A New Method for Lower Bounds of L-functions

In 1899, de la Vallee Poussin extended his method of proving the
Prime Number Theorem to showing that the Riemann zeta function has
a zero-free region of the form
$$
(\sigma) +it: \sigma > \frac{(c)}{(\log(|t|+2))}.
$$
Such lower bounds are expected to hold for any automorphic L-function.
In particular, we have a “standard” zero-free region for $L(s,\pi)$
for any cuspidal representation $\pi$ of $GL(n,{\bf A})$. Our goal
is to obtain a similar region for any $\pi$ and $r$ when $L(s,\pi,r)$
belongs to the Langlands-Shahidi list. Our new methods involve the
Maass-Selberg relations and the computation of Fourier coefficients of
Eisenstein series.

This is joint work with Eray Lapid and Peter Sarnak.