# Subconvexity Bounds and Simple Zeros of Modular L-Functions (15rit201)

## Organizers

Andrew Booker (University of Bristol)

(University of Mississippi)

(University of Lethbridge)

## Description

The Banff International Research Station will host the "Subconvexity bounds and simple zeros of modular L-functions" workshop in Banff from May 24 to May 31, 2015.

Analytic number theory is the branch of number theory that studies the natural numbers $\{ 1,2,3,4, \ldots, \}$ and the prime numbers $\{ 2,3,5,7, \ldots, \}$ via analysis and complex analysis. (The prime numbers are those natural numbers with exactly two positive divisors.) Prime numbers have been studied for thousands of years, by all civilizations. They are the building blocks of the natural numbers, since any natural number factors into a product of primes. Despite their simple definition, their occurrence among the other integers remains mysterious and the object of important conjectures in mathematics. In the last century, prime numbers have become very important for governments and industry.

The field of analytic number theory arose after publications of Dirichlet in 1837 and Riemann in 1859. In these works the authors showed that prime numbers can be studied via certain complex valued functions called L-functions. They showed that the values and zeros of L-functions influenced the behaviour of primes and primes in arithmetic progressions. In the last century, it became clear that statistical properties of L-functions were important in many arithmetic questions. Consequently, it is desirable to prove statistical results regarding the values of L-functions. This research in teams meeting shall focus on studying modular form L-functions $L_f(s)$ which are a special subclass of L-functions. We shall attempt to prove a non-trivial upper bound for $L_f(s)$ on the critical line and we shall also attempt to prove a good lower bound for the number of simple zeros of $L_f(s)$.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).