数论中一些和式的算术性质研究
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摘要
本文主要研究了数论中一些和式的算术性质。这些和式包括不完整区间上的特征和、多元多项式特征和、hyper-Kloosterman和、带特征的指数和、Dedekind和以及它们的推广和式。此外，还研究了半区间上的D．H．Lehmer问题以及以及一些Smarandache型函数的算术性质。具体来说，本文主要包括以下几方面的结果：
1．研究了带特征的完整三角和与两项指数和的高次均值，利用解析和初等的方法获得了它们四次均值的确切计算公式。
2．讨论了不完整区间上特征和的高次均值的渐近性质。首先，分别对四分之一区间上偶原特征和与奇原特征和进行了研究，获得了它们的2κ次均值的渐近公式；第二，研究了八分之一区间上偶原特征和的2κ次均值，并获得了一个较强的渐近公式；第三，对四分之一区间上的非主特征和的四次均值进行了研究，获得了一个较强的渐近公式；第四，讨论了四分之一区间上原特征和的一次均值，同样也获得了一些渐近公式；最后，利用我们获得的四分之一区间上特征和的一个恒等式，推广并证明了著名的欧拉数猜想。
3．讨论了多元多项式特征和的计算问题，对一类多元多项式特征和给出了确切的计算公式，并由此说明了Katz所获得的估计是最佳的。
4．对经典的Dedekind和与它的类似和式Cochrane和定义了一种特殊形式的均值，并研究了它们的渐近性质。定义了高维Cochrane和，得到了高维Cochrane和的阶估计与平方均值渐近公式。
5．研究了hyper-Kloosterman和的高次均值。在一定的条件限定下，给出了它的四次均值计算公式。
6．研究了D．H．Lehmer问题误差项的一种均值，并获得了一些有趣而奇特的结果。把D．H．Lehmer问题限定在半区间上。研究了半区间上D．H．Lehmer问题的误差项，获得了误差项平方均值的一个较强的渐近公式。
7．讨论了Smarandache函数的值分布性质和Smarandache幂函数的均值，得到了一些有趣的性质。
The main purpose of this dissertation is to study the arithmetical properties of some summations in number theory. These summations are Character sums over incomplete intervals, Dirichlet characters of polynomials in several variables, Exponential sums with characters, hyper-Kloosterman sums, Dedekind sums and their generalized sums. Besides these, D. H. Lehmer problems and some Smarandache type functions were also studied. The main achievements contained in this dissertation are follows:
1. Studied the hig.her power mean of complete trigonometric sums and two term exponential sums with Dirichlet characters, and obtained some exact calculation formulas for their fourth power mean by using analytical and elementary methods;
2. Studied the asymptotic properties of higher power mean of Dirichlet character sums over incomplete intervals. At first, I studied the odd and even primitive character sums over quarter interval, respectively, and obtained some asymptotic formulas of their 2κth power mean; Secondly, I studied the 2κth power mean of even primitive character sums over one eighth interval and got a sharper asymptotic formula for it; Thirdly, I studied the fourth power mean of nonprincipal character sums over quarter interval and obtained a sharper asymptotic formula too; Fourthly, I studied the first power mean of primitive character sums over quarter interval and got some asymptotic formulas for it; At last, applying the identity for character sums over quarter interval which I obtained, generalized and proved the famous Euler numbers conjecture;
3. Studied the evaluation problem of Dirichlet characters of polynomial in several variables and obtained some identities for a kind of Dirichlet characters of polynomial. This results explained the fact that Kats's estimation is the best possible;
4. Defined some special mean values for the classical Dedekind sums and its analogous sums which called Cochrane sums, and studied their asymptotic properties. Defined the high dimensional Cochrane sums and obtained its order estimation and asymptotic formula of its square mean value;
5. Studied the higher power mean of hyper-Kloosterman sums and got an calculation formula for their fourth power mean under some restrictions;
6. Studied a mean value of error term of D. H. Lehmer problem and obtained some interesting and strange results. Restricted D. H. Lehmer problem on a half interval. Studied the error term of the D. H. Lehmer problem over half interval and obtained a sharper asymptotic formula for its square mean value;
7. Studied the value distribution of Smarandache function and the mean value of Smarandache power function, and obtained some interesting properties.
