代数数域上算术函数的均值估计
摘要
历史上,在研究Fermat大定理和其它一些问题时,数学家们遇到了某些代数域中的代数整数不能唯一分解的困难.例如6=2.3=(1+51/2i).(1-51/2i),这两个分解是代数整数6在代数域K=Q((?))上完全不同的分解.这使得对Fermat大定理等问题的研究变得更复杂.
意识到这个问题后,在1845年Kummer给出了一个新的思想,提出了“理想数”的概念.对代数数域上的所有整数都可以嵌入到某个“理想数”中,这个“理想数”能够唯一分解成若干“素理想数”的乘积Kummer的‘‘理想数”的概念后来被称为代数数域整数环上的理想.这个思想现在已经发展为一个新的理论,即理想的唯一分解理论.也是现代代数数论与代数曲线理论的基础.
设K／Q是有理数域Q的代数扩张,且扩张次数为d.在代数数论对理想的研究中,理想的范是一个非常重要的概念.设a是数域K上的一个非零的整理想,且OK是代数数域K的整环.那么整理想a的范定义为ma=｜Oκ/α｜.它可以反映整理想许多的代数性质.
我们也可以用解析方法来研究代数数论.类似于Riemann zeta函数,Dedekind引进了一个新的函数,称为Dedekind zeta函数.对于扩张次数为d的代数数域K,它的Dedekind zeta函数(?)k(s)定义为其中a遍历K上所有的非零整理想,且nQ表示整理想a的范.
用aK(n)来表示K上范为n的整理想的个数,我们可以将Dedekind zeta函数重新写成它实际上是一个第n项系数为aK(n)的Dirichlet L函数.算术函数aκ(n)反映了数域K上的许多的代数性质.许多的数学家对此感兴趣,并作出了大量的研究.
Chandraseknaran和Good在文献[4]中证明了算术函数aK(n)是乘性函数,并且满足aκ(n)≤T(n)d,其中τ(n)是除数函数,且d=[K：Q].
很多作者(参见[4],[5],[38],[39],[42],[43],[44],[46]等)也对其作了深入的研究,给出了aκ(n,)的均值的渐近估计,并且给出了aκ(n)的l次均值估计.
本文中,我们讨论算术函数aK(n)在稀疏集上的均值估计,即考虑和式的估计,其中1≥2,m≥2为整数.
在第一章中,设K是Q的d次Galois扩张,利用代数数域中素理想分解定理,我们给出了算术函数aκ(n)的一种表示方法,构造出相应的L函数,利用分析中的方法,我们可以得到下面的结果
定理1.1.设K是Q的d≥2次Galois扩张,且l≥2为整数,当d是奇数时,我们有其中m=(C(d+1,2))1/d,Pm(t)是关于变量t的次数为m-1的多项式C(m,n)=m/(m-n)!n!,且ε>0是任意小的常数.
定理1.2.设K是Q的d≥2次Galois扩张,且2≥2为整数,当d是偶数时,我们有其中α=(C(d/2,1))l,β={(C(d+1,2))l-(C(d/2,1))l}/d,Pm(t)是关于变量t的次数为m-1的多项式,C(m,n)=m!/(m-n)!n!,且ε>0是任意小的常数.
上述定理假设了代数域K是Q的Galois扩张,我们要讨论其它的情况.在第二章中,我们讨论当K不是Q的Galois扩张时,和式(0.1)的估计.
设K3是Q三次非正规扩张,由不可约多项式f(x)=x3+ax2+bx+c给出.根据强Artin猜想,以及模形式Fourier系数的若干性质,利用对自守L函数相关的研究方法,我们对和式进行估计,得到以下结果
定理2.1.对于三次非正规扩张K3,有关系式其中c是常数,且ε>0是任意小常数.
定理2.2.对于三次非正规扩张K3,有关系式其中C1与C2为常数,且ε>0是任意小常数.
令K1与K2分别是两个不同的二次域.记aκi(n)(i=1,2)分别为在二次域K1与K2上范为n的整理想的个数.则它们的Dedekind zeta函数分别为
在第三章中,我们研究不同代数域上Dedekind zeta函数的系数乘积的均值估计,即关于卷积和的渐近估计.得到如下结果
定理3.1.设Ki=Q((?))(i=1,2)是判别式为di的二次域,且(d1,d2)=1.那么对任意的ε>0与整数1≥2,可以得到其中PK1,K2表示秩为4l-1-1的多项式.
定理3.2.设Ki=Q((?))(i=1,2)是判别式为di的二次域,且(Cd1,(d2)=1.那么对任意的ε>0与整数2≥2,可以得到其中PK1-1K2表示秩为M2+2M的多项式,且M=(3l-1)／2.我们还讨论了代数数域K上的k维除数问题.定义其中ai(i=1,2,…,k)为代数数域K上的非零整理想,且k≥1为整数.对有理数域Q上的Galois扩张,我们研究了和式在稀疏集上的分布.
