DOI QR코드

DOI QR Code

AN IDENTITY ON THE 2m-TH POWER MEAN VALUE OF THE GENERALIZED GAUSS SUMS

  • Liu, Feng (School of Mathematical Sciences Nanjing Normal University) ;
  • Yang, Quan-Hui (School of Mathematical Sciences Nanjing Normal University)
  • Received : 2011.06.30
  • Published : 2012.11.30

Abstract

In this paper, using analytic method and the properties of the Legendre's symbol, we prove an exact calculating formula on the $2m$-th power mean value of the generalized quadratic Gauss sums for $m{\geq}2$. This solves a conjecture of He and Zhang [On the 2k-th power mean value of the generalized quadratic Gauss sums, Bull. Korean Math. Soc. 48 (2011), no. 1, 9-15].

Acknowledgement

Supported by : National Natural Science Foundation of China

References

  1. Tom M. Apostol, Introduction to Analytic Number Theory, Spring-Verlag, New York, 1976.
  2. T. Cochrane and Z. Y. Zheng, Pure and mixed exponential sums, Acta Arith 91 (1999), no. 3, 249-278. https://doi.org/10.4064/aa-91-3-249-278
  3. Y. He and W. P. Zhang, On the 2k-th power mean value of the generalized quadratic Gauss sums, Bull. Korean Math. Soc. 48 (2011), no. 1, 9-15. https://doi.org/10.4134/BKMS.2011.48.1.009
  4. A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 204-207. https://doi.org/10.1073/pnas.34.5.204
  5. W. P. Zhang, Moments of generalized quadratic Gauss sums weighted by L-functions, J. Number Theory 92 (2002), no. 2, 304-314. https://doi.org/10.1006/jnth.2001.2715
  6. W. P. Zhang and H. Liu, On the general Gauss sums and their fourth power mean, Osaka J. Math. 42 (2005), no. 1, 189-199.