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On the mean values of Dedekind sums and Hardy sums

Huaning Liu
대한수학회 (The Korean Mathematical Society)|2009.01|대한수학회지 (Journal of the KMS)|제46권 제1호|pp. 187-213

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참고문헌

키워드

asymptotic formulaDedekind sumsmean valueHardy sums
주제분류수학

초록

For a positive integer $k$ and an arbitrary integer $h$, the classical Dedekind sums $s(h, k)$ is defined by <수식> where <수식>J. B. Conrey et al proved that <수식> For m≥2, C. Jia reduced the error terms to O(k^{2m-1}). While for m=1, W. Zhang showed <수식> In this paper we give some formulae on the mean value of the Dedekind sums and and Hardy sums, and generalize the above results.

Abstract

For a positive integer $k$ and an arbitrary integer $h$, the classical Dedekind sums $s(h, k)$ is defined by <수식> where <수식>J. B. Conrey et al proved that <수식> For m≥2, C. Jia reduced the error terms to O(k^{2m-1}). While for m=1, W. Zhang showed <수식> In this paper we give some formulae on the mean value of the Dedekind sums and and Hardy sums, and generalize the above results.

참고문헌 (22)

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  9. R. Dedekind, "Erlauterungen zu der Riemannschen Fragmenten uber die Grenzfalle der elliptischen Funktionen", Gesammelte Math. Werke, 1, pp.159~173 (1930)
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  11. R. R. Hall, "Dedekind sums and continued fractions", Acta Arith., 63(1), pp.79~90 (1993)
  12. G. H. Hardy, "An Introduction to the Theory of Numbers, Fifth edition" (1979), The Clarendon Press
  13. C. Jia, "On the mean value of Dedekind sums", J. Number Theory, 87(2), pp.173~188 (2001)
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  15. M. R. Pettet, "Three-term relations for Hardy sums", Num- ber Theory, 25(3), pp.328~339 (1987)
  16. J. Pommersheim, "Toric varieties, lattice points and Dedekind sums", Math. Ann., 295(1), pp.1~24 (1993)
  17. H. Rademacher, "Dedekind sums, The Carus Mathematical Mono- graphs", The Mathematical Association of America, (16) (1972)
  18. R. Sitaramachandrarao, "Dedekind and Hardy sums", Acta Arith., 48(4), pp.325~340 (1987)
  19. H. Walum, "An exact formula for an average of L-series", Illinois J. Math., 26(1), pp.1~3 (1982)
  20. W. Zhang, "On the mean values of Dedekind sums", J. Theor. Nombres Bordeaux, 8(2), pp.429~442 (1996)
  21. W. Zhang, "A note on the mean square value of the Dedekind sums", Acta Math. Hungar., 86(4), pp.275~289 (2000)
  22. W. Zhang, "A sum analogous to the Dedekind sum and its mean value formula", J. Number Theory, 89(1), pp.1~13 (2001)

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