대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제56권4호
  • /
  • Pages.815-827
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  • 2019
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  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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ON THE DENOMINATOR OF DEDEKIND SUMS

  • Louboutin, Stephane R. (Aix Marseille Universite)
  • 투고 : 2018.01.12
  • 심사 : 2019.04.25
  • 발행 : 2019.07.31
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초록

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

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