一些数论函数的均值估计
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摘要
众所周知,数论函数的均值估计问题在数论研究中占有十分重要的位置,许多著名的数论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必将对数论的发展起到重要的推动作用!
本文研究了一些特殊数论函数的均值估计问题,以及它们与一些重要函数之间的联系,有关和式的一系列加权均值。具体说来,本文的主要成果包括以下几方面:
1.Smarandache函数的均值问题,它与许多数论函数的均值有密切的关系。本文在第三章中研究了几类特殊Smarandache函数的均值问题,Smarandache函数与莫比乌斯函数之间的一个恒等式以及均值性质,Smarandache对序列以及它的性质,利用初等和解析的方法得出了一些新的渐近公式。
2.关于加性的Smarandache模拟函数的研究有着丰富的内容。在第四章中,本文研究了Smarandache单阶乘模拟函数以及Smarandache双阶乘模拟函数的均值性质,获得了一系列渐近公式。
3.研究了δ_k(n)函数倒数的性质及其均值估计问题,利用解析的方法得到关于δ_k(n)函数倒数的一个精确的计算公式。
It is well known that the mean value problems of number theory functions play an important role in the study of number theory, and they relate to many famous number theoretic problems. Therefore, any nontrivial progress in this field will contribute to the development of number theory.In this dissertation, we study the mean value problems of some special number theory functions, relation with some important functions and a series of asymptotic formulas about some hybrid functions. The main achievements contained in this dissertation are as follows:1. The study on the mean value of Smarandache functions play an important role in the study of number theory, and they relate to many famous number theoretic problems. In this dissertation, we study the mean value of some special number theory functions, an identity and mean value between Smarandache function and mobius function, Smarandache symmetric sequence and it' s properties, using elementary and analytic methods got a series of new asymptotic formulas.2. Additive analogue of smarandache functions enjoy their long history. In this chapter, We study the mean value of additive analogue of smarandache single factorial function and smarandache double factorial function, and several interesting asymptotic formulas are given.3. Properties of the reciprocal of δ_k(n) function and it' s mean value are study, using analytic methods give an exact calculating formula for the reciprocal of δ_k(n) function.
本文研究了一些特殊数论函数的均值估计问题,以及它们与一些重要函数之间的联系,有关和式的一系列加权均值。具体说来,本文的主要成果包括以下几方面:
1.Smarandache函数的均值问题,它与许多数论函数的均值有密切的关系。本文在第三章中研究了几类特殊Smarandache函数的均值问题,Smarandache函数与莫比乌斯函数之间的一个恒等式以及均值性质,Smarandache对序列以及它的性质,利用初等和解析的方法得出了一些新的渐近公式。
2.关于加性的Smarandache模拟函数的研究有着丰富的内容。在第四章中,本文研究了Smarandache单阶乘模拟函数以及Smarandache双阶乘模拟函数的均值性质,获得了一系列渐近公式。
3.研究了δ_k(n)函数倒数的性质及其均值估计问题,利用解析的方法得到关于δ_k(n)函数倒数的一个精确的计算公式。
It is well known that the mean value problems of number theory functions play an important role in the study of number theory, and they relate to many famous number theoretic problems. Therefore, any nontrivial progress in this field will contribute to the development of number theory.In this dissertation, we study the mean value problems of some special number theory functions, relation with some important functions and a series of asymptotic formulas about some hybrid functions. The main achievements contained in this dissertation are as follows:1. The study on the mean value of Smarandache functions play an important role in the study of number theory, and they relate to many famous number theoretic problems. In this dissertation, we study the mean value of some special number theory functions, an identity and mean value between Smarandache function and mobius function, Smarandache symmetric sequence and it' s properties, using elementary and analytic methods got a series of new asymptotic formulas.2. Additive analogue of smarandache functions enjoy their long history. In this chapter, We study the mean value of additive analogue of smarandache single factorial function and smarandache double factorial function, and several interesting asymptotic formulas are given.3. Properties of the reciprocal of δ_k(n) function and it' s mean value are study, using analytic methods give an exact calculating formula for the reciprocal of δ_k(n) function.
