关于数论中一些著名和式的均值研究
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摘要
众所周知，关于一些著名和式的均值分布问题在解析数论研究中占有十分重要的地位，许多著名的难题都与之密切相关．因而在这一领域取得任何实质性进展都将有可能对解析数论的发展起到重要的推动作用!
本文主要研究了数论中一些著名和式的均值估计问题，其中包括短区间上的特征和、带特征的二项指数和、Cochrane和、Kloosterman和以及它们的各种推广和式．此外，还研究了D．H．Lehmer问题、整数及其逆的推广等．具体说来，本文的主要成果包括以下几方面：
1．短区间上特征和的高次均值研究．研究了短区间[1，(?))上特征和与广义二次Gauss和、广义Kloosterman和的混合均值，得到了几个渐近公式．
2．超级Cochrane和的均值研究．研究了一种类似于Dedekind和的和式的高维形式——超级Cochrane和的均值性质，给出了其上界估计的一个推广；此外，还利用Gauss和与原特征的性质，以及Dirichlet L-函数的均值定理等研究了超级Cochrane和与超级Kloosterman和的混合均值．
3．广义Kloosterman和的均值研究．研究了广义二次Kloosterman和、广义Bernoulli数以及Gauss和的混合均值，此外，还研究了广义二次Kloosterman和与L'/L(1，x)的混合均值，获得了几个有趣的结果；建立了r次超级Kloosterman和与D．H．Lehmer问题之间的一个联系．
4．带特征的二项指数和的均值研究．研究了带特征的二项指数和、广义Bernoulli数以及Gauss和的混合均值，还研究了带特征的二项指数和与L'/L(1，x)的混合均值，并给出了几个渐近公式．
5．整数及其逆的推广．关于整数及其逆分布问题的研究有助于我们深入了解整数分布的性质，本章利用高维Kloosterman和以及三角和估计研究了整数及其逆的高次形式的均值性质，得到了一个渐近公式；这一结果后来被Shparlinski加以推广和改进．
It is well known that the mean value problems of some famous summations play an important role in the study of analytic number theory, and they relate to many famous number theoretic problems. Therefore, any nontrivial progress in this field will contribute to the development of analytic number theory.
In this dissertation, the mean value of some famous summations including character sums over a short interval, two-term exponential sums with Dirichlet character, Cochrane sums, Kloosterman sums and their generalized sums, are studied. Besides these, problems concerning an integer and its inverse are also studied. The main achievements contained in this dissertation are as follows:
1. The study of character sums over a short interval is one of the hottest issues in analytic number theory. In this dissertation, the hybrid mean value involving the high power mean of character sums over the short interval [1,(?)), general quadratic Gauss sums and general Kloosterman sums is studied, and sev-eral asymptotic formulae are given.
2. The study of Dedekind sums enjoys its long history. Cochrane sumsis studied, which is analogous to Dedekind sums, and generalized to the multiple varieties case as hyper Cochrane sums. An upper bound of hyper Cochrane sums is generalized. By using the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet L-functions, the hybrid mean value between hyper Cochrane sums and hyper Kloosterman sums is discussed.
3. There has been much progress in the study of Kloosterman sums in recent years. The hybrid mean value including the general quadratic Kloosterman sums is considered, and some connections between the r-th hyper Kloosterman sums and a problem of D.H. Lehmer are established.
4. The study of exponential sums also enjoys its long history. In this dissertation, the hybrid mean value of the two-term exponential sums with Dirichlet character is studied, and several interesting mean value formulae are obtained.
5. The study on the problem of an integer and its inverse will help us know more about the distributions of integers. In this dissertation, the high power mean of an integer and its inverse is studied, and an interesting asymptotic formula is given, which is sharpened by Shparlinski later.
