Zeta Functions

September 18 - 22, 2006, Moscow, Russia

Laboratoire J.-V. Poncelet

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Stephen Gelbart

Weizmann Institute, Israel

$L(s,\pi,r)$ in the Critical Strip of an automorphic form on a reductive group

Riemann's zeta-function satisfies a convexity bound on the critical line; moreover, $\zeta(s)$ is non-vanishing in a neighborhood of the edge of the critical strip.

Let $L(s,\pi,r)$ be a Langlands-Shahidi L-function; what generalizations of $\zeta(s)$ can be said about this $L(s,\pi,r)$? Does it satisfy a convexity bound on the critical line? Is it non-vanishing close to the edge of the critical strip? In joint work with Freydoon Shahidi and then Erez Lapid, we have used Eisenstein series to attack these questions. Parts of our proofs appeal to general arguments of ones introduced by Sarnak in 2004 for $\zeta$ and most of our exposition will be spent working out this case.