We show that twisted monomial Gauss sums modulo prime powers can be evaluated explicitly once the power is sufficiently large.
References
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. MR1625181 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. MR1625181
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. MR1956230 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. MR1956230
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. MR2778491 10.4134/BKMS.2011.48.1.009 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. MR2778491 10.4134/BKMS.2011.48.1.009
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. MR1409372 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. MR1409372
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. MR1765792 10.1093/qjmath/51.2.221 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. MR1765792 10.1093/qjmath/51.2.221
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. MR3002690 10.4134/BKMS.2012.49.6.1327 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. MR3002690 10.4134/BKMS.2012.49.6.1327
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]. MR2204426 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]. MR2204426
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]. MR2311594 10.4213/mzm3092 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]. MR2311594 10.4213/mzm3092
J.-L. Mauclaire, Sommes de Gauss modulo $p^{\alpha}$. I, Proc. Japan Acad. 59, Ser A (1983), 109–112. MR711310 10.3792/pjaa.59.109 euclid.pja/1195515688
J.-L. Mauclaire, Sommes de Gauss modulo $p^{\alpha}$. I, Proc. Japan Acad. 59, Ser A (1983), 109–112. MR711310 10.3792/pjaa.59.109 euclid.pja/1195515688
J.-L. Mauclaire, Sommes de Gauss modulo $p^{\alpha}$. II, Proc. Japan Acad. 59, Ser A (1983), 161–163. MR711325 10.3792/pjaa.59.161 euclid.pja/1195515641
J.-L. Mauclaire, Sommes de Gauss modulo $p^{\alpha}$. II, Proc. Japan Acad. 59, Ser A (1983), 161–163. MR711325 10.3792/pjaa.59.161 euclid.pja/1195515641
R. Odoni, On Gauss Sums (mod $p^n$), $n\geq 2$, Bull. London Math. Soc. 5 (1973), 325–327. MR327678 10.1112/blms/5.3.325 R. Odoni, On Gauss Sums (mod $p^n$), $n\geq 2$, Bull. London Math. Soc. 5 (1973), 325–327. MR327678 10.1112/blms/5.3.325
J-C. Puchta, Remark on a paper of Yu on Heilbronn's exponential sum, J. Number Theory 87 (2001), 239–241. MR1824145 10.1006/jnth.2000.2593 J-C. Puchta, Remark on a paper of Yu on Heilbronn's exponential sum, J. Number Theory 87 (2001), 239–241. MR1824145 10.1006/jnth.2000.2593
A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 203–210. MR27006 10.1073/pnas.34.5.204 A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 203–210. MR27006 10.1073/pnas.34.5.204
W. Zhang and H. Liu, On the general Gauss sums and their fourth power means, Osaka J. Math. 42 (2005), 189–199. MR2132010 euclid.ojm/1153494321
W. Zhang and H. Liu, On the general Gauss sums and their fourth power means, Osaka J. Math. 42 (2005), 189–199. MR2132010 euclid.ojm/1153494321