A reciprocity congruence for an analogue of the Dedekind sum and quadratic reciprocity

Jeffrey L. Meyer

Journal de théorie des nombres de Bordeaux (2000)

  • Volume: 12, Issue: 1, page 93-101
  • ISSN: 1246-7405

Abstract

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In the transformation formulas for the logarithms of the classical theta-functions, certain sums arise that are analogous to the Dedekind sums in the transformation of the logarithm of the eta-function. A new reciprocity law is established for one of these analogous sums and then applied to prove the law of quadratic reciprocity.

How to cite

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Meyer, Jeffrey L.. "A reciprocity congruence for an analogue of the Dedekind sum and quadratic reciprocity." Journal de théorie des nombres de Bordeaux 12.1 (2000): 93-101. <http://eudml.org/doc/248501>.

@article{Meyer2000,
abstract = {In the transformation formulas for the logarithms of the classical theta-functions, certain sums arise that are analogous to the Dedekind sums in the transformation of the logarithm of the eta-function. A new reciprocity law is established for one of these analogous sums and then applied to prove the law of quadratic reciprocity.},
author = {Meyer, Jeffrey L.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {theta functions; reciprocity law},
language = {eng},
number = {1},
pages = {93-101},
publisher = {Université Bordeaux I},
title = {A reciprocity congruence for an analogue of the Dedekind sum and quadratic reciprocity},
url = {http://eudml.org/doc/248501},
volume = {12},
year = {2000},
}

TY - JOUR
AU - Meyer, Jeffrey L.
TI - A reciprocity congruence for an analogue of the Dedekind sum and quadratic reciprocity
JO - Journal de théorie des nombres de Bordeaux
PY - 2000
PB - Université Bordeaux I
VL - 12
IS - 1
SP - 93
EP - 101
AB - In the transformation formulas for the logarithms of the classical theta-functions, certain sums arise that are analogous to the Dedekind sums in the transformation of the logarithm of the eta-function. A new reciprocity law is established for one of these analogous sums and then applied to prove the law of quadratic reciprocity.
LA - eng
KW - theta functions; reciprocity law
UR - http://eudml.org/doc/248501
ER -

References

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  1. [1] B.C. Berndt, Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan. J. Reine Angew. Math.303/304 (1978), 332-365. Zbl0384.10011MR514690
  2. [2] L.A. Goldberg, Transformations of theta-functions and analogues of dedekind sums. Ph.D. thesis, Thesis, University of Illinois, Urbana, 1981. 
  3. [3] H. Rademacher E. Grosswald, Dedekind sums. Carus Math. Monogr., vol. 16, Mathematical Association of America, Washington, D.C., 1972. Zbl0251.10020MR357299
  4. [4] H. Montgomery I. Niven, H. Zuckerman, An introduction to the theory of numbers. 5th ed., John Wiley and Sons, New York, 1991. Zbl0742.11001MR1083765
  5. [5] J. Lewittes, Analytic continuation of Eisenstein series. Trans. Amer. Math. Soc.171 (1972), 469-490. Zbl0253.10022MR306148
  6. [6] J.L. Meyer, Analogues of dedekind sums. Ph.D. thesis, University of Illinois, Urbana, 1997. Zbl0889.11013
  7. [7] _Properties of certain integer-valued analogues of Dedekind sums. Acta Arithmetica82 (1997), 229-242. Zbl0889.11013MR1482888
  8. [8] H. Rademacher, Topics in analytic number theory. Springer-Verlag, New York, 1973. Zbl0253.10002MR364103

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