本文主要研究了数论中一些著名和式的均值估计问题，其中包括短区间上的特征和、带特征的二项指数和、Cochrane和、Kloosterman和以及它们的各种推广和式．此外，还研究了D．H．Lehmer问题、整数及其逆的推广等．具体说来，本文的主要成果包括以下几方面：
1．短区间上特征和的高次均值研究．研究了短区间[1，(?))上特征和与广义二次Gauss和、广义Kloosterman和的混合均值，得到了几个渐近公式．
2．超级Cochrane和的均值研究．研究了一种类似于Dedekind和的和式的高维形式——超级Cochrane和的均值性质，给出了其上界估计的一个推广；此外，还利用Gauss和与原特征的性质，以及Dirichlet L-函数的均值定理等研究了超级Cochrane和与超级Kloosterman和的混合均值．
3．广义Kloosterman和的均值研究．研究了广义二次Kloosterman和、广义Bernoulli数以及Gauss和的混合均值，此外，还研究了广义二次Kloosterman和与L'/L(1，x)的混合均值，获得了几个有趣的结果；建立了r次超级Kloosterman和与D．H．Lehmer问题之间的一个联系．
4．带特征的二项指数和的均值研究．研究了带特征的二项指数和、广义Bernoulli数以及Gauss和的混合均值，还研究了带特征的二项指数和与L'/L(1，x)的混合均值，并给出了几个渐近公式．
5．整数及其逆的推广．关于整数及其逆分布问题的研究有助于我们深入了解整数分布的性质，本章利用高维Kloosterman和以及三角和估计研究了整数及其逆的高次形式的均值性质，得到了一个渐近公式；这一结果后来被Shparlinski加以推广和改进．
It is well known that the mean value problems of some famous summations play an important role in the study of analytic number theory, and they relate to many famous number theoretic problems. Therefore, any nontrivial progress in this field will contribute to the development of analytic number theory.
In this dissertation, the mean value of some famous summations including character sums over a short interval, two-term exponential sums with Dirichlet character, Cochrane sums, Kloosterman sums and their generalized sums, are studied. Besides these, problems concerning an integer and its inverse are also studied. The main achievements contained in this dissertation are as follows:
1. The study of character sums over a short interval is one of the hottest issues in analytic number theory. In this dissertation, the hybrid mean value involving the high power mean of character sums over the short interval [1,(?)), general quadratic Gauss sums and general Kloosterman sums is studied, and sev-eral asymptotic formulae are given.
2. The study of Dedekind sums enjoys its long history. Cochrane sumsis studied, which is analogous to Dedekind sums, and generalized to the multiple varieties case as hyper Cochrane sums. An upper bound of hyper Cochrane sums is generalized. By using the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet L-functions, the hybrid mean value between hyper Cochrane sums and hyper Kloosterman sums is discussed.
3. There has been much progress in the study of Kloosterman sums in recent years. The hybrid mean value including the general quadratic Kloosterman sums is considered, and some connections between the r-th hyper Kloosterman sums and a problem of D.H. Lehmer are established.
4. The study of exponential sums also enjoys its long history. In this dissertation, the hybrid mean value of the two-term exponential sums with Dirichlet character is studied, and several interesting mean value formulae are obtained.
5. The study on the problem of an integer and its inverse will help us know more about the distributions of integers. In this dissertation, the high power mean of an integer and its inverse is studied, and an interesting asymptotic formula is given, which is sharpened by Shparlinski later.
引文
[1] Apostol T.M., Introduction to Analytic Number Theory, New York, Springer-Verlag, 1976.
[2] Bump D., Duke W., Hoffstein J., Iwaniec H., An estimate for the Hecke eigen-values of Maass forms, International Mathematics Research Notices, 1992, 4:75-81.
[3] Burgess D.A., On Dirichlet characters of polynomials, Proc. London Math. Soc., 1963, 13: 537-548.
[4] Burgess D.A.. On a conjecture of Norton, Acta Arithmetica, 1975, 27: 265-267.
[5] Burgess D.A., Mean value of character sums, Mathematika, 1986, 33: 1-5.
[6] Burgess D.A., Mean value of character sums (Ⅱ), Mathematika, 1987, 33: 1-7.
[7] Chowla S., On Kloosterman's sum, Norkse Vid. Selbsk. Fak. Frondheim, 1967, 40: 70-72.
[8] Cochrane T., Zheng Z.Y., Pure and mixed exponential sums, Acta Arith-metica, 1999, 91: 249-278.
[9] Cochrane T., Zheng Z.Y., Bounds for certain exponential sums, Loo-Keng Hua: a great mathematician of the twentieth century, Asian J. Math., 2000, 4: 757-773.
[10] Cochrane T., Zheng Z.Y., Upper bounds on a two-term exponential sum, Science in China (Series A), 2001, 44: 1003-1015.
