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Mean value of some exponential sums and applications to Kloosterman sums. (English) Zbl 1255.11039

Let \(q,m,n,k\) be integers with \(q \geq 3\), \(k\geq 1\). Define \[ S(m,n,k;q) = \sideset\and {'}\to\sum_{a =1}^q e((ma + na^k)/q), \] where \(e(x) = e^{2\pi ix}\) and \(\sideset\and {'}\to\sum\) denotes summation under the condition \((a, q) =1\). It is easily seen that \[ \sideset\and {'}\to\sum_{ n=1}^q |S(m,n,k;q)|^r \] is multiplicative in \(q\) for any \(r\). The author evaluated in this paper the fourth power moment \[ \sideset\and {'}\to\sum_{ n=1}^q |S(m,n,k;q)|^4 \] for \(k =1, 2\) and \(k \equiv -1 \pmod {\varphi (q)}\). In the last case, \(S(m,n,k;q) = K(n,m;q)\), the classical Kloosterman sum. Hence a fourth power moment formula for \(K(n,m;q)\) follows from the main theorem in this paper. Also two other immediate corollaries for fourth power moments of hyper-Kloosterman sums are derived. The bulk of the paper concerns the enumeration of residue classes modulo prime powers.

MSC:

11L05 Gauss and Kloosterman sums; generalizations
11L03 Trigonometric and exponential sums (general theory)
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[1] Apostol, T. M., Introduction to Analytic Number Theory (1976), Springer-Verlag: Springer-Verlag New York · Zbl 0335.10001
[2] Bump, D.; Duke, W.; Hoffstein, J.; Iwaniec, H., An estimate for the Hecke eigenvalues of Maass forms, Int. Math. Res. Not., 4, 75-81 (1992) · Zbl 0760.11017
[3] Chowla, S., On Kloosterman’s sum, Norske Vid. Selsk. Forh. (Trondheim), 40, 70-72 (1967) · Zbl 0157.09001
[4] Dabrowski, R.; Fisher, B., A stationary phase formula for exponential sums over \(Z / p^m Z\) and applications to \(GL(3)\)-Kloosterman sums, Acta Arith., 80, 1-48 (1997) · Zbl 0893.11032
[5] Davenport, H.; Heilbronn, H., On an exponential sum, Proc. London Math. Soc., 41, 449-453 (1936) · JFM 62.0178.01
[6] Deligne, P., Applications de la formula des traces aux sommes trigonometriques, (Cohomologie Etale (SGA 4 1/2). Cohomologie Etale (SGA 4 1/2), Lecture Notes in Math., vol. 569 (1977), Springer-Verlag: Springer-Verlag New York), 168-232
[7] Estermann, T., On Kloosterman’s sum, Mathematika, 8, 83-86 (1961) · Zbl 0114.26302
[8] Hua, L. K., On exponential sums, Sci. Record (Peking) (N.S.), 1, 1-4 (1957) · Zbl 0083.04204
[9] Kloosterman, H. D., On the representation of numbers in the form \(a x^2 + b y^2 + c z^2 + d t^2\), Acta Math., 49, 407-464 (1926) · JFM 53.0155.01
[10] Loxton, J. H.; Smith, R. A., On Hua’s estimate for exponential sums, J. London Math. Soc. (2), 26, 15-20 (1982) · Zbl 0474.10030
[11] Loxton, J. H.; Vaughan, R. C., The estimation of complete exponential sums, Canad. Math. Bull., 28, 440-454 (1985) · Zbl 0575.10033
[12] Luo, W.; Rudnick, Z.; Sarnak, P., On Selberg’s eigenvalue conjecture, Geom. Funct. Anal., 5, 387-401 (1995) · Zbl 0844.11038
[13] Mordell, L. J., On a special polynomial congruence and exponential sums, (Calcutta Math. Soc. Golden Jubilee Commemoration Volume, Part 1 (1963), Calcutta Math. Soc.: Calcutta Math. Soc. Calcutta), 29-32 · Zbl 0306.10019
[14] Salié, H., Über die Kloostermanschen summen \(S(u, v; q)\), Math. Z., 34, 91-109 (1932), (in German) · JFM 57.0211.01
[15] Smith, R. A., On \(n\)-dimensional Kloosterman sums, J. Number Theory, 11, 324-343 (1979) · Zbl 0409.10024
[16] Ye, Y., Hyper-Kloosterman sums and estimation of exponential sums of polynomials of higher degrees, Acta Arith., 86, 255-267 (1998) · Zbl 0923.11117
[17] Ye, Y., Estimation of exponential sums of polynomials of higher degrees, II, Acta Arith., 93, 221-235 (2000) · Zbl 0953.11028
[18] Zhang, W., On the fourth power mean of the classical Kloosterman sums, (Proceedings of the 16th International Conference of the Jangjeon Mathematical Society (2005), Jangjeon Math. Soc.: Jangjeon Math. Soc. Hapcheon), 180-187
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