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An explicit formula for the fourth moment of certain exponential sums. (English) Zbl 1240.11090

Consider the generalised Kloosterman sum \[ S(m,n,\chi,\chi',q) = \mathop{{\sum}'}_{a=1}^q \chi(a)G(a,\chi') e\left({ma^k+na\over q}\right) \] for integers \(m, n, q, k\) with \(q, k\geq 1\) and Dirichlet characters \(\chi,\chi'\bmod q\) and the Gaussian sum \[ G(a,\chi')=\mathop{{\sum}'}_{u=1}^q \chi'(u)e\left({ua\over q}\right). \] The main result is an exact formula for the fourth moment \(M_k(q)\) of \(S(m,n,\chi,\chi',q)\) when averaged over the parameters \(m, \chi, \chi'\). The authors note that the sums are closely related to certain mixed exponential sums considered by H. N. Liu [Proc. Am. Math. Soc. 136, 1193–1203 (2008; Zbl 1145.11063)] because \(S(m,n,\chi,\chi',p^\alpha)= G(1,\chi')C(m,n,\chi\overline{\chi' }, k,p^\alpha)\) where \(C\) is Liu’s sum \[ C(m,n,\chi,k,q)=\mathop{{\sum}'}_{a=1}^q \chi(a) e\left({ma^k+na\over q}\right) \] and the theorem also follows from Liu’s result giving the fourth moment of \(C(m, n, \chi, k,q)\) averaged over \(\chi\) and \(m\).

MSC:

11L05 Gauss and Kloosterman sums; generalizations
11T23 Exponential sums
11L03 Trigonometric and exponential sums (general theory)

Citations:

Zbl 1145.11063
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References:

[1] T. M. Apostol, Introduction to Analytic Number Theory, Springer (New York, 1976). · Zbl 0335.10001
[2] R. W. Bruggeman, Fourier coefficients of cusp forms, Invent. Math., 45 (1978), 1–18. · Zbl 0363.10018 · doi:10.1007/BF01406220
[3] H. Davenport, On certain exponential sums, I, Reine Angew. Math., 169 (1933), 158–176. · JFM 59.0370.03
[4] H. Davenport and H. Heilbronn, On an exponential sum, Proc. London Math. Soc., 41 (1936), 449–453. · JFM 62.0178.01 · doi:10.1112/plms/s2-41.6.449
[5] J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math., 70 (1982), 219–288. · Zbl 0502.10021 · doi:10.1007/BF01390728
[6] L. K. Hua, On exponential sums, Sci. Record (Peking) (N.S.), 1 (1957), 1–4. · Zbl 0083.04204
[7] S. Kanemitsu, Y. Tanigawa, Y. Yi and W. P. Zhang, On general Kloosterman sums, Ann. Univ. Sci. Budapest. Sec. Comp., 22 (2003), 151–160. · Zbl 1108.11059
[8] H. D. Kloosterman, On the representation of numbers in the form ax 2+by 2+cz 2+dt 2, Acta Math., 49 (1926), 407–464. · JFM 53.0155.01 · doi:10.1007/BF02564120
[9] N. M. Korobov, Exponential Sums and their Applications, Kluwer (Dordrecht, 1992). · Zbl 0754.11022
[10] N. V. Kuznetsov, Paterson’s conjecture for cusp forms of weight zero and Linnik’s conjecture. Sums of Kloosterman sums, Mat. Sb., 3 (1980), 334–383. · Zbl 0427.10016
[11] H. N. Liu, Mean value of mixed exponential sums, Proc. Amer. Math Soc., 136 (2008), 1193–1203. · Zbl 1145.11063 · doi:10.1090/S0002-9939-07-09075-2
[12] M. B. Nathanson, Additive Number Theory: the Classical Bases, Graduate texts in Mathematics, 164, Springer (Berlin, 1996). · Zbl 0859.11002
[13] S. P. Varbanec, General Kloosterman sums over the ring of Gaussian integers, Ukrain. Math. Journal, 59 (2007), 1313–1341. · Zbl 1150.11028 · doi:10.1007/s11253-007-0090-4
[14] A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. USA, 34 (1948), 204–207. · Zbl 0032.26102 · doi:10.1073/pnas.34.5.204
[15] Y. Ye, Hyper-Kloosterman sums and estimation of exponential sums of polynomials of higher degrees, Acta Arith., 86 (1998), 255–267. · Zbl 0923.11117
[16] Y. Ye, Estimation of exponential sums of polynomials of higher degrees II, Acta Arithm., 93 (2000), 221–235. · Zbl 0953.11028
[17] Y. Yi and W. P. Zhang, On the generalization of a problem of D. H. Lehmer, Kyushu J. Math., 56 (2002), 235–241. · Zbl 1136.11323 · doi:10.2206/kyushujm.56.235
[18] W. P. Zhang, On the fourth power mean of the general Kloosterman sums, Indian J. Pure Appl. Math., 35 (2004), 237–242. · Zbl 1046.11055
[19] W. P. Zhang, On the general Kloosterman sum and its fourth power mean, J. Number Theory, 104 (2004), 156–161. · Zbl 1039.11052 · doi:10.1016/S0022-314X(03)00154-9
[20] W. P. Zhang and H. N. Liu, On the general Gauss sums and their fourth power mean, Osaka J. Math., 42 (2005), 189–199. · Zbl 1163.11337
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