On the denominator of Dedekind sums
Bull. Korean Math. Soc. 2019 Vol. 56, No. 4, 815-827
Published online July 9, 2019
Printed July 31, 2019
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
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