On the distribution of extreme values of zeta and $L$-functions in the strip $1/2<σ<1$

Details On the distribution of extreme values of zeta and $L$-functions in the strip $1/2<σ<1$ On the distribution of extreme values of zeta and $L$-functions in the strip $1/2<σ<1$

2010-05-25

arxiv.org/abs/1005.4640v2

Mathematics

Number Theory

45 pages. To appear in Int. Math. Res. Not

Scientific paper

We study the distribution of large (and small) values of several families of $L$-functions on a line $\text{Re(s)}=\sigma$ where $1/2<\sigma<1$. We consider the Riemann zeta function $\zeta(s)$ in the $t$-aspect, Dirichlet $L$-functions in the $q$-aspect, and $L$-functions attached to primitive holomorphic cusp forms of weight $2$ in the level aspect. For each family we show that the $L$-values can be very well modeled by an adequate random Euler product, uniformly in a wide range. We also prove new $\Omega$-results for quadratic Dirichlet $L$-functions (predicted to be best possible by the probabilistic model) conditionally on GRH, and other results related to large moments of $\zeta(\sigma+it)$.

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