关于几个数论函数的方程及其相关问题
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摘要
众所周知,各个时代的数学家对于一些特殊序列及函数的算术性质的研究都十分重视,已取得了许多数论方面的具有理论意义的研究成果.新问题的出现比老问题的解决更快.著名的美籍罗马尼亚数论专家Florentin Smarandache在译为《只有问题,没有解答》一书中提出了许多有趣的问题和猜想.许多学者都对此进行了深入的研究,并获得了不少具有重要理论价值的研究成果.
本文基于对以上所述问题的兴趣,主要研究了数论函数的对数均值估计,以及一些包含伪Smarandache函数的特殊方程的正整数解的问题等.具体说来,本文的主要成果包括以下几方面:
1、研究了函数LS(x)的均值,并得到一个较强的渐近公式.
2、利用初等方法研究了一个包含Smarandache函数与伪Smarandache函数的方程Z(n)+S(n)=kn解的存在性并给出了方程所有的正整数解.
3、分析了著名的Smarandache函数S(n)的互反函数S_c(n)与伪Smarandache函数Z(n)的对偶函数Z~*(n)的关系,证明了方程S_c(n)=Z~*(n)+n,S_c(n)+Z(n)=2n,有无穷多个解,并给出部分解的表达形式.
4、利用初等方法研究了倒置毗连平方序列中的完全平方数.
5、.利用初等方法研究了无穷级数(?)的收敛性质,并给出了一些有趣的等式.
6、提出了一个新的数论函数,并研究了此函数的一些特殊性质.
It is well known that various times mathematicians devoted much attention to the properties of some special sequences and the arithmetic function, and obtained many significant research results on Number Theory. The new question appearance is quicker than the unsolved problem solution. Famous American nationality Romania theory of numbers theorist Florentin Smarandache in the book named "Only problems, Not solutions" introduced many interesting questions and conjectures. Many researchers have studied these sequences and functions, and obtained many value research results.
I develop an interest in the function of F. Smarandache. In this dissertation we mainly study the average value of some arithmetical function and discuss solutions of several special equations concerning Pseudo Smarandache function. The major achievements are as listed below.
1. Study the average value of the function LS(x) and achieve an accurate asymptoticformulate for it.
2. We use the elementary methods to discuss the solvability of a function equation concerning the pseudo Smarandache function and give its all positive integer solutions.
3. Analyze the properties of the pseudo Smarandache function Z(n) and the Smarandachereciprocal function S_c(n), prove that S_c(n) + Z(n) = 2n, S_c(n) = Z~*(n) + n . They haveinfinite positive integer solutions.
4. We use the elementary methods to study perfect square number in the back concatenated square sequence.
5. We study the convergence of this infinity series through the elementary means, and give some interesting results.
6. Introduce a new number theory function and study its properties.
本文基于对以上所述问题的兴趣,主要研究了数论函数的对数均值估计,以及一些包含伪Smarandache函数的特殊方程的正整数解的问题等.具体说来,本文的主要成果包括以下几方面:
1、研究了函数LS(x)的均值,并得到一个较强的渐近公式.
2、利用初等方法研究了一个包含Smarandache函数与伪Smarandache函数的方程Z(n)+S(n)=kn解的存在性并给出了方程所有的正整数解.
3、分析了著名的Smarandache函数S(n)的互反函数S_c(n)与伪Smarandache函数Z(n)的对偶函数Z~*(n)的关系,证明了方程S_c(n)=Z~*(n)+n,S_c(n)+Z(n)=2n,有无穷多个解,并给出部分解的表达形式.
4、利用初等方法研究了倒置毗连平方序列中的完全平方数.
5、.利用初等方法研究了无穷级数(?)的收敛性质,并给出了一些有趣的等式.
6、提出了一个新的数论函数,并研究了此函数的一些特殊性质.
It is well known that various times mathematicians devoted much attention to the properties of some special sequences and the arithmetic function, and obtained many significant research results on Number Theory. The new question appearance is quicker than the unsolved problem solution. Famous American nationality Romania theory of numbers theorist Florentin Smarandache in the book named "Only problems, Not solutions" introduced many interesting questions and conjectures. Many researchers have studied these sequences and functions, and obtained many value research results.
I develop an interest in the function of F. Smarandache. In this dissertation we mainly study the average value of some arithmetical function and discuss solutions of several special equations concerning Pseudo Smarandache function. The major achievements are as listed below.
1. Study the average value of the function LS(x) and achieve an accurate asymptoticformulate for it.
2. We use the elementary methods to discuss the solvability of a function equation concerning the pseudo Smarandache function and give its all positive integer solutions.
3. Analyze the properties of the pseudo Smarandache function Z(n) and the Smarandachereciprocal function S_c(n), prove that S_c(n) + Z(n) = 2n, S_c(n) = Z~*(n) + n . They haveinfinite positive integer solutions.
