Arithmetic nature of special values of L-functions

Abstract: The study of L-functions has occupied a center stage in number theory since the work of Riemann and Dirichlet. A standard example of an L-function is the Riemann zeta-function, $\zeta(s)$, given by the series $\sum_{n=1}^{\infty} n^{-s}$ when $\Re(s)>1$. The aim of this talk will be to discuss the question of determining the arithmetic nature (that is, rational/irrational and algebraic/transcendental) of values of L-functions at positive integers. For example, it is expected that the values $\zeta(m)$ are transcendental for all integers $m >1$. However, the only known cases of this conjecture are the even zeta-values, which Euler had explicitly evaluated in the 1730s. Among the odd zeta-values, Apery proved that $\zeta(3)$ is irrational, whereas the irrationality of the remaining odd zeta-values remains a mystery.

In this talk, we will discuss various facets of this problem. If time permits, we will prove that for a fixed odd positive integer m, all the values $\zeta_K(m)$ are irrational as K varies over imaginary quadratic fields, with at most one possible exception. This is joint work with Ram Murty. This talk will be accessible to a wide audience.

number theory

Audience: researchers in the topic

CMI seminar series

Series comments: Please visit the seminar series homepage for streaming details. Timings of the seminar vary from week to week.