Ramanujan Auditorium

The elusive prediction of L-values

Loïc Merel

Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université Paris Cité

An early wonder of our mathematical life happens when we
come across the identity : \sum_{n=1}^\infty 1/n^2=π^2/6.
Even better, in Euler product form, \prod_p 1/(1-1/p^2)=π^2/6, where p
runs through the prime numbers. In the course of the ninetieth
century, it appeared that the (zeta) function of a complex variable
ζ(s)=\sum_{n=1}^\infty 1/n^s, and its variants, is key to understand
some of the subtle laws that govern prime numbers. Thus have the
hearts of analytic number theorists been set beating.

Meanwhile, algebraic number theorists have attempted to understand the
meaning of "π^2/6". And, out of arithmetic geometry, they have
defined a class of variants of ζ: the L-functions, series of the form
\sum_{k=1}^\infty a_k/k^s that can also be expressed as Euler
products. Their valuations at integers produce the mysterious
L-values, that we seek to understand. Well established conjectures now
predict what the correct replacement for π^2/6 should be. But is the
prediction complete? Contrary to what is often believed, not quite.

Explanation for all this, including why elaborate conjectures still
fall short, will not rely on general explanation of what L-functions
are, but on illustrative examples based on elliptic curves and
Dirichlet characters. An intriguing formula involving L-values will be
offered to help reflect on the "not quite".