Description |
Abstract:
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{\" 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 general audience and we shall be
explicitly defining the relevant terms.
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