USask Mathematics & Statistics Colloquium: Alia Hamieh

Abstract: Many of the central questions in number theory concern the distribution of arithmetic objects and the size of the quantities that encode them. Analytic number theory approaches such questions through the study of $L$-functions, generating series that encode arithmetic information and connect number theory with many other areas of mathematics. Questions concerning the size, zeros, and special values of $L$-functions are among the main problems of the subject. Since individual values of $L$-functions are often very difficult to understand, a fruitful approach is to study them on average. This leads naturally to the study of moments. The classical example is the $2k$-th moment of the Riemann zeta function
\[
I_k(T)=\int_0^T \left|\zeta\left(\frac12+it\right)\right|^{2k}\,dt.
\]
The study of these moments was originally motivated by questions concerning the size of the zeta function in the critical strip, and has since developed into a central theme in analytic number theory. Despite substantial progress, asymptotic formulae for $I_k(T)$ are known only in a few cases.

In this talk, we will begin with classical results and conjectures for moments of the Riemann zeta function and discuss some of their applications. We will then turn to moments in families of $L$-functions associated with modular forms, and explain how such averages can be used to obtain results on nonvanishing and subconvexity. Finally, we will discuss moments of Rankin-Selberg $L$-functions, including recent work on averages of the central values $L(1/2,f\otimes g)$ over families of modular forms.