[11] Dabrowski R., Fisher B., A stationary phase formula for exponential sums over Z/p~m Z and applications to GL(3)-Kloosterman sums, Acta Arithmetica, 1997, 80: 1-48.
[12] Darvenport H., Heibronn H., On an exponential sum, Proc. London Math. Soc, 1936, 41: 449-453.
[13] Davenport H., Multiplicative number theory, New York, Springer-Verlag, 1980.
[14] Duke W., Iwaniec H., A relation between cubic exponential Kloosterman sums, Contemporary Mathematics, 1993, 143: 255-258.
[15] Estermann T., On Kloosterman's sum, Mathematica, 1961, 8: 83-86.
[16] Hua L.K., On exponential sums, Sci Record(Peking)(N.S.), 1957, 1: 1-4.
[17]华罗庚，数论导引，北京，科学出版社，1979．
[18] Iwaniec H., Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, 1997, 17: 61-63.
[19] Kanemitsu S., Tanigawa Y., Yi Y., Zhang W.P., On general Kloosterman sums, Ann. Univ. Sci. Budapest. Sect. Comput., 2003, 22: 151-160.
[20] Katz N.M., Gauss sums, Kloosterman sums, and monodromy groups, Prince-ton, Annals of Mathematics Studies, Princeton University Press, 1988.
[21] Kloosterman H.D., On the representation of numbers in the form ax~2 + by~2 + cz~2 + dt~2. Acta Math., 1926, 49: 407-464.
[22] Liu H.N., Zhang W.P., On the hybrid mean value of Gauss sums and gen-eralized Bernoulli numbers, Proc. Japan Acad. Ser. A Math. Sci., 2004, 80: 113-115.
[23] Liu H.N.. Zhang X.B., On the mean value of L'/L (1,x), Journal of Mathemat-ical Analysis and Applications, 2006, 320: 562-577.
[24] Liu H. N., A note on the upper bound estimate of high-dimensional Cochrane sums, Journal of Number Theory, 2007, 125: 7-13.
[25] Liu H.N., Zhang W.P., Hybrid mean value of generalized Bernoulli numbers, general Kloosterman sums and Gauss sums, Journal of Korean Mathematical Society, 2007, 44: 11-24.
[26] Liu H.N., On a problem of D.H. Lehmer and hyper-Kloosterman sums, Ad-vances in Mathematics(China), 2007, 36(2): 245-252.
[27] Liu H.Y., Zhang W.P., On the general k-th Kloosterman sums and its fourth power mean, Chinese Ann. Math. Ser. B, 2004, 25: 97-102.
[28] Loxton J.H., Smith R.A., On Hua's estimate for exponential sums, J. London Math. Soc.(2), 1982, 26:15-20.
[29] Loxton J.H., Vaughan R.C., The estimate for complete exponential sums, Canada Math. Bull.(4), 1995, 26:442-454.
[30] Luo W.Z., Z. Rudnick, P. Sarnak, On Selberg's eigenvalue conjecture, Geo-metric and Functional Analysis, 1995, 5: 387-401.
[31] Malyshev A.V., A generalization of Kloostermann sums and their estimates (in Russian), Vestnik Leningrad Univ., 1960, 15: 59-75.
[32] Mordell L.J., On a special polynomial congruence and exponential sums, in "Calcutta Math.Soc.Golden Jubilee Commemoration Volume" Part 1, Cal-cutta: Calcutta Math.Soc, 1963, 29-32.
[33] Norton K.K., On character sums and power residues,Trans. Amer. Math. Soc., 1972, 167: 203-226.
[34]潘承洞，潘承彪，哥德巴赫猜想，北京，科学出版社，1981．
[35]潘承洞，潘承彪，初等数论，北京，北京大学出版社，1992．
[36]潘承洞，潘承彪，解析数论基础，北京，科学出版社，1999．
[37] P(?)lya G., ber die Verteilung der quadratische Reste und Nichtreste, (?)ber die Verteilung der quadratische Reste und Nichtreste, G(o|¨)ttingen Nachrichten, 1918, 167: 21-29
[38] Guy R.K., Unsolved Problems in Number Theory, New York, Springer-Verlag, 1994.