1．研究了带特征的完整三角和与两项指数和的高次均值，利用解析和初等的方法获得了它们四次均值的确切计算公式。
2．讨论了不完整区间上特征和的高次均值的渐近性质。首先，分别对四分之一区间上偶原特征和与奇原特征和进行了研究，获得了它们的2κ次均值的渐近公式；第二，研究了八分之一区间上偶原特征和的2κ次均值，并获得了一个较强的渐近公式；第三，对四分之一区间上的非主特征和的四次均值进行了研究，获得了一个较强的渐近公式；第四，讨论了四分之一区间上原特征和的一次均值，同样也获得了一些渐近公式；最后，利用我们获得的四分之一区间上特征和的一个恒等式，推广并证明了著名的欧拉数猜想。
3．讨论了多元多项式特征和的计算问题，对一类多元多项式特征和给出了确切的计算公式，并由此说明了Katz所获得的估计是最佳的。
4．对经典的Dedekind和与它的类似和式Cochrane和定义了一种特殊形式的均值，并研究了它们的渐近性质。定义了高维Cochrane和，得到了高维Cochrane和的阶估计与平方均值渐近公式。
5．研究了hyper-Kloosterman和的高次均值。在一定的条件限定下，给出了它的四次均值计算公式。
6．研究了D．H．Lehmer问题误差项的一种均值，并获得了一些有趣而奇特的结果。把D．H．Lehmer问题限定在半区间上。研究了半区间上D．H．Lehmer问题的误差项，获得了误差项平方均值的一个较强的渐近公式。
7．讨论了Smarandache函数的值分布性质和Smarandache幂函数的均值，得到了一些有趣的性质。
The main purpose of this dissertation is to study the arithmetical properties of some summations in number theory. These summations are Character sums over incomplete intervals, Dirichlet characters of polynomials in several variables, Exponential sums with characters, hyper-Kloosterman sums, Dedekind sums and their generalized sums. Besides these, D. H. Lehmer problems and some Smarandache type functions were also studied. The main achievements contained in this dissertation are follows:
1. Studied the hig.her power mean of complete trigonometric sums and two term exponential sums with Dirichlet characters, and obtained some exact calculation formulas for their fourth power mean by using analytical and elementary methods;
2. Studied the asymptotic properties of higher power mean of Dirichlet character sums over incomplete intervals. At first, I studied the odd and even primitive character sums over quarter interval, respectively, and obtained some asymptotic formulas of their 2κth power mean; Secondly, I studied the 2κth power mean of even primitive character sums over one eighth interval and got a sharper asymptotic formula for it; Thirdly, I studied the fourth power mean of nonprincipal character sums over quarter interval and obtained a sharper asymptotic formula too; Fourthly, I studied the first power mean of primitive character sums over quarter interval and got some asymptotic formulas for it; At last, applying the identity for character sums over quarter interval which I obtained, generalized and proved the famous Euler numbers conjecture;
3. Studied the evaluation problem of Dirichlet characters of polynomial in several variables and obtained some identities for a kind of Dirichlet characters of polynomial. This results explained the fact that Kats's estimation is the best possible;
4. Defined some special mean values for the classical Dedekind sums and its analogous sums which called Cochrane sums, and studied their asymptotic properties. Defined the high dimensional Cochrane sums and obtained its order estimation and asymptotic formula of its square mean value;
5. Studied the higher power mean of hyper-Kloosterman sums and got an calculation formula for their fourth power mean under some restrictions;
6. Studied a mean value of error term of D. H. Lehmer problem and obtained some interesting and strange results. Restricted D. H. Lehmer problem on a half interval. Studied the error term of the D. H. Lehmer problem over half interval and obtained a sharper asymptotic formula for its square mean value;
7. Studied the value distribution of Smarandache function and the mean value of Smarandache power function, and obtained some interesting properties.
