Abstract
The Landsberg–Schaar relation is a classical identity between quadratic Gauss sums, often used as a stepping stone to prove the law of quadratic reciprocity. The Landsberg–Schaar relation itself is usually proved by carefully taking a limit in the functional equation for Jacobi’s theta function. In this article, we present a direct proof, avoiding any analysis.
Similar content being viewed by others
References
Bellman, R.: A Brief Introduction to Theta Functions. Holt, Rinehart and Winston, Inc., New York (1961)
Berndt, B.C., Evans, R.J., Williams, K.S.: Gauss and Jacobi Sums. Wiley, New York (1998)
Boylan, H., Skoruppa, N.-P.: A quick proof of reciprocity for Hecke Gauss sums. J. Number Theory. 133, 110–114 (2013)
Dickson, L.E.: Introduction to the Theory of Numbers. Dover Publications, New York (1957)
Estermann, T.: On the sign of the Gaussian sum. J. Lond. Math. Soc. 2, 66–67 (1945)
Gauss, C.F.: Summatio quarandum serierium singularium. Comment. Soc. Reg. Sci. Gottingensis 1. https://gdz.sub.uni-goettingen.de/id/PPN602151724 (1811)
Hecke, E.: Lectures on the Theory of Algebraic Numbers. Springer, New York (1981)
Husemoller, D., Milnor, J.: Symmetric Bilinear Forms. Ergeb. Math. Grenzgeb, vol. 73. Springer, New York (1971)
Landsberg, G.: Zur Theorie der Gaussschen Summen und der linearen Transformation der Thetafunctionen. J. Reine Angew Math. 111, 234–253 (1893)
Murty, M.R., Pacelli, A.: Quadratic reciprocity via theta functions, Ramanujan Math. Society Lecture Notes, vol. 1, pp. 107–116 (2005)
Murty, M.R., Pathak, S.: Evaluation of the quadratic Gauss sum. Math. Stud. 86, 139–150 (2017)
Schaar, M.: Mémoire sur la théorie des résidus quadratiques Acad. R. Sci. Lett. Beaux Arts Belgique 24. https://www.biodiversitylibrary.org/ia/mmoiresdelacad24acad#page/467/mode/1up (1850)
Sylvester, J.J.: Question 7382. In: Mathematical Questions with their solutions, from the “Educational Times”. Hodgson, vol. 41, p. 21. https://ia801409.us.archive.org/5/items/mathematicalque10millgoog/ (1884)
Acknowledgements
The author is extremely grateful to Mike Eastwood for his support and encouragement concerning this article, and most especially for his firm belief that an elementary proof of the Landsberg–Schaar relation should exist! The author would also like to thank Bruce Berndt for reading an earlier draft, Ram Murty for some encouraging remarks, David Roberts for tracking down Gauss’ original evaluation of his eponymous sums and the anonymous referee for suggesting valuable improvements to the article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Moore, B. A proof of the Landsberg–Schaar relation by finite methods. Ramanujan J 53, 653–665 (2020). https://doi.org/10.1007/s11139-019-00195-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-019-00195-4