在第四章中,我们得到以下结果
定理4.1.令K是关于Q的cd≥2次Galois扩张,当d是奇数时,我们可以得到其中k≥2是整数,m=(k2d+k)/2,Pm(t)是关于变量t的m-1次多项式,且ε>0是任意小的常数.
定理4.2.设K是二次域,且k≥2为整数.则当k≥3,我们可以得到更精确的余项其中m=k2+k,Pm(t)是关于变量t的m-1次多项式,且ε>0是任意小的常数.
In history, there is a trouble to be solved when studying the Fermat's last theorem and some others, that is, the algebraic integral numbers have no unique factorization in some algebraic number fields. For example,6=2·3=(1+51/2i)·(1-51/2i), these two decompositions are essentially different decompositions of the al-gebraic integer6in the field K=Q(51/2). This trouble made the research about Fermat's last theorem more complex.
Realizing the failure of unique factorization, in1845, Kummer gave a new idea that the integer in a number field would have to admit an embedding into a bigger domain of "ideal numbers", where unique decomposition into "ideal prime numbers" would hold. Kummer's concept of "ideal numbers" was later replaced by that of ideals of Ok, the ring of all integers in a number field K. This idea has become the theory of ideals now, which is called the theory of unique decomposition of ideals in algebra, and is the basic theory in modern algebraic number theory and algebraic curves.
Let K/Q be an algebraic extension over the rational number field Q with degree d. The concept of the norm of ideals palys an important role in studying the ideals in algebraic number theory. Assume that a is a non-zero ideal in a field K, and Ok the ring of integers of K. Then the norm of o is defined by (?)a=｜OK/a｜. It reflects many algebraic properties of the ideals.
We can also do the research in the algebraic number theory by using analytic method. Similarly to Riemann zeta function, Dedekind introduced a new function which is called Dedekind zeta function. For the algebraic number field K with degree d. Its Dedekind zeta function (?)k(s) is defined by where a runs over the non-zero integral ideals of K, and (?)a denotes the norm of the integral ideal a.
Denote ακ(n) the number of integral ideals in K with norm equal to n, then we can rewrite the Dedekind zeta function as It is actually a Dirichlet L-series with coefficients ακ(n) in the n-th terms. The arithmetic function ακ(n) reflects a lot of algebraic properties of the field K, it is an important arithmetic function in algebraic number theory. Many mathematicians were interested in and made a study of it.
Chandraseknaran and Good [4] showed that the arithmetic function ακ(n) is a multiplicative function, and satisfies ακ(n)≤τn)d, where τ(k) is the divisor function, and d=[K:Q].
Many authors(see [4],[5],[38],[39],[42],[43],[44],[46] etc.) determined the asymptotic estimation of ακ(n) and the l-th mean value of ακ(n).
In this paper, we will focus our attention on the estimation of mean val-ues of the arithmetic function ακ(n) over sparse sets, that is, the estimation of the sum where l≥2,m≥2are integers.
In Chapter1of this dissertation, we assume that K is a Galois extension of Q of degree d, by using the factorization of ideals in algebraic number field, we get the formula of aK(n), and construct the related L function, by using some analytic methods, we get the following results
Theorem1.1. Let K be a Galois extension of Q with degree d>2, and l>2an integer, when d is odd, we obtain where m=(C (d+1,2))l/d, Pm(t) is a polynomial in t with degree m-1, C(m, n)=m!/(m-n)!n!, and e>0is an arbitrarily small constant.
Theorem1.2. Let K be a Galois extension of Q with degree d>2, and l>2an integer, when d is even, we have where a=(C (d/2/,1))l,β=(C(d+1,2))-(C(d/2,1))ld, Pm(t) is a polynomial in t with degree m-l,C(m,n)=m!/(m-n)!n!, and ε>0is an arbitrarily small constant.
We suppose that K is the Galois extension of Q in the above discussion, now we will discuss other cases. In Chapter2, we consider the estimation of the sum (0.1) when K is a non-Galois extension of Q.
Assume that K3is a non-normal cubic extension of Q, which is deter-mined by the irreducible polynomial f(x)=x3+ax2+bx+c. According to Strong Artin conjecture, the properties of the Fourier coefficients of modular forms, by using the tools of studying L functions, we discuss the estimation of the sum
The following are the results
Theorem2.1. For the non-normal cubic field K we have the relation where c is a constant, and ε>0is an arbitrarily small constant.
Theorem2.2. For the non-normal cubic field K3, we have the relation where C1and C2are constants, and ε>0is an arbitrarily small constant.
Let K1and K2be different quadratic fields. Denote aKi(n)(i=1,2) be the number of integral ideals in the fields K1and K2with norm n respectively. Then their Dedekind zeta functions are
In Chapter3, we consider the multiplication of the coefficients of the Dedekind zeta function in different fields, that is, the asymptotic estimation of the convolution sum
We get the following results
Theorem3.1. Let Ki=Q(di1/2)(i=1,2) be the quadratic field of discriminant di. Assume that (d1,d2)=1. Then for any ε>0and any integer l>2, we have where PK1,K2denotes a suitable polynomial of degree4l-1-1.