引文
[1] T. M. Apostol, Introduction to Analytic Number Theory. New York: SpringerVerlag, 1976.
[2] T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory. New York: Springer-Verlag, 1976.
[3] D. Bump, W. Duke, J. Hoffstein and H. Iwaniec, An estimate for the Hecke eigenvalues of Maass forms. International Mathematics Research Notices, 1992, 4: 75-81.
[4] D. A. Burgess, On Dirichlet characters of polynomials. Proc. London Math. Soc, 1963, 13: 537-548.
[5] D. A. Burgess, On a conjecture of Norton. Acta Arithmetica, 1975, 27: 265-267.
[6] D. A. Burgess, Mean value of character sums. Mathematika, 1986, 33: 1-5.
[7] D. A. Burgess, Mean value of character sums Ⅱ. Mathematika, 1987, 34: 1-7.
[8] L. Carlitz, The reciprocity theorem of Dedekind sums. Pacific J. Math., 1953, 3: 513-522.
[9] J. H. H. Chalk and R. A. Smith, On Bombieri's estimate for exponential sums. Acta Arithmetica, 1971, 18: 191-212.
[10] 潘承洞，潘承彪．哥德巴赫猜想．北京：科学出版社，1981．
[11] 潘承洞，潘承彪．初等数论．北京：北京大学出版社，1992．
[12] 潘承洞，潘承彪．解析数论基础．北京：科学出版社，1999．
[13] S. Chowla, On Kloosterman's sum. Norkse Vid. Selbsk. Fak. Frondheim, 1967, 40: 70-72.
[14] T. Cochrane and Z. Y. Zheng, Bounds for certain exponential sums. Loo-Keng Hua: a great mathematician of the twentieth century. Asian J. Math., 2000, 4: 757-773.
[15] T. Cochrane and Z. Y. Zheng, Upper bounds on a two-term exponential sum. Science in China (Series A), 2001, 44: 1003-1015.
[16] J. B. Conrey, E. Fransen, R. Klein and C. Scott, Mean values of Dedekind sums. Journal of Number Theory, 1982, 41: 27-31.
[17] R. Dabrowski and B. Fisher, A stationary phase formula for exponential sums over Z/p~mZ and applications to GL(3)-Kloosterman sums. Acta Arithmetica, 1997, 80: 1-48.
[18] H. Darvenport and H. Heibronn, On an exponential sum. Proc. London Math. Soc., 1936, 41: 449-453.
[19] P. Deligne, Applications de la formule des traces aux sommes trigonometriques, in: Cohomologie Etale (SGA4 1/2), Lectures Notes in Math. 569. Berlin: Springer, 1977, 168-232.
[20] R. L. Duncan, Application of Uniform Distribution to the Fibonacci Numbers. The Fibonacci Quarterly, 1967, 5: 137-140.
[21] P. Erdos, Problem 6674. Amer. Math. Monthly, 1991, 98: 965.
[22] T. Estermann, On Kloostermann's sum. Mathematica, 1961, 8: 83-86.
[23] G. D. Liu, Identities and congruences involving higher-order Euler-Bernoulli numbers and polynomials. The Fibonacci Quarterly, 2001, 39(3): 279-284.
[24] G. D. Liu, The solution of problem for Euler numbers. Acta Mathematica Sinica (Chinese), 2004, 47(4): 825-828.
[25] R. K. Guy, Unsolved problem in number theory. New York: Springer-Verlag, 1994.
[26] G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number n. Quart. J. Math., 1917, 48: 76-92.
[27] L. K. Hun, On exponential sums. Sci Record(Peking)(N. S.), 1957, 1: 1-4.
