A proof of the Landsberg–Schaar relation by finite methods

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.

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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.

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Correspondence to Ben Moore.

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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

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Keywords

  • Gauss sums
  • Quadratic reciprocity
  • Landsberg–Schaar
  • Hecke reciprocity

Mathematics Subject Classification

  • 11L05