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A sum analogous to Kloosterman sum and its fourth power mean. (English) Zbl 1484.11167

Summary: The main purpose of this paper is using the analytic methods and the properties of the Legendre’s symbol and quadratic residue mod \(p\) to study the computational problem of the fourth power mean of a sum analogous to Kloosterman sum, and give a sharp asymptotic formula for it.

MSC:

11L03 Trigonometric and exponential sums (general theory)
11L05 Gauss and Kloosterman sums; generalizations
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References:

[1] T. M. Apostol, <i>Introduction to Analytic Number Theory</i>, Springer-Verlag, New York, 1976. · Zbl 0335.10001
[2] A. Weil, i>On some exponential sums</i, Proc. Nat. Acad. Sci. U. S. A., 34, 203-210 (1948) · Zbl 0031.39703 · doi:10.1073/pnas.34.5.203
[3] W. P. Zhang, i>On the fourth power mean of the general Kloosterman sums</i, J. Number Theory, 169, 315-326 (2016) · Zbl 1409.11057 · doi:10.1016/j.jnt.2016.05.018
[4] W. P. Zhang, i>On the fourth power mean of the general Kloosterman sums</i, Indian J. Pure Ap. Mat., 35, 237-242 (2004) · Zbl 1046.11055
[5] J. H. Li; Y. N. Liu, i>Some new identities involving Gauss sums and general Kloosterman sums</i, Acta Math. Sinica (Chinese Series), 56, 413-416 (2013) · Zbl 1299.11060
[6] Y. T. Zhang, i>Bounded gaps between primes</i, Ann. Math., 179, 1121-1174 (2014) · Zbl 1290.11128 · doi:10.4007/annals.2014.179.3.7
[7] X. X. Lv; W. P. Zhang, i>A new hybrid power mean involving the generalized quadratic Gauss sums</i> <i>and sums analogous to Kloosterman sums</i, Lith. Math. J., 57, 359-366 (2017) · Zbl 1373.11060 · doi:10.1007/s10986-017-9366-z
[8] S. Chern, i>On the power mean of a sum analogous to the Kloosterman sum</i, Bull. Math. Soc. Sci. Math. Roumanie, 62, 77-92 (2019) · Zbl 1463.11136
[9] W. P. Zhang, i>On the fourth and sixth power mean of the classical Kloosterman sums</i, J. Number Theory, 131, 228-238 (2011) · Zbl 1218.11074 · doi:10.1016/j.jnt.2010.08.008
[10] W. P. Zhang; S. M. Shen, i>A note on the fourth power mean of the generalized Kloosterman sums</i, J. Number Theory, 174, 419-426 (2017) · Zbl 1355.11082 · doi:10.1016/j.jnt.2016.11.020
[11] T. Estermann, i>On Kloostermann’s sums</i, Mathematica, 8, 83-86 (1961) · Zbl 0114.26302
[12] H. D. Kloosterman, i>On the representation of numbers in the form ax</i><sup>2</sup> +<i>by</i><sup>2</sup> +<i>cz</i><sup>2</sup> +<i>dt</i><sup>2</sup, Acta Math., 49, 407-464 (1926) · JFM 53.0155.01 · doi:10.1007/BF02564120
[13] H. Iwaniec, i>Topics in Classical Automorphic Forms</i, Grad. Stud. Math., 17, 61-63 (1997) · Zbl 0905.11023
[14] H. Salié, i>Uber die Kloostermanschen Summen S</i> (<i>u</i, v; q), Math. Z., 34, 91-109 (1931) · JFM 57.0211.01
[15] A. V. Malyshev, i>A generalization of Kloosterman sums and their estimates, (in Russian)</i, Vestnik Leningrad University, 15, 59-75 (1960) · Zbl 0102.03801
[16] W. P. Zhang, i>On the general Kloosterman sums and its fourth power mean</i, J. Number Theory, 104, 156-161 (2004) · Zbl 1039.11052 · doi:10.1016/S0022-314X(03)00154-9
[17] W. P. Zhang; X. X. Li, i>The fourth power mean of the general</i> 2<i>-dimensional Kloostermann sums</i> mod <i>p</i, Acta Math. Sinica, English Series, 33, 861-867 (2017) · Zbl 1427.11076 · doi:10.1007/s10114-016-6347-9
[18] W. P. Zhang; X. X. Lv, i>The fourth power mean of the general</i> 3<i>-dimensional Kloostermann sums</i> mod <i>p</i, Acta Math. Sinica, English Series, 35, 369-377 (2019) · Zbl 1454.11144 · doi:10.1007/s10114-018-7455-5
[19] Y. Ye, i>Identities of incomplete Kloostermann sums</i, Proc. Amer. Math. Soc., 127, 2591-2600 (1999) · Zbl 0929.11023 · doi:10.1090/S0002-9939-99-05037-6
[20] R. A. Smith, i>On n-dimensional Kloostermann sums</i, J. Number Theory, 11, 324-343 (1979) · Zbl 0409.10024 · doi:10.1016/0022-314X(79)90006-4
[21] W. Luo, i>Bounds for incomplete Kloostermann sums</i, J. Number Theory, 75, 41-46 (1999) · Zbl 0923.11118 · doi:10.1006/jnth.1998.2340
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