[28] H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, volume 17. Providence: American Mathematical Society, 1997, 60-63.
[29] J. H. Ren and W. P. Zhang, On the average of Dirichlet L-functions. Journal of Sichuan University Natural Science Edition, 1990, 26(Special Issue): 75-79.
[30] J. Wang, On the Jacobi sums modulo p~n. Journal of Number theory, 1991, 39: 50-64.
[31] N. M. Katz, Gauss sums, Kloosterman sums, and monodromy groups. Princeton: Annals of Mathematics Studies, Princeton University Press, 1988.
[32] N. M. Katz, Estimates for "nonsingular" mutiplicative character sums. International Mathematics Research Notices, 2002, 7: 333-349.
[33] H. D. Kloosterman, On the representation of numbers in the form ax~2+by~2+cz~2+dt~2. Acta Math., 1926, 49: 407-464.
[34] L. Kuipers, Remark on a paper by R. L. Duncan concerning the uniform distribution rood 1 of the Logarithms of the Fibonacci numbers. The Fibonacci Quarterly, 1969, 5: 465-466.
[35] E. Lehmer, On Congruences Involving Bernoulli Numbers and the quotients of Fermat and Wilson. Annals of Math., 1938, 39: 350-360.
[36] R. Lidl and H. Niederreiter, Finite Fields. Addison-Wesley: Reading, MA, 1983.
[37] J. H. Loxton and R. A. Smith, On Hua's estimate for exponential sums. J. London Math. Soc. (2), 1982, 26: 15-20.
[38] J. H. Loxton and R. C. Vanghan, The estimate for complete exponential sums. Canada Math. Bull. (4), 1995, 26: 442-454.
[39] Minggao Lu, estimate of a complete trigonometric sum. Sci. Sinica Ser. A, 1985, 28: 561-578.
[40] 华罗庚．数论导引．北京：科学出版社，1979．
[41] 谢敏，张文鹏．关于广义Dedekind和的2k次均值．数学学报．2001，44：85-94．
[42] H. L. Montgomery and R. C. Vaughan, Mean value of character sums. Canadian Journal of Mathematics, 1979, 31(3): 476-487.
[43] L. J. Mordell, On a special polynomiaI congruence and exponential sums, in "Calcutta Math. Soc. Golden Jubilee Commemoration Volume" Part 1. Calcutta: Calcutta Math. Soc., 1963, 29-32.
[44] K. K. Norton. On character sums and power residues. Trans. Amer. Math. Soc., 1972, 167: 203-226.
[45] 潘承洞，潘承彪．解析数论基础．北京：科学出版社，1999．
[46] Juan C. Peral, Character sums and explicit estimates for L-functions. Contemporary Mathematics, 1995, 189: 449-459.
[47] Yuan Pingzhi, A conjecture on Euler numbers. Proc. Japan Acad. (Ser. A), 2004, 80: 180-181.
[48] G. Polya, Uber die Verteilung der quadratische Reste und Nichtreste. Gottingen Nachrichten, 1918, 21-29.
[49] H. Rademacher, On the transformation of log η(Υ). J. Indian. Math. soc., 1955, 19: 25-30.
[50] N. Robbins, Applications of Fibonacci Numbers. Kluwer Academic Publishers, 1986, pp. 77-88.
[51] K. G. Richard, Unsolved Problems in Number Theory. New York: SpringerVerlag, 1994.
[52] H. Salie, Uber die Kloostermanschen Summen S(u, v; q). Mathematische Zeitschrift, 1931, 34: 91-109.
[53] H. N. Shapiro, Introduction to the theory of numbers. New York: John Wiley and Sons, Inc., 1983, 304-305, 307-308.
[54] F. Smarandache, Collected Papers,Vol. Ⅲ. Bucharest: Tempus Publishing House, 1998.
[55] R. A. Smith, On n-dimensional Kloosterman sums. Journal of Number Theory, 1979, 11: 324-343.
