Functiones et Approximatio Commentarii Mathematici

Twisted monomial Gauss sums modulo prime powers

Vincent Pigno and Christopher Pinner

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We show that twisted monomial Gauss sums modulo prime powers can be evaluated explicitly once the power is sufficiently large.

Article information

Funct. Approx. Comment. Math., Volume 51, Number 2 (2014), 285-301.

First available in Project Euclid: 26 November 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11L05: Gauss and Kloosterman sums; generalizations
Secondary: 11L07: Estimates on exponential sums 11L03: Trigonometric and exponential sums, general

exponential sums Gauss sums


Pigno, Vincent; Pinner, Christopher. Twisted monomial Gauss sums modulo prime powers. Funct. Approx. Comment. Math. 51 (2014), no. 2, 285--301. doi:10.7169/facm/2014.51.2.4.

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  • T. Apostol, Introduction to Analytic Number Theory, Springer 1976.
  • B.C. Berndt, R.J. Evans and K.S. Williams, Gauss and Jacobi Sums, Canadian Math. Soc. series of monographs and advanced texts, vol. 21, Wiley, New York 1998.
  • T. Cochrane and Z. Zheng, Pure and mixed exponential sums, Acta Arith. 91 (1999), no. 3, 249–278.
  • T. Cochrane and Z. Zheng, Bounds for certain exponential sums, Asian J. Math. 4 (2000), no. 4, 757–774.
  • T. Cochrane and Z. Zheng, A survey on pure and mixed exponential sums modulo prime powers, Number theory for the millennium, I (Urbana, IL, 2000), 273–300, A K Peters, Natick, MA, 2002.
  • X. Guo and T. Wang, On the generalized $k$-th Gauss sums, to appear Hacet. J. Math. Stat.
  • Y. He and W. Zhang, On the $2k$-th power mean value of the generalized quadratic Gauss sum, Bull Korean Math. Soc. 48 (2011), 9–15.
  • D.R. Heath-Brown, An estimate for Heilbronn's exponential sum, Analytic Number Theory: Proceedings of a conference in honor of Heini Halberstam, Birkhäuser, Boston, MA, (1996), 451–463.
  • D.R. Heath-Brown and S. Konyagin, New bounds for Gauss sums derived from $k$th powers, and for Heilbronn's exponential sum, Quart. J. Math. 51 (2000), 221–235.
  • F. Liu and Q.-H. Yang, An identity on the $2m$-th power mean value of the generalized Gauss sums, Bull. Korean Math. Soc. 49 (2012), no. 6, 1327–1334.
  • Yu.V. Malykhin, Bounds for exponential sums modulo $p^2$, Journal of Mathematical Sciences 146, No. 2, (2007), 5686-5696 [Translated from Fundamentalnaya i Prikladnaya Matematika 11, No. 6, (2005), 81–94].
  • Yu.V. Malykhin, Estimates of trigonometric sums modulo $p^r$, Mathematical Notes 80 (2006), No. 5, 748–752. [Translated from Matematicheskie Zametki 80 (2006), No. 5, 793–796].
  • J.-L. Mauclaire, Sommes de Gauss modulo $p^{\alpha}$. I, Proc. Japan Acad. 59, Ser A (1983), 109–112.
  • J.-L. Mauclaire, Sommes de Gauss modulo $p^{\alpha}$. II, Proc. Japan Acad. 59, Ser A (1983), 161–163.
  • R. Odoni, On Gauss Sums (mod $p^n$), $n\geq 2$, Bull. London Math. Soc. 5 (1973), 325–327.
  • J-C. Puchta, Remark on a paper of Yu on Heilbronn's exponential sum, J. Number Theory 87 (2001), 239–241.
  • A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 203–210.
  • W. Zhang and H. Liu, On the general Gauss sums and their fourth power means, Osaka J. Math. 42 (2005), 189–199.