| 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 resultsTheorem1.1. 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 sumThe following are the resultsTheorem2.1. 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 areIn Chapter3, we consider the multiplication of the coefficients of the Dedekind zeta function in different fields, that is, the asymptotic estimation of the convolution sumWe get the following resultsTheorem3.1. 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 resultsTheorem4.1. 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. |
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