[56] A. V. Sokolovskii, On a theorem of Sarkozy. Acta Arithmetica, 1982, 41: 27-31.
[57] Tabirca Sabin, About S-Multiplicative Functions. Octogon, 1999, 7: 169-170.
[58] I. M. Vinogradov, On the distribution of residues and non-residues of powers. Journal of the Physico-Mathematical society of Perm., 1918, 1: 94-96.
[59] A. Weil, On some exponential sums. Proc. Nat. Acad. Sci. U. S. A., 1948, 34: 204-207.
[60] W. P. Zhang, A problem of D. H. Lehmer and its generalization (Ⅱ). Compositio Mathematica, 1994, 91: 47-56.
[61] W. P. Zhang, On the difference between an integer and its inverse modulo n. Journal of Number Theory, 1995, 52: 1-6.
[62] W. P. Zhang, On the mean values of Dedekind sums. Jouunal de Theorie des Nombres de Bordeaux, 1996, 8: 429-442.
[63] W. P. Zhang, Some identities involving the Fibonacci numbers. The Fibonacci Quarterly, 1997, 35: 225-229.
[64] W. P. Zhang, Some Identities Involving The Euler and The Central Factorial Numbers. The Fibonacci Quarterly, 1998, 36(2): 154-157.
[65] W. P. Zhang, Y. Yi and X. L. He, On the 2k-th power mean of Dirichlet L-functions with the weight of general Kloosterman sums. Journal of Number Theory, 2000, 84: 199-213.
[66] W. P. Zhang, A note on the mean square value of the Dedekind sums. Acta Math. Hangar., 2000, 86: 275-289.
[67] W. P. Zhang, On the general Dedekind sums and one kind identities of Dirichlet L-functions. Acta Mathematica Sinica, 2001, 44: 269-272.
[68] W. P. Zhang and Y. Yi, On Dirichlet characters of polynomials Bull. London Math. Soc, 2002, 34: 469-473.
[69] W. P. Zhang, On a Cochrane sum and its hybrid mean value formula. Journal of Mathematical Analysis and Applications, 2002, 267: 89-96.
[70] W. P. Zhang, On a Cochrane sum and its hybrid mean value formula (Ⅱ). Journal of Mathematical Analysis and Applications, 2002, 276: 446-457.
[71] W. P. Zhang, On a sum analogous to Dedekind sum and its mean square value formula. International J. of Math. and Math. Sci., 2002, 32: 47-55.
[72] W. P. Zhang, On a problem of D. H. Lehmer and Kloosterman sums. Monatshefte fur Mathematik, 2003, 139: 247-257.
[73] W. P. Zhang, On the difference between an integer and its inverse modulo n (Ⅱ). Science in China (Series A), 2003, 46: 229-238.
[74] W. P. Zhang and H. N. Liu, A Note on the Cochrane sum and its hybrid mean value formula. Journal of Mathematical Analysis and Applications, 2003, 288: 646-659.
[75] W. P. Zhang, Z. B. Xu and Y. Yi, A problem of D. H. Lehmer and its mean square value formula. Journal of Number theory, 2003, 103: 193-203.
[76] W. P. Zhang, A problem of D. H. Lehmer and its mean square value formula. Japanese Journal of Mathematics, 2003, 29: 109-116.
[77] W. P. Zhang and W. L. Yao, A note on the Diriehlet characters of polynomials. Acta Arithmetica, 2004, 115: 225-229.
[78] W. P. Zhang, On the general Kloosterman sum and its fourth power mean. Journal of Number Theory, 2004, 104: 156-161.
[79] W. Z. Luo, Z. Rudnick and P. Sarnak, On Selberg's eigenvalue conjecture. Geometric and Functional Analysis, 1995, 5: 387-401.
[80] Y. B. Ye, Estimation of exponential sums of polynomials of higher degrees Ⅱ. Acta Arithmetica, 2000, XCⅢ. 3: 221-235.
