# Colloquium abstracts

**Saurabh Kumar Singh**

Indian Statistical Institute, Kolkata

August 23, 2018

**Sub-convexity problems: Some history and recent developments**: Bounding automorphic $L$-functions on the critical line $ ext{Re}(s) = 1/2$ is a central problem in the analytic theory of $L$-functions. The functional equation and the Phragmen-Lindel{" o}f principle from complex analysis yield the convexity bound $L(1/2+it,pi)ll C(pi,t)^{1/4+varepsilon}$ where $C(pi,t)$ is the ``analytic conductor" of the $L$-function. Lindel{" o}f hypothesis, which is a consequence of the Grand Riemann Hypothesis (GRH), predicts that the bound $C(pi,t)^varepsilon$ for any $varepsilon>0$. Any bound with exponent smaller than $1/4$ is called a sub-convexity bound. In this context the Weyl exponent $1/6$, which is one-third of the way down from convexity towards Lindel{" o}f, is a known barrier which has been achieved only for a handful of families. First sub-convexity bound is proved by Hardy-Littlewood and Weyl independently for the Riemann zeta function. In this talk we shall talk about some recent developments and new techniques. This talk is meant for a gen