Euler-Kronecker constants: from Ramanujan to Ihara (partly joint work with Kevin Ford and Florian Luca)

Zeta functions usually have a pole, say at s=n, and a residue there that
provides a lot of information about the associated object. The next step is
to consider the constant in the Taylor series around s=n. For the Riemann
zeta function
this gives the Euler constant. The ratio of the constant and the residue
is called the Euler-Kronecker
constant. Lately this constant has been intensively studied by
mathematicians such as Y. Ihara,
Kumar Murty and Tsfasman (especially for the Dedekind zeta function of a
cyclotomic number field).
  In my talk I will discuss various contexts in which the Euler-Kronecker
constants show up and
discuss the truth/falsity (*) of some conjectures of Ramanujan and Ihara.

(*) Come to the talk to find out which case applies.