Theorem3.2. Let Ki=Q(di1/2)(i=1,2) be the quadratic field of discriminant di. Assume that (d1,d2)=1. Then for any e>0and any integer l≥1, we have where Pk1,k2denotes a suitable polynomial of degree M2+2M, and M-(3'-l)/2.
We also discussed the k dimensional divisor problem in algebraic number field K. Define where αi(i=1,2,???,k) are the non-zero ideals in algebraic number field K, k≥1is an integer. We discuss the distribution of the sum in the sparse sets on some Galois extensions over Q.
In Chapter4, we get the results
Theorem4.1. Let K be a Galois extension of Q with degree d>2, when d is odd, we obtain where k>2is an integer, m=(k2d+k)/2, Pm(t) is a polynomial in t with degree m-1, and e>0is an arbitrarily small constant.
Theorem4.2. Let K be a quadratic field, k>2be an integer. Then we have when k>3, we can get a more precise error term as where m.=k2+k, Pm(t) is a polynomial in t with degree m-1, and ε>0is an arbitrarily small constant.
意识到这个问题后,在1845年Kummer给出了一个新的思想,提出了“理想数”的概念.对代数数域上的所有整数都可以嵌入到某个“理想数”中,这个“理想数”能够唯一分解成若干“素理想数”的乘积Kummer的‘‘理想数”的概念后来被称为代数数域整数环上的理想.这个思想现在已经发展为一个新的理论,即理想的唯一分解理论.也是现代代数数论与代数曲线理论的基础.
设K／Q是有理数域Q的代数扩张,且扩张次数为d.在代数数论对理想的研究中,理想的范是一个非常重要的概念.设a是数域K上的一个非零的整理想,且OK是代数数域K的整环.那么整理想a的范定义为ma=｜Oκ/α｜.它可以反映整理想许多的代数性质.
我们也可以用解析方法来研究代数数论.类似于Riemann zeta函数,Dedekind引进了一个新的函数,称为Dedekind zeta函数.对于扩张次数为d的代数数域K,它的Dedekind zeta函数(?)k(s)定义为其中a遍历K上所有的非零整理想,且nQ表示整理想a的范.
用aK(n)来表示K上范为n的整理想的个数,我们可以将Dedekind zeta函数重新写成它实际上是一个第n项系数为aK(n)的Dirichlet L函数.算术函数aκ(n)反映了数域K上的许多的代数性质.许多的数学家对此感兴趣,并作出了大量的研究.
Chandraseknaran和Good在文献[4]中证明了算术函数aK(n)是乘性函数,并且满足aκ(n)≤T(n)d,其中τ(n)是除数函数,且d=[K：Q].
很多作者(参见[4],[5],[38],[39],[42],[43],[44],[46]等)也对其作了深入的研究,给出了aκ(n,)的均值的渐近估计,并且给出了aκ(n)的l次均值估计.
本文中,我们讨论算术函数aK(n)在稀疏集上的均值估计,即考虑和式的估计,其中1≥2,m≥2为整数.
在第一章中,设K是Q的d次Galois扩张,利用代数数域中素理想分解定理,我们给出了算术函数aκ(n)的一种表示方法,构造出相应的L函数,利用分析中的方法,我们可以得到下面的结果
定理1.1.设K是Q的d≥2次Galois扩张,且l≥2为整数,当d是奇数时,我们有其中m=(C(d+1,2))1/d,Pm(t)是关于变量t的次数为m-1的多项式C(m,n)=m/(m-n)!n!,且ε>0是任意小的常数.
定理1.2.设K是Q的d≥2次Galois扩张,且2≥2为整数,当d是偶数时,我们有其中α=(C(d/2,1))l,β={(C(d+1,2))l-(C(d/2,1))l}/d,Pm(t)是关于变量t的次数为m-1的多项式,C(m,n)=m!/(m-n)!n!,且ε>0是任意小的常数.
上述定理假设了代数域K是Q的Galois扩张,我们要讨论其它的情况.在第二章中,我们讨论当K不是Q的Galois扩张时,和式(0.1)的估计.
设K3是Q三次非正规扩张,由不可约多项式f(x)=x3+ax2+bx+c给出.根据强Artin猜想,以及模形式Fourier系数的若干性质,利用对自守L函数相关的研究方法,我们对和式进行估计,得到以下结果
定理2.1.对于三次非正规扩张K3,有关系式其中c是常数,且ε>0是任意小常数.
定理2.2.对于三次非正规扩张K3,有关系式其中C1与C2为常数,且ε>0是任意小常数.