[81] Yi Yuan and Zhang Wenpeng, On the 2k-th Power Mean of Dirichlet L-function With the Weight of Gauss Sums. Advances in Mathematics (in Chinese), 2000, 31: 518-526.
[2] T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory. New York: Springer-Verlag, 1976.
[3] D. Bump, W. Duke, J. Hoffstein and H. Iwaniec, An estimate for the Hecke eigenvalues of Maass forms. International Mathematics Research Notices, 1992, 4: 75-81.
[4] D. A. Burgess, On Dirichlet characters of polynomials. Proc. London Math. Soc, 1963, 13: 537-548.
[5] D. A. Burgess, On a conjecture of Norton. Acta Arithmetica, 1975, 27: 265-267.
[6] D. A. Burgess, Mean value of character sums. Mathematika, 1986, 33: 1-5.
[7] D. A. Burgess, Mean value of character sums Ⅱ. Mathematika, 1987, 34: 1-7.
[8] L. Carlitz, The reciprocity theorem of Dedekind sums. Pacific J. Math., 1953, 3: 513-522.
[9] J. H. H. Chalk and R. A. Smith, On Bombieri's estimate for exponential sums. Acta Arithmetica, 1971, 18: 191-212.
[10] 潘承洞，潘承彪．哥德巴赫猜想．北京：科学出版社，1981．
[11] 潘承洞，潘承彪．初等数论．北京：北京大学出版社，1992．
[12] 潘承洞，潘承彪．解析数论基础．北京：科学出版社，1999．
[13] S. Chowla, On Kloosterman's sum. Norkse Vid. Selbsk. Fak. Frondheim, 1967, 40: 70-72.
[14] T. Cochrane and Z. Y. Zheng, Bounds for certain exponential sums. Loo-Keng Hua: a great mathematician of the twentieth century. Asian J. Math., 2000, 4: 757-773.
[15] T. Cochrane and Z. Y. Zheng, Upper bounds on a two-term exponential sum. Science in China (Series A), 2001, 44: 1003-1015.
[16] J. B. Conrey, E. Fransen, R. Klein and C. Scott, Mean values of Dedekind sums. Journal of Number Theory, 1982, 41: 27-31.
[17] R. Dabrowski and B. Fisher, A stationary phase formula for exponential sums over Z/p~mZ and applications to GL(3)-Kloosterman sums. Acta Arithmetica, 1997, 80: 1-48.
[18] H. Darvenport and H. Heibronn, On an exponential sum. Proc. London Math. Soc., 1936, 41: 449-453.
[19] P. Deligne, Applications de la formule des traces aux sommes trigonometriques, in: Cohomologie Etale (SGA4 1/2), Lectures Notes in Math. 569. Berlin: Springer, 1977, 168-232.
[20] R. L. Duncan, Application of Uniform Distribution to the Fibonacci Numbers. The Fibonacci Quarterly, 1967, 5: 137-140.
[21] P. Erdos, Problem 6674. Amer. Math. Monthly, 1991, 98: 965.
[22] T. Estermann, On Kloostermann's sum. Mathematica, 1961, 8: 83-86.
[23] G. D. Liu, Identities and congruences involving higher-order Euler-Bernoulli numbers and polynomials. The Fibonacci Quarterly, 2001, 39(3): 279-284.
[24] G. D. Liu, The solution of problem for Euler numbers. Acta Mathematica Sinica (Chinese), 2004, 47(4): 825-828.
[25] R. K. Guy, Unsolved problem in number theory. New York: Springer-Verlag, 1994.
[26] G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number n. Quart. J. Math., 1917, 48: 76-92.
[27] L. K. Hun, On exponential sums. Sci Record(Peking)(N. S.), 1957, 1: 1-4.
[28] H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, volume 17. Providence: American Mathematical Society, 1997, 60-63.
[29] J. H. Ren and W. P. Zhang, On the average of Dirichlet L-functions. Journal of Sichuan University Natural Science Edition, 1990, 26(Special Issue): 75-79.
