An asymptotic distribution for |L' /L(1, χ)|

Let χ be a Dirichlet character modulo q, let L(s, χ) be the attached Dirichlet L-
function, and let L'(s, χ) denotes its derivative with respect to the complex variable s. The
value L(1, χ) has received considerable attention, due to its algebraical
or geometrical interpretation. Let us mention, in particular, the Birch and Swinnerton-Dyer
conjectures, the Kolyvagin Theorem and the Gross-Stark conjecture. Less is known about
L '/L evaluated also at the point s = 1, though these values are known to be fundamental
in studying the distribution of primes since Dirichlet in 1837. In this talk, we show that the
values |L' /L(1, χ)| behave according to a distribution law. The key to this result is to give an
asymptotic formula of the 2k-th power mean value of |L' /L(1, χ)| when χ ranges over
the primitive Dirichlet characters modulo q for q a prime.