St\'ephane R. Louboutin Aix Marseille Universit\'e

Abstract : It is well known that the denominator of the Dedekind sum $s(c,d)$ divides $2\gcd (d,3)d$ and that no smaller denominator independent of $c$ can be expected. In contrast, here we prove that we usually get a smaller denominator in $S(H,d)$, the sum of the $s(c,d)$'s over all the $c$'s in a subgroup $H$ of order $n>1$ in the multiplicative group $({\mathbb Z}/d{\mathbb Z})^*$. First, we prove that for $p>3$ a prime, the sum $2S(H,p)$ is a rational integer of the same parity as $(p-1)/2$. We give an application of this result to upper bounds on relative class numbers of imaginary abelian number fields of prime conductor. Finally, we give a general result on the denominator of $S(H,d)$ for non necessarily prime $d$'s. We show that its denominator is a divisor of some explicit divisor of $2d\gcd (d,3)$.

Keywords : Dedekind sum, Dirichlet character, mean square value $L$-functions, relative class number