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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 $2d\gcd (d,3)$.
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참고문헌 (11)
참고문헌 신청
T. M. Apostol / 1976 / Modular Functions and Dirichlet Series in Number Theory / Springer-Verlag
P. J. Cho / 2012 / Dihedral and cyclic extensions with large class numbers / J. Theor. Nombres Bordeaux 24(3):583~603
Stéphane Louboutin / 1993 / Quelques formules exactes pour des moyennes de fonctions L de Dirichlet / Canadian Mathematical Bulletin 36(2):190~196
Stéphane R. Louboutin / 2016 / Dedekind Sums, Mean Square Value of $L$-Functions at $s=1$ and Upper Bounds on Relative Class Numbers / Bulletin Polish Acad. Sci. Math. 64(2,3):165~174
T. Metsankyla / 1974 / Class Numbers and μ-Invariants of Cyclotomic Fields / Proceedings of the American Mathematical Society 43(2):299~300
P. J. Cho / 2012 / Dihedral and cyclic extensions with large class numbers / J. Theor. Nombres Bordeaux 24(3):583~603
Stéphane Louboutin / 1993 / Quelques formules exactes pour des moyennes de fonctions L de Dirichlet / Canadian Mathematical Bulletin 36(2):190~196
Stéphane R. Louboutin / 2016 / Dedekind Sums, Mean Square Value of $L$-Functions at $s=1$ and Upper Bounds on Relative Class Numbers / Bulletin Polish Acad. Sci. Math. 64(2,3):165~174
T. Metsankyla / 1974 / Class Numbers and μ-Invariants of Cyclotomic Fields / Proceedings of the American Mathematical Society 43(2):299~300
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