4. We use the elementary methods to study perfect square number in the back concatenated square sequence.
5. We study the convergence of this infinity series through the elementary means, and give some interesting results.
6. Introduce a new number theory function and study its properties.
引文
[1] Dumitrescu V.Seleacu. The Smarandache function [M].Erhus University press, 1996
[2] Florian Luca. The average Smarandache function [J]. Smarandache Notions Journal,2001,V12(1-2-3):19-27
[3] Florian Luca. On a series involving S(1)·S(2)…S(n) [J]. Smarandache Notions Journal,1999,V10(1-2-3):128-129
[4] 张文鹏．初等数论[M]．西安：陕西师范大学出版社，2007
[5] 潘承洞，潘承彪．解析数论基础[M]．北京：科学出版社，1999
[6] Apostol Tom M. Introduction to analytical number theory [M]. Spring-Verlag, New York,1976
[7] 刘燕妮，潘晓玮．两个包含Smarandache函数的方程及其解[J]．黑龙江大学学报(自然科学版)，2006，23(6)：857-858
[8] 乐茂华．关于Smarandache函数的一个猜想[J]．黑龙江大学学报(自然科学版)，2007，24(5)：687-688
[9] 徐哲峰．关于Smarandache函数的值分布[J]．数学学报，2006，49(5)：1009-1012
[10] 杜凤英．关于Smarandache函数S(n)的一个猜想[J]．纯粹数学与应用数学，2007，23(2)：205-208
[11] Lou Yuanbing. On the pseudo Smarandache function [J]. Scientia Magna. 2007, V3(4):48-50
[12] David Gorski. The pseudo Smarandache function [J]. Smarandache Notions Journal,2002, V13(1-2-3): 140-149
[13] 关文吉，郑亚妮．关于伪Smarandache函数的方程[J]．纺织高校基础科学学报，2008，21(2)：151-153
[14] 张爱玲．关于伪Smarandache函数的一个方程及其正整数解[J]．西北大学学报(自然科学版)，2008，34(4)：531-532
[15] J.Sandor. On a dual of the pseudo Smarandache function[J]. Smarandache Notions Journal, 2002, V13(1-2-3): 18-23
[16] 张文鹏．关于F．Smarandache函数的两个问题[J]．西北大学学报(自然科学版)，2008，38(2)：173-175
[17] 张爱玲．关于F．Smarandache函数的一个问题[J]．纯粹数学与应用数学，2008，24(2)：385-387
[18] A.Murthy.Smarandache reciprocal function and an elementary inequality [J]. Smarandache Notions Journal, 2000,V12(1-2-3):307-309.
[19] Ding Liping. On the Smarandache reciprocal function and its mean value [J]. Scientia Magna,2008,V4(1):120-123
[20] Ren Zhibin. On an equation involving the Smarandache reciprocal function and its positive integer solution [J]. Scientia Magna, 2008,V4(1):23-25
[2] Florian Luca. The average Smarandache function [J]. Smarandache Notions Journal,2001,V12(1-2-3):19-27
[3] Florian Luca. On a series involving S(1)·S(2)…S(n) [J]. Smarandache Notions Journal,1999,V10(1-2-3):128-129
[4] 张文鹏．初等数论[M]．西安：陕西师范大学出版社，2007
[5] 潘承洞，潘承彪．解析数论基础[M]．北京：科学出版社，1999
[6] Apostol Tom M. Introduction to analytical number theory [M]. Spring-Verlag, New York,1976
[7] 刘燕妮，潘晓玮．两个包含Smarandache函数的方程及其解[J]．黑龙江大学学报(自然科学版)，2006，23(6)：857-858
[8] 乐茂华．关于Smarandache函数的一个猜想[J]．黑龙江大学学报(自然科学版)，2007，24(5)：687-688
[9] 徐哲峰．关于Smarandache函数的值分布[J]．数学学报，2006，49(5)：1009-1012
[10] 杜凤英．关于Smarandache函数S(n)的一个猜想[J]．纯粹数学与应用数学，2007，23(2)：205-208
[11] Lou Yuanbing. On the pseudo Smarandache function [J]. Scientia Magna. 2007, V3(4):48-50
[12] David Gorski. The pseudo Smarandache function [J]. Smarandache Notions Journal,2002, V13(1-2-3): 140-149
[13] 关文吉，郑亚妮．关于伪Smarandache函数的方程[J]．纺织高校基础科学学报，2008，21(2)：151-153
[14] 张爱玲．关于伪Smarandache函数的一个方程及其正整数解[J]．西北大学学报(自然科学版)，2008，34(4)：531-532
[15] J.Sandor. On a dual of the pseudo Smarandache function[J]. Smarandache Notions Journal, 2002, V13(1-2-3): 18-23
[16] 张文鹏．关于F．Smarandache函数的两个问题[J]．西北大学学报(自然科学版)，2008，38(2)：173-175
[17] 张爱玲．关于F．Smarandache函数的一个问题[J]．纯粹数学与应用数学，2008，24(2)：385-387
[18] A.Murthy.Smarandache reciprocal function and an elementary inequality [J]. Smarandache Notions Journal, 2000,V12(1-2-3):307-309.
[19] Ding Liping. On the Smarandache reciprocal function and its mean value [J]. Scientia Magna,2008,V4(1):120-123
[20] Ren Zhibin. On an equation involving the Smarandache reciprocal function and its positive integer solution [J]. Scientia Magna, 2008,V4(1):23-25