引文
[1] T. M. Apostol, Introduction to Analytic Number Theory. New York: Springer-Verlag, 1976.
[2] Jozsef Sandor, On an generalization of the Smarandache function, Notes Numb. Th. Discr. Math. 5(1999), pp, 41-51.
[3] 潘承洞,潘承彪.哥德巴赫猜想.北京:科学出版社,1981.
[4] 潘承洞,潘承彪.初等数论.北京:北京大学出版社,1992.
[5] 潘承洞,潘承彪.解析数论基础.北京:科学出版社,1999.
[6] K. G. Richard, Unsolved Problems in Number Theory. New York: Springer-Verlag, 1994.
[7] Charles Ashbacher. Collection of Problems on Smarandache Notions[M]. Erhus University Press: Vail, 1996.
[8] Krassimir T.Atanassov. On Some of the Smarandache's Problems[M]. American Research Press: Lupton, AZ USA, 1999.
[9] Kenichiro Kashihara. Comments and Topics On Smarandache Notions and Problems[M]. Erhus University Press, 1996.
[10] G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number n. Quart. J. Math., 1917, 48: 76-92.
[11] Richard Bellman. Analytic Number Theory-An Introduction[M]. University of Southern California, 1980.
[12] Kenneth Ireland and Michael Rosen. A Classical Introduction to Modern Number Theory (Vol. Ⅱ)
[13] M. Ram Murty. Problems in Analytic Number Theory[M]. New York: Springer-Verlag, 2001.
[14] Dumitrescu C and Seleacu V. Some notions and questions in Number Theory. Glendale: Erhus University Press, 1994.
[15] Felice Russo. Five properties of the Smarandache Double Factorial Function. Smarandache Notions Journal, 2002, 13: 183-185.
[16] 华罗庚.数论导引.北京:科学出版社,1979.
[17] Jing Gao and Huaning Liu. On the mean value of Smarandache double factorial function. Smarandache Notions Journal, 2004, 14: 193-195.
[18] Ibstedt. Surfining On the Ocean of Number-A New Smarandache Notions and Similar Topics[hl]. New Mexico: Erhus University Press, 1997.
[19] F. Mertens. Ein Beitray Zur Analytischen Zahlentheorie. Crelle's Journal[J]. 1874, 78: 46-62.
[20] K. G. Richard, Unsolved Problems in Number Theory. New York: Springer-Verlag, 1994.
[21] 柯召等.数论讲义[M].北京:高等教育出版社,1986.
[22] F. Smarandache.Only Problems, Not Solutions[M]. Chicago: Xiquan Publishing House, 1993.
[23] 闵嗣鹤.数论的方法[M]_北京:科学出版社,1981.
[24] 冯克勤.代数数论[M].北京:科学出版社,2000.
[25] Zhang Wenpeng: On the symmetric sequence and its some properties. A note on the Smarandache Notions Journal, 2002, 13: 150-152.
[26] J. HERZOG and T. MAXSEIN. On the behavior of a certain error-term. Arch Math, 1988, 50: 145-155.
[27] Jozsef Sandor, On an additive analogue of the function S, Smarandance Notes Numb. Th. Discr. Math. 5(2002), pp. 266-270.
[28] Mark Farris and Patrick Mitchell, Bounding the smarandache functoin, Smaramche Notes Journal 2002, 13: 37-42.
[29] Hardy G H and Wright E M. An Introduction to the Theory of Numbers. Oxford: Oxford University Press, 1979.
[30] T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory. New York: Springer-Verlag, 1976.
[31] H. N. Shapiro, Introduction to the theory of numbers. New York: John Wiley and Sons, Inc., 1983.
[32] Zhang Wengpeng. Research on Smarandache problems in Number Theory. The United States of America. 2004.
[33] Pan Chengdong and Pan Chengbiao. Foundation of Analytic Number Theory. Beijing: Science Press, 1977.