[30] J. Wang, On the Jacobi sums modulo p~n. Journal of Number theory, 1991, 39: 50-64.
[31] N. M. Katz, Gauss sums, Kloosterman sums, and monodromy groups. Princeton: Annals of Mathematics Studies, Princeton University Press, 1988.
[32] N. M. Katz, Estimates for "nonsingular" mutiplicative character sums. International Mathematics Research Notices, 2002, 7: 333-349.
[33] H. D. Kloosterman, On the representation of numbers in the form ax~2+by~2+cz~2+dt~2. Acta Math., 1926, 49: 407-464.
[34] L. Kuipers, Remark on a paper by R. L. Duncan concerning the uniform distribution rood 1 of the Logarithms of the Fibonacci numbers. The Fibonacci Quarterly, 1969, 5: 465-466.
[35] E. Lehmer, On Congruences Involving Bernoulli Numbers and the quotients of Fermat and Wilson. Annals of Math., 1938, 39: 350-360.
[36] R. Lidl and H. Niederreiter, Finite Fields. Addison-Wesley: Reading, MA, 1983.
[37] J. H. Loxton and R. A. Smith, On Hua's estimate for exponential sums. J. London Math. Soc. (2), 1982, 26: 15-20.
[38] J. H. Loxton and R. C. Vanghan, The estimate for complete exponential sums. Canada Math. Bull. (4), 1995, 26: 442-454.
[39] Minggao Lu, estimate of a complete trigonometric sum. Sci. Sinica Ser. A, 1985, 28: 561-578.
[40] 华罗庚．数论导引．北京：科学出版社，1979．
[41] 谢敏，张文鹏．关于广义Dedekind和的2k次均值．数学学报．2001，44：85-94．
[42] H. L. Montgomery and R. C. Vaughan, Mean value of character sums. Canadian Journal of Mathematics, 1979, 31(3): 476-487.
[43] L. J. Mordell, On a special polynomiaI congruence and exponential sums, in "Calcutta Math. Soc. Golden Jubilee Commemoration Volume" Part 1. Calcutta: Calcutta Math. Soc., 1963, 29-32.
[44] K. K. Norton. On character sums and power residues. Trans. Amer. Math. Soc., 1972, 167: 203-226.
[45] 潘承洞，潘承彪．解析数论基础．北京：科学出版社，1999．
[46] Juan C. Peral, Character sums and explicit estimates for L-functions. Contemporary Mathematics, 1995, 189: 449-459.
[47] Yuan Pingzhi, A conjecture on Euler numbers. Proc. Japan Acad. (Ser. A), 2004, 80: 180-181.
[48] G. Polya, Uber die Verteilung der quadratische Reste und Nichtreste. Gottingen Nachrichten, 1918, 21-29.
[49] H. Rademacher, On the transformation of log η(Υ). J. Indian. Math. soc., 1955, 19: 25-30.
[50] N. Robbins, Applications of Fibonacci Numbers. Kluwer Academic Publishers, 1986, pp. 77-88.
[51] K. G. Richard, Unsolved Problems in Number Theory. New York: SpringerVerlag, 1994.
[52] H. Salie, Uber die Kloostermanschen Summen S(u, v; q). Mathematische Zeitschrift, 1931, 34: 91-109.
[53] H. N. Shapiro, Introduction to the theory of numbers. New York: John Wiley and Sons, Inc., 1983, 304-305, 307-308.
[54] F. Smarandache, Collected Papers,Vol. Ⅲ. Bucharest: Tempus Publishing House, 1998.
[55] R. A. Smith, On n-dimensional Kloosterman sums. Journal of Number Theory, 1979, 11: 324-343.
[56] A. V. Sokolovskii, On a theorem of Sarkozy. Acta Arithmetica, 1982, 41: 27-31.
[57] Tabirca Sabin, About S-Multiplicative Functions. Octogon, 1999, 7: 169-170.
