## Number Theory - Siddhi Pathak (Queen's University)

### Tuesday, January 15th, 2019

**Time:** 1:00-2:00 p.m. **Place:** Jeffery Hall 422

**Speaker:** Siddhi Pathak (Queen's University)

**Title:** Dedekind zeta function at odd positive integers.

**Abstract:** Let $\zeta(s)$ denote the Riemann zeta-function. Thanks to Euler's evaluation and Lindemann's theorem on transcendence of $\pi$, we understand that $\zeta(2k)$ is transcendental for any positive integer $k$. However, the arithmetic nature of the values of $\zeta(s)$ at odd positive integers remains a mystery. Recently, significant progress was made concerning the irrationality of these values, with perhaps the most remarkable theorem being that infinitely many of $\zeta(2k+1)$ are irrational, which was shown by T. Rivoal in 2000.

Similarly, one can inquire regarding the arithmetic nature of values of the Dedekind zeta-function $\zeta_K(s)$ attached to a number field $K$. When $K$ is totally real, the values $\zeta_K(2k)$ were proven to be algebraic multiples of powers of $\pi$ by Siegel and Klingen, independently. But this question remains unsolved in all other cases. In this talk, we discuss how our current knowledge allows us to deduce certain irrationality results for $\zeta_K(2k+1)$, where $K$ is an imaginary quadratic field. This is joint work with Prof. M. Ram Murty.