令K1与K2分别是两个不同的二次域.记aκi(n)(i=1,2)分别为在二次域K1与K2上范为n的整理想的个数.则它们的Dedekind zeta函数分别为
在第三章中,我们研究不同代数域上Dedekind zeta函数的系数乘积的均值估计,即关于卷积和的渐近估计.得到如下结果
定理3.1.设Ki=Q((?))(i=1,2)是判别式为di的二次域,且(d1,d2)=1.那么对任意的ε>0与整数1≥2,可以得到其中PK1,K2表示秩为4l-1-1的多项式.
定理3.2.设Ki=Q((?))(i=1,2)是判别式为di的二次域,且(Cd1,(d2)=1.那么对任意的ε>0与整数2≥2,可以得到其中PK1-1K2表示秩为M2+2M的多项式,且M=(3l-1)／2.我们还讨论了代数数域K上的k维除数问题.定义其中ai(i=1,2,…,k)为代数数域K上的非零整理想,且k≥1为整数.对有理数域Q上的Galois扩张,我们研究了和式在稀疏集上的分布.
在第四章中,我们得到以下结果
定理4.1.令K是关于Q的cd≥2次Galois扩张,当d是奇数时,我们可以得到其中k≥2是整数,m=(k2d+k)/2,Pm(t)是关于变量t的m-1次多项式,且ε>0是任意小的常数.
定理4.2.设K是二次域,且k≥2为整数.则当k≥3,我们可以得到更精确的余项其中m=k2+k,Pm(t)是关于变量t的m-1次多项式,且ε>0是任意小的常数.
In history, there is a trouble to be solved when studying the Fermat's last theorem and some others, that is, the algebraic integral numbers have no unique factorization in some algebraic number fields. For example,6=2·3=(1+51/2i)·(1-51/2i), these two decompositions are essentially different decompositions of the al-gebraic integer6in the field K=Q(51/2). This trouble made the research about Fermat's last theorem more complex.
Realizing the failure of unique factorization, in1845, Kummer gave a new idea that the integer in a number field would have to admit an embedding into a bigger domain of "ideal numbers", where unique decomposition into "ideal prime numbers" would hold. Kummer's concept of "ideal numbers" was later replaced by that of ideals of Ok, the ring of all integers in a number field K. This idea has become the theory of ideals now, which is called the theory of unique decomposition of ideals in algebra, and is the basic theory in modern algebraic number theory and algebraic curves.
Let K/Q be an algebraic extension over the rational number field Q with degree d. The concept of the norm of ideals palys an important role in studying the ideals in algebraic number theory. Assume that a is a non-zero ideal in a field K, and Ok the ring of integers of K. Then the norm of o is defined by (?)a=｜OK/a｜. It reflects many algebraic properties of the ideals.
We can also do the research in the algebraic number theory by using analytic method. Similarly to Riemann zeta function, Dedekind introduced a new function which is called Dedekind zeta function. For the algebraic number field K with degree d. Its Dedekind zeta function (?)k(s) is defined by where a runs over the non-zero integral ideals of K, and (?)a denotes the norm of the integral ideal a.
Denote ακ(n) the number of integral ideals in K with norm equal to n, then we can rewrite the Dedekind zeta function as It is actually a Dirichlet L-series with coefficients ακ(n) in the n-th terms. The arithmetic function ακ(n) reflects a lot of algebraic properties of the field K, it is an important arithmetic function in algebraic number theory. Many mathematicians were interested in and made a study of it.
Chandraseknaran and Good [4] showed that the arithmetic function ακ(n) is a multiplicative function, and satisfies ακ(n)≤τn)d, where τ(k) is the divisor function, and d=[K:Q].
Many authors(see [4],[5],[38],[39],[42],[43],[44],[46] etc.) determined the asymptotic estimation of ακ(n) and the l-th mean value of ακ(n).
In this paper, we will focus our attention on the estimation of mean val-ues of the arithmetic function ακ(n) over sparse sets, that is, the estimation of the sum where l≥2,m≥2are integers.
In Chapter1of this dissertation, we assume that K is a Galois extension of Q of degree d, by using the factorization of ideals in algebraic number field, we get the formula of aK(n), and construct the related L function, by using some analytic methods, we get the following results
Theorem1.1. Let K be a Galois extension of Q with degree d>2, and l>2an integer, when d is odd, we obtain where m=(C (d+1,2))l/d, Pm(t) is a polynomial in t with degree m-1, C(m, n)=m!/(m-n)!n!, and e>0is an arbitrarily small constant.
Theorem1.2. Let K be a Galois extension of Q with degree d>2, and l>2an integer, when d is even, we have where a=(C (d/2/,1))l,β=(C(d+1,2))-(C(d/2,1))ld, Pm(t) is a polynomial in t with degree m-l,C(m,n)=m!/(m-n)!n!, and ε>0is an arbitrarily small constant.
We suppose that K is the Galois extension of Q in the above discussion, now we will discuss other cases. In Chapter2, we consider the estimation of the sum (0.1) when K is a non-Galois extension of Q.
