##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory

## Dorian Goldfeld

#### Columbia University

## A standard zero free region for Rankin-Selberg L-functions on GL(n)

##### Abstract:

For $n > 1$, let $\pi, \pi'$ be two irreducible cuspidal automorphic representations of GL(n, A) where A denotes the adeles over Q. Let $L(s, \pi \times \pi')$ be the Rankin-Selberg L-function. If one of $\pi$ or $\pi'$ is self dual then it was shown by Moreno and Sarnak that the Rankin-Selberg L-function does not vanish at s = c+it when 1-c is less than a positive fixed constant times a negative power of log(|t| +2). This is also called a standard zero free region. A standard zero free region for the Riemann zeta function was first obtained by de la Vallee Poussin (prime number theorem). Currently, the best known zero free region for Rankin Selberg L-functions on GL(n) (in the non self dual case) is due to Brumley who has proved 1-c is less than a fixed constant times a negative power of |t| +2. In joint work with Xiaoqing Li we obtain a standard zero free region in the non self dual case.

Host: Alina Bucur

### November 6, 2014

### 1:00 PM

### AP&M 7321

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