Alladi Ramakrishnan Hall

Ph.D Thesis defense talk

Tapas Chatterjee

IMSc

A conjecture of Milnor asserts that the
$\mathhbb Q$- dimension of the vector space generated by Hurwitz zeta
values $\zeta(k,a/q)$, where $k>1, q>1$ fixed integers and $a$ varies over
co-prime residue classes modulo $q$, is $\varphi(q)$. This conjecture has
been investigated in recent works of Gun, Murty and Rath. In the
first half of the talk, we will discuss various ramifications and
generalizations of the conjecture of Milnor. In the later half, we will
discuss about the existence of infinitely many zeros of certain
generalized Hurwitz zeta functions in their half plane of absolute
convergence. Time permitting, we will discuss about zero-free regions of
such functions and deduce a variant of a conjecture of Erd\"{o}s.