[39] Sali(?) H., Uber die Kloostermanschen Summen S(u,v;q), Mathematische Zeitschrift, 1931, 34: 91-109.
[40] Shparlinski I.E., On the distribution of points on multidimensional modular hyperbolas, Proc. Japan Acad. Ser. A Math. Sci., 2007, 2: 5-9.
[41] Siegel C.L., Uber die Classenzahl quadratischer Zahlkorper, Acta Arith-metica. 1935, 1: 83-86.
[42] Sokolovskii A.V., On a theorem of Sarkozy, Acta Arithmetics, 1982, 41: 27-31.
[43] Smith R.A., On n-dimensional Kloosterman sums, Journal of Number Theory, 1979, 11: 324-343.
[44] Takeo F., On Kronecker's limit formula for Dirichlet series with periodic co-efficients, Acta Arithmetica, 1990, 55: 59-73.
[45] Vinogradov I.M., On the distribution of residues and non-residues of powers, Journal of the Physico-Mathematical society of Perm., 1918, 1: 94-96.
[46] Weil A., On some exponential sums, Proc. Nat. Acad. Sci. U.S.A., 1948, 34: 204-207.
[47] Weinstein L., The hyper-Kloosterman sum, Enseign. Math., 1981, 27: 29-40.
[48]谢敏，张文鹏，关于广义Dedekind和的2k次均值，数学学报，2001，44：86-94．
[49] Xu Z.F., Zhang W.P., On the order of the high-dimensional Cocharane sum and its mean value, Journal of Number Theory, 2006, 117: 131-145.
[50] Xu Z.F., Zhang W.P., On the 2kth power mean of the character sums over short intervals, Acta Arithmetica, 2006,121: 149-160.
[51] Xu Z.F., Zhang W.P., On a problem of D.H.Lehmer over short intervals, Journal of Mathematical Analysis and Applications, 2006, 320: 756-770.
[52] Xu Z.F., Zhang T.P., Zhang W.P., On the mean value of the two-term ex-ponential sums with Dirichlet characters, Journal of Number Theory, 2007, 123(2): 352-362.
[53] Xu Z.F., Zhang W.P., The mean value of Hardy sums in short intervals, Proceedings of the Royal Society of Edinburgh, 2007, 137(4): 885-894.
[54] Ye Y.B., Estimation of exponential sums of polynomials of higher degrees (Ⅱ), Acta Arithmetica, 2000, XCⅢ.3: 221-235.
[55] Yi Y., Zhang W.P., On the generalization of a problem of D.H. Lehmer, Kyushu Journal of Mathematics, 2002,56: 235-241.
[56]张天平，张文鹏，关于超级Cochrane和的混合型均值，数学年刊(A辑)，2005，26：469-476．
[57] Zhang T.P., Zhang W.P., On the r-th hyper-Kloosterman sums and its hybrid mean value, Journal of Korean Mathematical Society, 2006, 43: 1199-1217.
[58] Zhang T.P., Zhang W.P., A generalization on the difference between an inte-ger and its inverse modulo q (Ⅰ), Acta Mathematica Sinica, 2008, 24: 215-222.
[59]张文鹏，关于D．H．Lehmer问题，科学通报，1992，31：1351-1354．
[60] Zhang W.P., On a problem of D.H. Lehmer and its generalizaion (Ⅱ), Chinese Journal of Contemporary Mathematics, 1993, 86: 307-316.
[61] Zhang W.P., On a problem of D.H. Lehmer and its generalizaion (Ⅱ), Com-positio Mathematica, 1994,91: 47-56.
[62] Zhang W.P., On the difference between an integer and its inverse modulo n, Journal of Number Theory, 1995,52:1-6.
[63] Zhang W.P., On the mean values of Dedekind sums,Journal de Theoriedes Nombres de Bordeaux, 1996, 8: 429-442.
[64] Zhang W.P., Yi Y., He X.L., 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.
[65] Zhang W.P., On the general Dedekind sums and one kind identities of Dirich-let L-functions, Acta Mathematica Sinica, 2001, 44: 269-272.
[66] Zhang W.P., Yi Y., On the upper bound estimate of Cochrane sums, Soochow Journal of Mathematics, 2002,28: 297-304.
[67] Zhang W.P., On a Cochrane sum and its hybrid mean value formula, Journal of Mathematical Analysis and Applications, 2002, 267: 89-96.
