On the denominator of Dedekind sums
Bull. Korean Math. Soc.
Published online July 9, 2019
Stéphane Louboutin
Uinersité Aix-Marseille
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
MSC numbers : 11F20, 11M20, 11R29
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