Assume that K3is a non-normal cubic extension of Q, which is deter-mined by the irreducible polynomial f(x)=x3+ax2+bx+c. According to Strong Artin conjecture, the properties of the Fourier coefficients of modular forms, by using the tools of studying L functions, we discuss the estimation of the sum
The following are the results
Theorem2.1. For the non-normal cubic field K we have the relation where c is a constant, and ε>0is an arbitrarily small constant.
Theorem2.2. For the non-normal cubic field K3, we have the relation where C1and C2are constants, and ε>0is an arbitrarily small constant.
Let K1and K2be different quadratic fields. Denote aKi(n)(i=1,2) be the number of integral ideals in the fields K1and K2with norm n respectively. Then their Dedekind zeta functions are
In Chapter3, we consider the multiplication of the coefficients of the Dedekind zeta function in different fields, that is, the asymptotic estimation of the convolution sum
We get the following results
Theorem3.1. Let Ki=Q(di1/2)(i=1,2) be the quadratic field of discriminant di. Assume that (d1,d2)=1. Then for any ε>0and any integer l>2, we have where PK1,K2denotes a suitable polynomial of degree4l-1-1.
Theorem3.2. Let Ki=Q(di1/2)(i=1,2) be the quadratic field of discriminant di. Assume that (d1,d2)=1. Then for any e>0and any integer l≥1, we have where Pk1,k2denotes a suitable polynomial of degree M2+2M, and M-(3'-l)/2.
We also discussed the k dimensional divisor problem in algebraic number field K. Define where αi(i=1,2,???,k) are the non-zero ideals in algebraic number field K, k≥1is an integer. We discuss the distribution of the sum in the sparse sets on some Galois extensions over Q.
In Chapter4, we get the results
Theorem4.1. Let K be a Galois extension of Q with degree d>2, when d is odd, we obtain where k>2is an integer, m=(k2d+k)/2, Pm(t) is a polynomial in t with degree m-1, and e>0is an arbitrarily small constant.
Theorem4.2. Let K be a quadratic field, k>2be an integer. Then we have when k>3, we can get a more precise error term as where m.=k2+k, Pm(t) is a polynomial in t with degree m-1, and ε>0is an arbitrarily small constant.
引文
[1]T. M. Apostol, Introduction to analytic number theory, UTM, Springer-Verlag, New York,1976.
[2]M. Artin, Algebra, Prentice Hall, Inc., Englewood Cliffs, NJ,1991.
[3]J. W. S. Cassels and A. Frohlich (Eds.), Algebraic Number Theory, Brighton, September 1-17, 1965, Academic Press/Thompson Book, Lon-don/Washington,1967.
[4]K. Chandraseknaran and A. Good, On the number of integeral ideals in Galois extensions, Monatsh. Math.95(1983),99-109.
[5]K. Chandrasekharan and R. Narasimhan, The approximate functional equa-tion for a class of zeta functions, Math. Ann.152(1963),30-64.
[6]J. W. Cogdell, Lectures on L functions, converse theorems, and functoriality for GLn, Lectures on automorphic L functions, ASM, Providence, RI,2004.
[7]H. Cohn, A Classical invitation to algebraic numbers and class field, New-York,1978.
[8]P. Deligne and J.-P. Serre, "Formes modulaires de poids", Ann. Sci. Ecole Norm. Sup., Ser.4,7(1974),507-530.
[9]F. Diamond and J. Shurman, A first course in modular forms, GTM 228, Springer-Verlag, New York,2005.
[10]E. Drza and L. Varukhina, On mean values of some arithmeitc functions in number fields, Diecrete mathematics 308(2008),4892-4899.
[11]K. Q. Feng, Algebraic number fields, Science Press, Beijing, 2000 (in Chinese).
[12]O. M. Fomenko, Distribution of lattice points on surfaces of second order, J. Math. Sci.83(1997),795-815.
[13]O. M. Fomenko, Mean values connected with the Dedekind zeta function, J. Math. Sci.150(3)(2008),2115-2122.
[14]O. M. Fomenko, The mean number of solutions of certain congruences, J. Math. Sci.105(2001),2257-2268.
[15]A. Frohlich and M. J. Taylor, Algebraic number theory. Cambridge Studies in Advanced Mathematics 27, Cambridge University Press, Cambridge,1993.
[16]S. Gelbert and H. Jacquet, A relation between automorphic representations of GL{2) and GL(3), Ann. Sci. Ecole Norm. Sup.11(4)(1978),471-542.
[17]A. Good, The square mean of Dirichlet series associated with cusp forms, Mathematika 29(2)(1982),278-295.
[18]D. R. Heath-Brown, On the density of the zeros of the Dedekind zeta function, Acta. Arith.33(1977),169-181.
[19]D. R. Heath-Brown, The fourth power moment of the Riemann zeta funtion, Proc. London Math. Soc.38(1979),385-422.
[20]D. R. Heath-Brown, The growth rate of the Dedekind zcta-function on the critical line, Acta Arith.49(1988),323-339.