[68] Zhang W.P., On a Cochrane sum and its hybrid mean value formula (Ⅱ), Journal of Mathematical Analysis and Applications, 2002, 276: 446-457.
[69] Zhang W.P., Moments of generalized quadratic Gauss sums weighted by L-functions, Journal of Number Theory, 2002, 92: 304-314.
[70] Zhang W.P., Deng Y.P., A hybrid mean value of the inversion of L- functions and general quadratic Gauss sums, Nagoya Math. J., 2002, 167: 1-15.
[71] Zhang W.P., Liu H.N., A note on the Cochrane sum and its hybrid mean value formula, Journal of Mathematical Analysis and Applications, 2003, 288: 646-659.
[72] Zhang W.P., On a problem of D. H. Lehmer and Kloosterman sums, Monat-shefte fur Mathematik, 2003, 139: 247-257.
[73] Zhang W.P., On the difference between an integer and its inverse modulo n (Ⅱ), Science in China (Series A), 2003, 46: 229-238.
[74] Zhang W.P., On the general Kloosterman sums and its fourth power mean, Journal of Number Theory, 2004,104: 156-161.
[75] Zhang W.P., The first power mean of the inversion of L- functions weighted by quadratic Gauss sums, Acta Mathematica Sinica, 2004, 20: 283-292.
[76] Zhang W.P., Wang X.Y., On the fourth power mean of the character sums over short intervals, Acta Mathematica Sinica, 2007, 23 : 153-164.
[2] Bump D., Duke W., Hoffstein J., Iwaniec H., An estimate for the Hecke eigen-values of Maass forms, International Mathematics Research Notices, 1992, 4:75-81.
[3] Burgess D.A., On Dirichlet characters of polynomials, Proc. London Math. Soc., 1963, 13: 537-548.
[4] Burgess D.A.. On a conjecture of Norton, Acta Arithmetica, 1975, 27: 265-267.
[5] Burgess D.A., Mean value of character sums, Mathematika, 1986, 33: 1-5.
[6] Burgess D.A., Mean value of character sums (Ⅱ), Mathematika, 1987, 33: 1-7.
[7] Chowla S., On Kloosterman's sum, Norkse Vid. Selbsk. Fak. Frondheim, 1967, 40: 70-72.
[8] Cochrane T., Zheng Z.Y., Pure and mixed exponential sums, Acta Arith-metica, 1999, 91: 249-278.
[9] Cochrane T., Zheng Z.Y., Bounds for certain exponential sums, Loo-Keng Hua: a great mathematician of the twentieth century, Asian J. Math., 2000, 4: 757-773.
[10] Cochrane T., Zheng Z.Y., Upper bounds on a two-term exponential sum, Science in China (Series A), 2001, 44: 1003-1015.
[11] Dabrowski R., Fisher B., A stationary phase formula for exponential sums over Z/p~m Z and applications to GL(3)-Kloosterman sums, Acta Arithmetica, 1997, 80: 1-48.
[12] Darvenport H., Heibronn H., On an exponential sum, Proc. London Math. Soc, 1936, 41: 449-453.
[13] Davenport H., Multiplicative number theory, New York, Springer-Verlag, 1980.
[14] Duke W., Iwaniec H., A relation between cubic exponential Kloosterman sums, Contemporary Mathematics, 1993, 143: 255-258.
[15] Estermann T., On Kloosterman's sum, Mathematica, 1961, 8: 83-86.
[16] Hua L.K., On exponential sums, Sci Record(Peking)(N.S.), 1957, 1: 1-4.
[17]华罗庚，数论导引，北京，科学出版社，1979．
[18] Iwaniec H., Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, 1997, 17: 61-63.
[19] Kanemitsu S., Tanigawa Y., Yi Y., Zhang W.P., On general Kloosterman sums, Ann. Univ. Sci. Budapest. Sect. Comput., 2003, 22: 151-160.
[20] Katz N.M., Gauss sums, Kloosterman sums, and monodromy groups, Prince-ton, Annals of Mathematics Studies, Princeton University Press, 1988.
[21] Kloosterman H.D., On the representation of numbers in the form ax~2 + by~2 + cz~2 + dt~2. Acta Math., 1926, 49: 407-464.
