Published on *Max Planck Institute for Mathematics* (https://www.mpim-bonn.mpg.de)

Posted in

- Talk [1]

Speaker:

Junxian Li
Affiliation:

MPIM
Date:

Wed, 2019-10-02 14:30 - 15:30 A celebrated theorem of Selberg indicates that the typical size of

$|\zeta(\frac{1}{2}+it)|$ is $\exp( \sqrt{\frac{1}{2}\log \log

T})$. What about extreme values of $\zeta(\frac{1}{2}+it)$? If

we choose $t$ randomly from $[0, T]$, then it has been shown recently

that $\max_{ t\in[0,T]} | \zeta( \frac{1}{2}+it)| \gg \exp (

\sqrt{\frac{2\log T\log\log \log T}{\log \log T}})$. What if we

choose $t$ at some discrete points? In this talk, we are interested in

choosing $t$ to be the imaginary part of the zeros of some other

$L$-functions on the critical line. In particular, under GRH, we can

show that $\max_{L(\rho,\chi)=0, T\leq Im \rho\leq 2T} | \zeta( \rho)| \gg \exp (

c\sqrt{\frac{\log T}{\log \log T}}), c>0$, where $L(s, \chi)$ is a Dirichlet $L$-function.

**Links:**

[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39

[2] https://www.mpim-bonn.mpg.de/node/3444

[3] https://www.mpim-bonn.mpg.de/node/246