[21]D. R. Heath-Brown, The twelfth power moment of the Rimeann zeta-function, Q. J. Math.29(1978),443-462.
[22]D. Hilbert, The theory of algebraic number fields, Springer-Verlag, Berlin, 1998.
[23]M. N. Huxley and N. Watt, The number of ideals in a quadratic field Ⅱ, Israel J. Math. Part A 120(2000),125-153.
[24]A. Ivic, Exponent pairs and the zeta function of Riemann, Studia Sci. Math. Hungar.15(1-3)(1980),157-181.
[25]A. Ivic, The Riemann zeta-function. theory and applications, Dover Publica-tions, Inc., Mineola, NY,2003.
[26]H. Iwaniec and E. Kowalski, Analytic number theory, Amer. Math. Soc. Col-loquium Publ.53, Amer. Math. Soc., Providence RI,2004.
[27]H. Iwaniec, Topics in classical automorphic forms, GSM 17, Amer. Math. Soc, Providence, RI,1997.
[28]H. Iwaniec, On mean values for Dirichlet polynomials and the Riemann zeta-function, J. London Math. Soc.22(2)(1980),39-45.
[29]M. Jutila, Lectures on a method in the theory of exponential sums, Tata Inst. Fund. Res. Lectures on Math, and Phys.80. Springer-Verlag, Berlin, 1987.
[30]A. A. Karatsuba, Basic analytic number theory, Springer-Verlag, Berlin,1993.
[31]H. Kim, Functoriality and number of solutions of congruences, Acta Arith. 128(3)(2007),235-243.
[32]H. Kim, Functoriality for the exterior square of GL4 and symmetric fourth of GL2, J. Amer. Math. Soc.16(2003),139-183.
[33]H. Kim and F. Shahidi, Symmetric cube L-functions for GL2 are entire, Ann. of Math.150(2)(1999),645-662.
[34]H. Kim and F. Shahidi, Funcorial products for GL2 x GL3 and functorial symmetric cube for GL2, Ann. of Math.155(2002),837-893.
[35]H. Kim and F. Shahidi, Cuspidality of symmetric power with applications, Duke Math. J.112(2002),177-197.
[36]M. Kiihleitner and W. G. Nowak, An omega theorem for a class of arithmetic functions, Math. Nachr.165(1994),79-98.
[37]M. Kiihleitner and W. G. Nowak, The average number of solutions of the Diophantine equation U2+V2=W3 and related arithematic function, Acta Math. Hungar.104(2004),225-240.
[38]E. Landau, Einfuhrung in die elementare and analytische theorie der alge-braischen zahlen und der ideale, Teubner,1927.
[39]H. X. Lao, On the distribution of integral ideals and Hecke Grofiencharacters, Chin. Ann. Math.31B(3)(2010),385-392.
[40]S. Lang, Algebraic number theory (Second edition). GTM 110, Springer-Verlag, New York,1994.
[41]S. Lang, Introduction to modular forms, Springer-Verlag, Berlin Heidelberg, New York, 1976.
[42]G. S. Lii, Mean values connected with the Dedekind zeta-function of a non-normal cubic field, Cent. Eur. J. Math. 11(2)(2013),274-282.
[43]G. S. Lii, On mean values of some arithmetic functions in number fields, Acta. Math. Hungar.132(4)(2010), 348-357.
[44]G. S. Lu and Y. H. Wang, Note on the number of integral ideals in Galois extension, Sci. China Ser. A 53(9)(2010),2417-2424.
[45]G. S. Lu and Z. S. Yang, On the number of Integral ideals in number fields submitted to Monatshefte fiir Mathematik.
[46]G. S. Lii and Z. S. Yang, The average behavior of the coefficients of Dedekind zeta function over square numbers, Journal of Number Theory 131(2011), 1924-1938.
[47]T. Meurman, On the mean square of the Riemann zeta function, Quart J Math (2) 38(1987),337-343.
[48]T. Meurman, The mean twelfth power of Dirichlet L functions on the critical line, Ann. Acd. Sci. Fenn. Ser. A Math. Dissertationes 52(1984),44 pp.
[49]T. Mitsui, On the prime ideal theorem, J. Math. Soc. Japan. 20(1968),233-247.
[50]M. Miiller, on the asymptotic behavior of the ideal counting function in quadratic number fields, Monatsh. Math.108(1989),301-323.
[51]W. Miiller, On the distribution of ideals in cubic number fields, Monatsh. Math.106(1988),211-219.
[52]W. Narkiewicz, Elementary and analytic theory of algebraic numbers,2nd ed., Springer, Polish Scientific Publishers & PWN, Warszawa, 1990.
[53]J. Neukirch, Algebraic number theory, Science Press, Beijing, 2007.
[54]W. G. Nowak, On the distribution of integral ideals in algebraic number theory fields, Math. Nachr.161(1993),59-74.
[55]C. D. Pan and C. B. Pan, Fundamentals of analytic number theory, Science Press, Beijing, 1991 (in Chinese).
