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 $\text{Re}(s) = 1/2$ is a central problem in the analytic theory of $L$-functions. The functional equation and the Phragmen-Lindel\"of 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.
Indian Statistical Institute, Kolkata
August 23, 2018
Sub-convexity problems: Some history and recent developments: Bounding automorphic $L$-functions on the critical line $\text{Re}(s) = 1/2$ is a central problem in the analytic theory of $L$-functions. The functional equation and the Phragmen-Lindel\"of 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 general audience and we shall be explicitly defining the relevant terms.