Gauss sums can be used to calculate the number of solutions of polynomial equations over finite fields, and thus can be used to calculate certain zeta functions.
Quadratic Gauss sum
Elliptic Gauss sum
Jacobi sum
Kummer sum
Kloosterman sum
Gaussian period
Hasse–Davenport relation
Chowla–Mordell theorem
Stickelberger's theorem
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Apostol, Tom M.
; Evans, R. J.; Williams, K. S. (1998). Gauss and Jacobi Sums. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley. ISBN 0-471-12807-4. Zbl 0906.11001.
Berndt, B. C.
Ireland, Kenneth; Rosen, Michael (1990). A Classical Introduction to Modern Number Theory. . Vol. 84 (2nd ed.). Springer-Verlag. ISBN 0-387-97329-X. Zbl 0712.11001.
Graduate Texts in Mathematics
Section 3.4 of ; Kowalski, Emmanuel (2004), Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, Providence, RI: American Mathematical Society, ISBN 978-0-8218-3633-0, MR 2061214, Zbl 1059.11001