[56]E. I. Panteleeva, Dirichlet divisor problem in number fields, Matematicheskie Zametki 44(4)(1988),494-505.
[57]E. I. Panteleeva, On the mean values of certain arithmetic functions, Matem-aticheskie Zametki 55(2)(1994),109-117.
[58]H. Rademacher, On the Phragmen-Linderof theorem and some applications, Math Z 72(1959),192-204.
[59]A. Sankaranarayanan, Fundamental properties of symmetric square L func-tions I. Illinois J. Math.46(1)(2002),23-43.
[60]J. P. Serre, Linear representations of finite groups, GTM 42, Springer-Verlag, New York,1971.
[61]H. P. F. Swinnerton-Dyer, A brife guide to algebraic number theory, London Math. Soc. Stud. Texts,50, Cambridge University Press, Cambridge, 2001.
[62]E. C. Titchmarsh, The theory of the Riemann zeta-function, Clarendon Press, Oxford,1986.
[63]H. Weber, Lehrbuch der Algebra, Vol. Ⅱ, Braunschweig,1896.
[2]M. Artin, Algebra, Prentice Hall, Inc., Englewood Cliffs, NJ,1991.
[3]J. W. S. Cassels and A. Frohlich (Eds.), Algebraic Number Theory, Brighton, September 1-17, 1965, Academic Press/Thompson Book, Lon-don/Washington,1967.
[4]K. Chandraseknaran and A. Good, On the number of integeral ideals in Galois extensions, Monatsh. Math.95(1983),99-109.
[5]K. Chandrasekharan and R. Narasimhan, The approximate functional equa-tion for a class of zeta functions, Math. Ann.152(1963),30-64.
[6]J. W. Cogdell, Lectures on L functions, converse theorems, and functoriality for GLn, Lectures on automorphic L functions, ASM, Providence, RI,2004.
[7]H. Cohn, A Classical invitation to algebraic numbers and class field, New-York,1978.
[8]P. Deligne and J.-P. Serre, "Formes modulaires de poids", Ann. Sci. Ecole Norm. Sup., Ser.4,7(1974),507-530.
[9]F. Diamond and J. Shurman, A first course in modular forms, GTM 228, Springer-Verlag, New York,2005.
[10]E. Drza and L. Varukhina, On mean values of some arithmeitc functions in number fields, Diecrete mathematics 308(2008),4892-4899.
[11]K. Q. Feng, Algebraic number fields, Science Press, Beijing, 2000 (in Chinese).
[12]O. M. Fomenko, Distribution of lattice points on surfaces of second order, J. Math. Sci.83(1997),795-815.
[13]O. M. Fomenko, Mean values connected with the Dedekind zeta function, J. Math. Sci.150(3)(2008),2115-2122.
[14]O. M. Fomenko, The mean number of solutions of certain congruences, J. Math. Sci.105(2001),2257-2268.
[15]A. Frohlich and M. J. Taylor, Algebraic number theory. Cambridge Studies in Advanced Mathematics 27, Cambridge University Press, Cambridge,1993.
[16]S. Gelbert and H. Jacquet, A relation between automorphic representations of GL{2) and GL(3), Ann. Sci. Ecole Norm. Sup.11(4)(1978),471-542.
[17]A. Good, The square mean of Dirichlet series associated with cusp forms, Mathematika 29(2)(1982),278-295.
[18]D. R. Heath-Brown, On the density of the zeros of the Dedekind zeta function, Acta. Arith.33(1977),169-181.
[19]D. R. Heath-Brown, The fourth power moment of the Riemann zeta funtion, Proc. London Math. Soc.38(1979),385-422.
[20]D. R. Heath-Brown, The growth rate of the Dedekind zcta-function on the critical line, Acta Arith.49(1988),323-339.
[21]D. R. Heath-Brown, The twelfth power moment of the Rimeann zeta-function, Q. J. Math.29(1978),443-462.
[22]D. Hilbert, The theory of algebraic number fields, Springer-Verlag, Berlin, 1998.
[23]M. N. Huxley and N. Watt, The number of ideals in a quadratic field Ⅱ, Israel J. Math. Part A 120(2000),125-153.
[24]A. Ivic, Exponent pairs and the zeta function of Riemann, Studia Sci. Math. Hungar.15(1-3)(1980),157-181.
[25]A. Ivic, The Riemann zeta-function. theory and applications, Dover Publica-tions, Inc., Mineola, NY,2003.
[26]H. Iwaniec and E. Kowalski, Analytic number theory, Amer. Math. Soc. Col-loquium Publ.53, Amer. Math. Soc., Providence RI,2004.
[27]H. Iwaniec, Topics in classical automorphic forms, GSM 17, Amer. Math. Soc, Providence, RI,1997.
[28]H. Iwaniec, On mean values for Dirichlet polynomials and the Riemann zeta-function, J. London Math. Soc.22(2)(1980),39-45.