[34] 杜瑞芝.数学史辞典[M].山东教育出版社,2000.
[2] Jozsef Sandor, On an generalization of the Smarandache function, Notes Numb. Th. Discr. Math. 5(1999), pp, 41-51.
[3] 潘承洞,潘承彪.哥德巴赫猜想.北京:科学出版社,1981.
[4] 潘承洞,潘承彪.初等数论.北京:北京大学出版社,1992.
[5] 潘承洞,潘承彪.解析数论基础.北京:科学出版社,1999.
[6] K. G. Richard, Unsolved Problems in Number Theory. New York: Springer-Verlag, 1994.
[7] Charles Ashbacher. Collection of Problems on Smarandache Notions[M]. Erhus University Press: Vail, 1996.
[8] Krassimir T.Atanassov. On Some of the Smarandache's Problems[M]. American Research Press: Lupton, AZ USA, 1999.
[9] Kenichiro Kashihara. Comments and Topics On Smarandache Notions and Problems[M]. Erhus University Press, 1996.
[10] G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number n. Quart. J. Math., 1917, 48: 76-92.
[11] Richard Bellman. Analytic Number Theory-An Introduction[M]. University of Southern California, 1980.
[12] Kenneth Ireland and Michael Rosen. A Classical Introduction to Modern Number Theory (Vol. Ⅱ)
[13] M. Ram Murty. Problems in Analytic Number Theory[M]. New York: Springer-Verlag, 2001.
[14] Dumitrescu C and Seleacu V. Some notions and questions in Number Theory. Glendale: Erhus University Press, 1994.
[15] Felice Russo. Five properties of the Smarandache Double Factorial Function. Smarandache Notions Journal, 2002, 13: 183-185.
[16] 华罗庚.数论导引.北京:科学出版社,1979.
[17] Jing Gao and Huaning Liu. On the mean value of Smarandache double factorial function. Smarandache Notions Journal, 2004, 14: 193-195.
[18] Ibstedt. Surfining On the Ocean of Number-A New Smarandache Notions and Similar Topics[hl]. New Mexico: Erhus University Press, 1997.
[19] F. Mertens. Ein Beitray Zur Analytischen Zahlentheorie. Crelle's Journal[J]. 1874, 78: 46-62.
[20] K. G. Richard, Unsolved Problems in Number Theory. New York: Springer-Verlag, 1994.
[21] 柯召等.数论讲义[M].北京:高等教育出版社,1986.
[22] F. Smarandache.Only Problems, Not Solutions[M]. Chicago: Xiquan Publishing House, 1993.
[23] 闵嗣鹤.数论的方法[M]_北京:科学出版社,1981.
[24] 冯克勤.代数数论[M].北京:科学出版社,2000.
[25] Zhang Wenpeng: On the symmetric sequence and its some properties. A note on the Smarandache Notions Journal, 2002, 13: 150-152.
[26] J. HERZOG and T. MAXSEIN. On the behavior of a certain error-term. Arch Math, 1988, 50: 145-155.
[27] Jozsef Sandor, On an additive analogue of the function S, Smarandance Notes Numb. Th. Discr. Math. 5(2002), pp. 266-270.
[28] Mark Farris and Patrick Mitchell, Bounding the smarandache functoin, Smaramche Notes Journal 2002, 13: 37-42.
[29] Hardy G H and Wright E M. An Introduction to the Theory of Numbers. Oxford: Oxford University Press, 1979.
[30] T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory. New York: Springer-Verlag, 1976.
[31] H. N. Shapiro, Introduction to the theory of numbers. New York: John Wiley and Sons, Inc., 1983.
[32] Zhang Wengpeng. Research on Smarandache problems in Number Theory. The United States of America. 2004.
[33] Pan Chengdong and Pan Chengbiao. Foundation of Analytic Number Theory. Beijing: Science Press, 1977.
[34] 杜瑞芝.数学史辞典[M].山东教育出版社,2000.