of the approximate functional equation, with improved constants in the error terms.