Alladi Ramakrishnan Hall

On holomorphy and special values of Artin $L$-functions

Dhananjaya Sahu

IMSc, a constituent institute of HBNI

Let $K/F$ be a Galois extension of number fields with Galois group $G$.
We begin by examining the descent of real zeros of the Dedekind zeta function
$zeta_K$ to the Dedekind zeta functions of subfields of $K$.
This descent will allow us to get lower bound of residue of $\zeta_K(s)$ at $s = 1$ for certain number fields.
In the next part,
we explore the connection between the holomorphy of the Artin
$L$-functions $L(s, \chi, K/F)$ at a point $s_0$ and order of $\zeta_K(s)$ at $s_0$ for any character $\chi$ of $G$.