[22] Liu H.N., Zhang W.P., On the hybrid mean value of Gauss sums and gen-eralized Bernoulli numbers, Proc. Japan Acad. Ser. A Math. Sci., 2004, 80: 113-115.
[23] Liu H.N.. Zhang X.B., On the mean value of L'/L (1,x), Journal of Mathemat-ical Analysis and Applications, 2006, 320: 562-577.
[24] Liu H. N., A note on the upper bound estimate of high-dimensional Cochrane sums, Journal of Number Theory, 2007, 125: 7-13.
[25] Liu H.N., Zhang W.P., Hybrid mean value of generalized Bernoulli numbers, general Kloosterman sums and Gauss sums, Journal of Korean Mathematical Society, 2007, 44: 11-24.
[26] Liu H.N., On a problem of D.H. Lehmer and hyper-Kloosterman sums, Ad-vances in Mathematics(China), 2007, 36(2): 245-252.
[27] Liu H.Y., Zhang W.P., On the general k-th Kloosterman sums and its fourth power mean, Chinese Ann. Math. Ser. B, 2004, 25: 97-102.
[28] Loxton J.H., Smith R.A., On Hua's estimate for exponential sums, J. London Math. Soc.(2), 1982, 26:15-20.
[29] Loxton J.H., Vaughan R.C., The estimate for complete exponential sums, Canada Math. Bull.(4), 1995, 26:442-454.
[30] Luo W.Z., Z. Rudnick, P. Sarnak, On Selberg's eigenvalue conjecture, Geo-metric and Functional Analysis, 1995, 5: 387-401.
[31] Malyshev A.V., A generalization of Kloostermann sums and their estimates (in Russian), Vestnik Leningrad Univ., 1960, 15: 59-75.
[32] Mordell L.J., On a special polynomial congruence and exponential sums, in "Calcutta Math.Soc.Golden Jubilee Commemoration Volume" Part 1, Cal-cutta: Calcutta Math.Soc, 1963, 29-32.
[33] Norton K.K., On character sums and power residues,Trans. Amer. Math. Soc., 1972, 167: 203-226.
[34]潘承洞，潘承彪，哥德巴赫猜想，北京，科学出版社，1981．
[35]潘承洞，潘承彪，初等数论，北京，北京大学出版社，1992．
[36]潘承洞，潘承彪，解析数论基础，北京，科学出版社，1999．
[37] P(?)lya G., ber die Verteilung der quadratische Reste und Nichtreste, (?)ber die Verteilung der quadratische Reste und Nichtreste, G(o|¨)ttingen Nachrichten, 1918, 167: 21-29
[38] Guy R.K., Unsolved Problems in Number Theory, New York, Springer-Verlag, 1994.
[39] Sali(?) H., Uber die Kloostermanschen Summen S(u,v;q), Mathematische Zeitschrift, 1931, 34: 91-109.
[40] Shparlinski I.E., On the distribution of points on multidimensional modular hyperbolas, Proc. Japan Acad. Ser. A Math. Sci., 2007, 2: 5-9.
[41] Siegel C.L., Uber die Classenzahl quadratischer Zahlkorper, Acta Arith-metica. 1935, 1: 83-86.
[42] Sokolovskii A.V., On a theorem of Sarkozy, Acta Arithmetics, 1982, 41: 27-31.
[43] Smith R.A., On n-dimensional Kloosterman sums, Journal of Number Theory, 1979, 11: 324-343.
[44] Takeo F., On Kronecker's limit formula for Dirichlet series with periodic co-efficients, Acta Arithmetica, 1990, 55: 59-73.
[45] Vinogradov I.M., On the distribution of residues and non-residues of powers, Journal of the Physico-Mathematical society of Perm., 1918, 1: 94-96.
[46] Weil A., On some exponential sums, Proc. Nat. Acad. Sci. U.S.A., 1948, 34: 204-207.
[47] Weinstein L., The hyper-Kloosterman sum, Enseign. Math., 1981, 27: 29-40.
[48]谢敏，张文鹏，关于广义Dedekind和的2k次均值，数学学报，2001，44：86-94．
[49] Xu Z.F., Zhang W.P., On the order of the high-dimensional Cocharane sum and its mean value, Journal of Number Theory, 2006, 117: 131-145.
