# Article

Full entry | PDF   (0.1 MB)
Keywords:
general $k$-th Kloosterman sum; Dirichlet $L$-function; the mean square value; asymptotic formula
Summary:
For the general modulo $q\geq 3$ and a general multiplicative character $\chi$ modulo $q$, the upper bound estimate of $|S(m, n, 1, \chi , q)|$ is a very complex and difficult problem. In most cases, the Weil type bound for $|S(m, n, 1, \chi , q)|$ is valid, but there are some counterexamples. Although the value distribution of $|S(m, n, 1, \chi , q)|$ is very complicated, it also exhibits many good distribution properties in some number theory problems. The main purpose of this paper is using the estimate for $k$-th Kloosterman sums and analytic method to study the asymptotic properties of the mean square value of Dirichlet $L$-functions weighted by Kloosterman sums, and give an interesting mean value formula for it, which extends the result in reference of W. Zhang, Y. Yi, X. He: On the $2k$-th power mean of Dirichlet L-functions with the weight of general Kloosterman sums, Journal of Number Theory, 84 (2000), 199–213.
References:
[1] Apostol, T. M.: Introduction to Analytic Number Theory. Springer, New York (1976). MR 0434929 | Zbl 0335.10001
[2] Chowla, S.: On Kloosterman's sum. Norske Vid. Selsk. Forhdl. 40 (1967), 70-72. MR 0228452 | Zbl 0157.09001
[3] Cochrane, T., Pinner, C.: A further refinement of Mordell's bound on exponential sums. Acta Arith. 116 (2005), 35-41. DOI 10.4064/aa116-1-4 | MR 2114903 | Zbl 1082.11050
[4] Cochrane, T., Zheng, Z.: Exponential sums with rational function entries. Acta Arith. 95 (2000), 67-95. MR 1787206 | Zbl 0956.11018
[5] Cochrane, T., Pinner, C.: Using Stepanov's method for exponential sums involving rational functions. J. Number Theory 116 (2006), 270-292. DOI 10.1016/j.jnt.2005.04.001 | MR 2195926 | Zbl 1093.11058
[6] Deshouillers, J.-M., Iwaniec, H.: Kloosterman sums and Fourier coefficients of cusp forms. Invent. Math. 70 (1982), 219-288. DOI 10.1007/BF01390728 | MR 0684172 | Zbl 0502.10021
[7] Estermann, T.: On Kloosterman's sum. Mathematika, Lond. 8 (1961), 83-86. DOI 10.1112/S0025579300002187 | MR 0126420 | Zbl 0114.26302
[8] Iwaniec, H., Kowalski, E.: Analytic Number Theory. Colloquium Publicastions. American Mathematical Society 53. Providence, RI: American Mathematical Society (2004). MR 2061214 | Zbl 1059.11001
[9] Malyshev, A. V.: A generalization of Kloosterman sums and their estimates. Russian Vestnik Leningrad Univ. 15 (1960), 59-75. MR 0125084
[10] Weil, A.: On some exponential sums. Proc. Natl. Acad. Sci. USA 34 (1948), 204-207. DOI 10.1073/pnas.34.5.204 | MR 0027006 | Zbl 0032.26102
[11] Zhang, W., Yi, Y., He, X.: On the $2k$-th power mean of Dirichlet $L$-functions with the weight of general Kloosterman sums. J. Number Theory 84 (2000), 199-213. DOI 10.1006/jnth.2000.2515 | MR 1795790 | Zbl 0958.11061

Partner of