aequationes mathematicae

, Volume 30, Issue 1, pp 134–141 | Cite as

The triplication formula for Gauss sums

  • John Greene
  • Dennis Stanton
Research Papers


A new proof of the triplication formula for Gauss sums is given. It mimics an old proof of the analogous result for gamma functions. The techniques are formal and rely upon the character properties of fields. A new character sum evaluation is given.

AMS (1980) subject classification

Primary 10G05, 33A15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Askey, R.,The q-gamma and q-beta functions. Applicable Anal.8 (1978/79), 125–141.Google Scholar
  2. [2]
    Boyarsky, M.,p-adic gamma functions and Dwork cohomology. Trans. Amer. Math. Soc.257 (1980), 359–568.Google Scholar
  3. [3]
    Davenport, H. andHasse, H.,Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklishen Fällen. J. Reine Angew. Math.172 (1934), 151–182.Google Scholar
  4. [4]
    Edwards, J.,Integral calculus, vol. 2. Macmillan, London, 1922.Google Scholar
  5. [5]
    Erdélyi, A.,Higher transcendental functions, vol. I. McGraw Hill, New York-Toronto-London, 1953.Google Scholar
  6. [6]
    Gel'fand, I., Graev, M., andPyateckii-sapiro, I.,Representation theory and automorphic functions. Saunders, 1969.Google Scholar
  7. [7]
    Gross, B. andKoblitz, N.,Gauss sums and the p-adic γ-function. Ann. of Math. (2)109 (1979), 569–581.Google Scholar
  8. [8]
    Ireland, K. andRosen, M.,Elements of number theory. Bogden and Quigley, Tarrytown-on-Hudson, 1972.Google Scholar
  9. [9]
    Jacobi, C.,Über die Kreistheilung und ihre Anwendung auf die Zahlentheorie. J. Reine Angew. Math.30 (1846), 166–182.Google Scholar
  10. [10]
    Koblitz, N.,q-extensions of the p-adic gamma function. Trans. Amer. Math. Soc.260 (1980), 449–457.Google Scholar
  11. [11]
    Koblitz, N.,q-extensions of the q-adic gamma function, II. Trans. Amer. Math. Soc.273 (1982), 111–130.Google Scholar
  12. [12]
    Liouville, J.,Détermination des valeurs d'une class remarquable d'intégrales definies multiples, et demonstration nouvelle d'une cèlébre formule de Gauss concernant les fonctions gamma de Legendre. J. Math. (2nd series)1 (1856), 82–88.Google Scholar
  13. [13]
    Sears, D.,On the transformation theory of basic hypergeometric functions. Proc. London Math. Soc. (2)53 (1951), 158–180.Google Scholar
  14. [14]
    Yamamoto, K.,On a conjecture of Hasse concerning multiplicative relations of Gaussian sums. J. Combin. Theory Ser.A 1 (1966), 476–489.Google Scholar

Copyright information

© Birkhäuser Verlag 1986

Authors and Affiliations

  • John Greene
    • 1
  • Dennis Stanton
    • 1
  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisU.S.A.

Personalised recommendations