[29]M. Jutila, Lectures on a method in the theory of exponential sums, Tata Inst. Fund. Res. Lectures on Math, and Phys.80. Springer-Verlag, Berlin, 1987.
[30]A. A. Karatsuba, Basic analytic number theory, Springer-Verlag, Berlin,1993.
[31]H. Kim, Functoriality and number of solutions of congruences, Acta Arith. 128(3)(2007),235-243.
[32]H. Kim, Functoriality for the exterior square of GL4 and symmetric fourth of GL2, J. Amer. Math. Soc.16(2003),139-183.
[33]H. Kim and F. Shahidi, Symmetric cube L-functions for GL2 are entire, Ann. of Math.150(2)(1999),645-662.
[34]H. Kim and F. Shahidi, Funcorial products for GL2 x GL3 and functorial symmetric cube for GL2, Ann. of Math.155(2002),837-893.
[35]H. Kim and F. Shahidi, Cuspidality of symmetric power with applications, Duke Math. J.112(2002),177-197.
[36]M. Kiihleitner and W. G. Nowak, An omega theorem for a class of arithmetic functions, Math. Nachr.165(1994),79-98.
[37]M. Kiihleitner and W. G. Nowak, The average number of solutions of the Diophantine equation U2+V2=W3 and related arithematic function, Acta Math. Hungar.104(2004),225-240.
[38]E. Landau, Einfuhrung in die elementare and analytische theorie der alge-braischen zahlen und der ideale, Teubner,1927.
[39]H. X. Lao, On the distribution of integral ideals and Hecke Grofiencharacters, Chin. Ann. Math.31B(3)(2010),385-392.
[40]S. Lang, Algebraic number theory (Second edition). GTM 110, Springer-Verlag, New York,1994.
[41]S. Lang, Introduction to modular forms, Springer-Verlag, Berlin Heidelberg, New York, 1976.
[42]G. S. Lii, Mean values connected with the Dedekind zeta-function of a non-normal cubic field, Cent. Eur. J. Math. 11(2)(2013),274-282.
[43]G. S. Lii, On mean values of some arithmetic functions in number fields, Acta. Math. Hungar.132(4)(2010), 348-357.
[44]G. S. Lu and Y. H. Wang, Note on the number of integral ideals in Galois extension, Sci. China Ser. A 53(9)(2010),2417-2424.
[45]G. S. Lu and Z. S. Yang, On the number of Integral ideals in number fields submitted to Monatshefte fiir Mathematik.
[46]G. S. Lii and Z. S. Yang, The average behavior of the coefficients of Dedekind zeta function over square numbers, Journal of Number Theory 131(2011), 1924-1938.
[47]T. Meurman, On the mean square of the Riemann zeta function, Quart J Math (2) 38(1987),337-343.
[48]T. Meurman, The mean twelfth power of Dirichlet L functions on the critical line, Ann. Acd. Sci. Fenn. Ser. A Math. Dissertationes 52(1984),44 pp.
[49]T. Mitsui, On the prime ideal theorem, J. Math. Soc. Japan. 20(1968),233-247.
[50]M. Miiller, on the asymptotic behavior of the ideal counting function in quadratic number fields, Monatsh. Math.108(1989),301-323.
[51]W. Miiller, On the distribution of ideals in cubic number fields, Monatsh. Math.106(1988),211-219.
[52]W. Narkiewicz, Elementary and analytic theory of algebraic numbers,2nd ed., Springer, Polish Scientific Publishers & PWN, Warszawa, 1990.
[53]J. Neukirch, Algebraic number theory, Science Press, Beijing, 2007.
[54]W. G. Nowak, On the distribution of integral ideals in algebraic number theory fields, Math. Nachr.161(1993),59-74.
[55]C. D. Pan and C. B. Pan, Fundamentals of analytic number theory, Science Press, Beijing, 1991 (in Chinese).
[56]E. I. Panteleeva, Dirichlet divisor problem in number fields, Matematicheskie Zametki 44(4)(1988),494-505.
[57]E. I. Panteleeva, On the mean values of certain arithmetic functions, Matem-aticheskie Zametki 55(2)(1994),109-117.
[58]H. Rademacher, On the Phragmen-Linderof theorem and some applications, Math Z 72(1959),192-204.
[59]A. Sankaranarayanan, Fundamental properties of symmetric square L func-tions I. Illinois J. Math.46(1)(2002),23-43.
[60]J. P. Serre, Linear representations of finite groups, GTM 42, Springer-Verlag, New York,1971.
[61]H. P. F. Swinnerton-Dyer, A brife guide to algebraic number theory, London Math. Soc. Stud. Texts,50, Cambridge University Press, Cambridge, 2001.
[62]E. C. Titchmarsh, The theory of the Riemann zeta-function, Clarendon Press, Oxford,1986.
[63]H. Weber, Lehrbuch der Algebra, Vol. Ⅱ, Braunschweig,1896.