A proof of the Landsberg–Schaar relation by finite methods


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.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.


  1. 1.

    Bellman, R.: A Brief Introduction to Theta Functions. Holt, Rinehart and Winston, Inc., New York (1961)

  2. 2.

    Berndt, B.C., Evans, R.J., Williams, K.S.: Gauss and Jacobi Sums. Wiley, New York (1998)

  3. 3.

    Boylan, H., Skoruppa, N.-P.: A quick proof of reciprocity for Hecke Gauss sums. J. Number Theory. 133, 110–114 (2013)

  4. 4.

    Dickson, L.E.: Introduction to the Theory of Numbers. Dover Publications, New York (1957)

  5. 5.

    Estermann, T.: On the sign of the Gaussian sum. J. Lond. Math. Soc. 2, 66–67 (1945)

  6. 6.

    Gauss, C.F.: Summatio quarandum serierium singularium. Comment. Soc. Reg. Sci. Gottingensis 1. (1811)

  7. 7.

    Hecke, E.: Lectures on the Theory of Algebraic Numbers. Springer, New York (1981)

  8. 8.

    Husemoller, D., Milnor, J.: Symmetric Bilinear Forms. Ergeb. Math. Grenzgeb, vol. 73. Springer, New York (1971)

  9. 9.

    Landsberg, G.: Zur Theorie der Gaussschen Summen und der linearen Transformation der Thetafunctionen. J. Reine Angew Math. 111, 234–253 (1893)

  10. 10.

    Murty, M.R., Pacelli, A.: Quadratic reciprocity via theta functions, Ramanujan Math. Society Lecture Notes, vol. 1, pp. 107–116 (2005)

  11. 11.

    Murty, M.R., Pathak, S.: Evaluation of the quadratic Gauss sum. Math. Stud. 86, 139–150 (2017)

  12. 12.

    Schaar, M.: Mémoire sur la théorie des résidus quadratiques Acad. R. Sci. Lett. Beaux Arts Belgique 24. (1850)

  13. 13.

    Sylvester, J.J.: Question 7382. In: Mathematical Questions with their solutions, from the “Educational Times”. Hodgson, vol. 41, p. 21. (1884)

Download references


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

Correspondence to Ben Moore.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Moore, B. A proof of the Landsberg–Schaar relation by finite methods. Ramanujan J (2020).

Download citation


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

Mathematics Subject Classification

  • 11L05