Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2019; 56(4): 815-827

Online first article July 9, 2019      Printed July 31, 2019

https://doi.org/10.4134/BKMS.b180043

Copyright © The Korean Mathematical Society.

On the denominator of Dedekind sums

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

MSC numbers: Primary 11F20; Secondary 11M20, 11R29