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http://dx.doi.org/10.4134/BKMS.b180043

ON THE DENOMINATOR OF DEDEKIND SUMS  

Louboutin, Stephane R. (Aix Marseille Universite)
Publication Information
Bulletin of the Korean Mathematical Society / v.56, no.4, 2019 , pp. 815-827 More about this Journal
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;
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