On the mean value of Dedekind sum weighted by the quadratic Gauss sum

Tingting Wang; Wenpeng Zhang

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 2, page 357-367
  • ISSN: 0011-4642

Abstract

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Various properties of classical Dedekind sums S ( h , q ) have been investigated by many authors. For example, Wenpeng Zhang, On the mean values of Dedekind sums, J. Théor. Nombres Bordx, 8 (1996), 429–442, studied the asymptotic behavior of the mean value of Dedekind sums, and H. Rademacher and E. Grosswald, Dedekind Sums, The Carus Mathematical Monographs No. 16, The Mathematical Association of America, Washington, D.C., 1972, studied the related properties. In this paper, we use the algebraic method to study the computational problem of one kind of mean value involving the classical Dedekind sum and the quadratic Gauss sum, and give several exact computational formulae for it.

How to cite

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Wang, Tingting, and Zhang, Wenpeng. "On the mean value of Dedekind sum weighted by the quadratic Gauss sum." Czechoslovak Mathematical Journal 63.2 (2013): 357-367. <http://eudml.org/doc/260618>.

@article{Wang2013,
abstract = {Various properties of classical Dedekind sums $S(h, q)$ have been investigated by many authors. For example, Wenpeng Zhang, On the mean values of Dedekind sums, J. Théor. Nombres Bordx, 8 (1996), 429–442, studied the asymptotic behavior of the mean value of Dedekind sums, and H. Rademacher and E. Grosswald, Dedekind Sums, The Carus Mathematical Monographs No. 16, The Mathematical Association of America, Washington, D.C., 1972, studied the related properties. In this paper, we use the algebraic method to study the computational problem of one kind of mean value involving the classical Dedekind sum and the quadratic Gauss sum, and give several exact computational formulae for it.},
author = {Wang, Tingting, Zhang, Wenpeng},
journal = {Czechoslovak Mathematical Journal},
keywords = {Dedekind sum; quadratic Gauss sum; mean value; identity; Dedekind sum; quadratic Gauss sum; mean value; class number; quadratic field},
language = {eng},
number = {2},
pages = {357-367},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the mean value of Dedekind sum weighted by the quadratic Gauss sum},
url = {http://eudml.org/doc/260618},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Wang, Tingting
AU - Zhang, Wenpeng
TI - On the mean value of Dedekind sum weighted by the quadratic Gauss sum
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 2
SP - 357
EP - 367
AB - Various properties of classical Dedekind sums $S(h, q)$ have been investigated by many authors. For example, Wenpeng Zhang, On the mean values of Dedekind sums, J. Théor. Nombres Bordx, 8 (1996), 429–442, studied the asymptotic behavior of the mean value of Dedekind sums, and H. Rademacher and E. Grosswald, Dedekind Sums, The Carus Mathematical Monographs No. 16, The Mathematical Association of America, Washington, D.C., 1972, studied the related properties. In this paper, we use the algebraic method to study the computational problem of one kind of mean value involving the classical Dedekind sum and the quadratic Gauss sum, and give several exact computational formulae for it.
LA - eng
KW - Dedekind sum; quadratic Gauss sum; mean value; identity; Dedekind sum; quadratic Gauss sum; mean value; class number; quadratic field
UR - http://eudml.org/doc/260618
ER -

References

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  2. Apostol, T. M., Modular Functions and Dirichlet Series in Number Theory, Springer, New York (1976). (1976) Zbl0332.10017MR0422157
  3. Conrey, J. B., Fransen, E., Klein, R., Scott, C., 10.1006/jnth.1996.0014, J. Number Theory 56 (1996), 214-226. (1996) Zbl0851.11028MR1373548DOI10.1006/jnth.1996.0014
  4. Hua, L. K., Introduction to Number Theory, Science Press, Peking Chinese (1964). (1964) Zbl0221.10002MR0194380
  5. Jia, Ch., 10.1006/jnth.2000.2580, J. Number Theory 87 (2001), 173-188. (2001) Zbl0976.11044MR1824141DOI10.1006/jnth.2000.2580
  6. Rademacher, H., On the transformation of log η ( τ ) , J. Indian Math. Soc., n. Ser. 19 (1955), 25-30. (1955) Zbl0064.32703MR0070660
  7. Rademacher, H., Grosswald, E., Dedekind Sums, The Carus Mathematical Monographs No. 16, The Mathematical Association of America, Washington, D.C. (1972). (1972) Zbl0251.10020MR0357299
  8. Weil, A., 10.1073/pnas.34.5.204, Proc. Natl. Acad. Sci. USA 34 (1948), 204-207. (1948) Zbl0032.26102MR0027006DOI10.1073/pnas.34.5.204
  9. Zhang, W., 10.1023/A:1006724724840, Acta Math. Hung. 86 (2000), 275-289. (2000) Zbl0963.11049MR1756252DOI10.1023/A:1006724724840
  10. Zhang, W., 10.5802/jtnb.179, J. Théor. Nombres Bordx. 8 (1996), 429-442. (1996) Zbl0871.11033MR1438480DOI10.5802/jtnb.179

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