[50] Xu Z.F., Zhang W.P., On the 2kth power mean of the character sums over short intervals, Acta Arithmetica, 2006,121: 149-160.
[51] Xu Z.F., Zhang W.P., On a problem of D.H.Lehmer over short intervals, Journal of Mathematical Analysis and Applications, 2006, 320: 756-770.
[52] Xu Z.F., Zhang T.P., Zhang W.P., On the mean value of the two-term ex-ponential sums with Dirichlet characters, Journal of Number Theory, 2007, 123(2): 352-362.
[53] Xu Z.F., Zhang W.P., The mean value of Hardy sums in short intervals, Proceedings of the Royal Society of Edinburgh, 2007, 137(4): 885-894.
[54] Ye Y.B., Estimation of exponential sums of polynomials of higher degrees (Ⅱ), Acta Arithmetica, 2000, XCⅢ.3: 221-235.
[55] Yi Y., Zhang W.P., On the generalization of a problem of D.H. Lehmer, Kyushu Journal of Mathematics, 2002,56: 235-241.
[56]张天平，张文鹏，关于超级Cochrane和的混合型均值，数学年刊(A辑)，2005，26：469-476．
[57] Zhang T.P., Zhang W.P., On the r-th hyper-Kloosterman sums and its hybrid mean value, Journal of Korean Mathematical Society, 2006, 43: 1199-1217.
[58] Zhang T.P., Zhang W.P., A generalization on the difference between an inte-ger and its inverse modulo q (Ⅰ), Acta Mathematica Sinica, 2008, 24: 215-222.
[59]张文鹏，关于D．H．Lehmer问题，科学通报，1992，31：1351-1354．
[60] Zhang W.P., On a problem of D.H. Lehmer and its generalizaion (Ⅱ), Chinese Journal of Contemporary Mathematics, 1993, 86: 307-316.
[61] Zhang W.P., On a problem of D.H. Lehmer and its generalizaion (Ⅱ), Com-positio Mathematica, 1994,91: 47-56.
[62] Zhang W.P., On the difference between an integer and its inverse modulo n, Journal of Number Theory, 1995,52:1-6.
[63] Zhang W.P., On the mean values of Dedekind sums,Journal de Theoriedes Nombres de Bordeaux, 1996, 8: 429-442.
[64] Zhang W.P., Yi Y., He X.L., 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.
[65] Zhang W.P., On the general Dedekind sums and one kind identities of Dirich-let L-functions, Acta Mathematica Sinica, 2001, 44: 269-272.
[66] Zhang W.P., Yi Y., On the upper bound estimate of Cochrane sums, Soochow Journal of Mathematics, 2002,28: 297-304.
[67] Zhang W.P., On a Cochrane sum and its hybrid mean value formula, Journal of Mathematical Analysis and Applications, 2002, 267: 89-96.
[68] Zhang W.P., On a Cochrane sum and its hybrid mean value formula (Ⅱ), Journal of Mathematical Analysis and Applications, 2002, 276: 446-457.
[69] Zhang W.P., Moments of generalized quadratic Gauss sums weighted by L-functions, Journal of Number Theory, 2002, 92: 304-314.
[70] Zhang W.P., Deng Y.P., A hybrid mean value of the inversion of L- functions and general quadratic Gauss sums, Nagoya Math. J., 2002, 167: 1-15.
[71] Zhang W.P., Liu H.N., A note on the Cochrane sum and its hybrid mean value formula, Journal of Mathematical Analysis and Applications, 2003, 288: 646-659.
[72] Zhang W.P., On a problem of D. H. Lehmer and Kloosterman sums, Monat-shefte fur Mathematik, 2003, 139: 247-257.
[73] Zhang W.P., On the difference between an integer and its inverse modulo n (Ⅱ), Science in China (Series A), 2003, 46: 229-238.
[74] Zhang W.P., On the general Kloosterman sums and its fourth power mean, Journal of Number Theory, 2004,104: 156-161.
[75] Zhang W.P., The first power mean of the inversion of L- functions weighted by quadratic Gauss sums, Acta Mathematica Sinica, 2004, 20: 283-292.
[76] Zhang W.P., Wang X.Y., On the fourth power mean of the character sums over short intervals, Acta Mathematica Sinica, 2007, 23 : 153-164.