[58] I. M. Vinogradov, On the distribution of residues and non-residues of powers. Journal of the Physico-Mathematical society of Perm., 1918, 1: 94-96.
[59] A. Weil, On some exponential sums. Proc. Nat. Acad. Sci. U. S. A., 1948, 34: 204-207.
[60] W. P. Zhang, A problem of D. H. Lehmer and its generalization (Ⅱ). Compositio Mathematica, 1994, 91: 47-56.
[61] W. P. Zhang, On the difference between an integer and its inverse modulo n. Journal of Number Theory, 1995, 52: 1-6.
[62] W. P. Zhang, On the mean values of Dedekind sums. Jouunal de Theorie des Nombres de Bordeaux, 1996, 8: 429-442.
[63] W. P. Zhang, Some identities involving the Fibonacci numbers. The Fibonacci Quarterly, 1997, 35: 225-229.
[64] W. P. Zhang, Some Identities Involving The Euler and The Central Factorial Numbers. The Fibonacci Quarterly, 1998, 36(2): 154-157.
[65] W. P. Zhang, Y. Yi and X. L. He, On the 2k-th power mean of Dirichlet L-functions with the weight of general Kloosterman sums. Journal of Number Theory, 2000, 84: 199-213.
[66] W. P. Zhang, A note on the mean square value of the Dedekind sums. Acta Math. Hangar., 2000, 86: 275-289.
[67] W. P. Zhang, On the general Dedekind sums and one kind identities of Dirichlet L-functions. Acta Mathematica Sinica, 2001, 44: 269-272.
[68] W. P. Zhang and Y. Yi, On Dirichlet characters of polynomials Bull. London Math. Soc, 2002, 34: 469-473.
[69] W. P. Zhang, On a Cochrane sum and its hybrid mean value formula. Journal of Mathematical Analysis and Applications, 2002, 267: 89-96.
[70] W. P. Zhang, On a Cochrane sum and its hybrid mean value formula (Ⅱ). Journal of Mathematical Analysis and Applications, 2002, 276: 446-457.
[71] W. P. Zhang, On a sum analogous to Dedekind sum and its mean square value formula. International J. of Math. and Math. Sci., 2002, 32: 47-55.
[72] W. P. Zhang, On a problem of D. H. Lehmer and Kloosterman sums. Monatshefte fur Mathematik, 2003, 139: 247-257.
[73] W. P. Zhang, On the difference between an integer and its inverse modulo n (Ⅱ). Science in China (Series A), 2003, 46: 229-238.
[74] W. P. Zhang and H. N. Liu, A Note on the Cochrane sum and its hybrid mean value formula. Journal of Mathematical Analysis and Applications, 2003, 288: 646-659.
[75] W. P. Zhang, Z. B. Xu and Y. Yi, A problem of D. H. Lehmer and its mean square value formula. Journal of Number theory, 2003, 103: 193-203.
[76] W. P. Zhang, A problem of D. H. Lehmer and its mean square value formula. Japanese Journal of Mathematics, 2003, 29: 109-116.
[77] W. P. Zhang and W. L. Yao, A note on the Diriehlet characters of polynomials. Acta Arithmetica, 2004, 115: 225-229.
[78] W. P. Zhang, On the general Kloosterman sum and its fourth power mean. Journal of Number Theory, 2004, 104: 156-161.
[79] W. Z. Luo, Z. Rudnick and P. Sarnak, On Selberg's eigenvalue conjecture. Geometric and Functional Analysis, 1995, 5: 387-401.
[80] Y. B. Ye, Estimation of exponential sums of polynomials of higher degrees Ⅱ. Acta Arithmetica, 2000, XCⅢ. 3: 221-235.
[81] Yi Yuan and Zhang Wenpeng, On the 2k-th Power Mean of Dirichlet L-function With the Weight of Gauss Sums. Advances in Mathematics (in Chinese), 2000, 31: 518-526.