两类虚二次域类数之间关系的一个注记

就素数d=5,13,29,53,173,293时给出了虚二次数域■(-k)与■(-kd)(k>2,k∈■)的类数之间的关系.     

实二次域上理想类的zeta-函数在-1处的值

用连分数给出了实二次域理想类的zeta-函数-1处值的一个具体的计算公式.     

实二次域上理想类的zeta-函数在-1处的值

用连分数给出了实二次域理想类的zeta-函数-1处值的一个具体的计算公式.     

实二次数域的一类L-函数在中心点的值

为了证明L（1/2,x）≠0,本文定义了实二次域的一类L-函数并给出其在在中心点≥处的值的表达式．     

关于简单实二次域的两个猜想

本文在假定广义Riemann猜想成立的条件下,证实了R.Mollin和H.Williams以及S.Louboutin分别在文献[2]、[3]中提出的两个有关简单实二次域的猜想.     

实二次域的Kronecker极限公式(Ⅱ)

本文讨论实二次域的带DirichIet特征的Kronecker极限公式,并由此得出某些虚二次域的类数公式。     

虚二次域的L-函数在中心点的值

给出了虚二次数域的一类L-函数在中心点的值,这为研究Dirichlet L-函数在中心点的算术意义提供了一条新的途径.     

类数公式的机器证明——Chowla一个猜想的注记

设素数 p=4N² + 1 (N为一个正整数 )使得实二次数域Q(√p)的类数为1 ,作为追随吴文俊数学定理机器证明方向的一次尝试 ,用微机证明了一些虚二次数域Q√( -qp)的类数公式 ,这里 q =3,7,1 1 ,1 9,2 3,31 ,43和 47.     

实二次域的一个同余式

本文证明了关于实二次域的类数和某类特征和的同余式,同时给出某类实二次域的类数可除性的一个判别法则．     

Dedekind zeta函数与Dedekind和

Dedekind和表示两个实二次数域的Dedekind zeta函数在-1处值的积,给出了不同于Siegel的表示公式. 为应用,得到ζ_K(-1)的一个多项式表示:1/45 (26n³-41n±9), n≡±2(mod5),这里K=Q(Ö5q),素数q = 4n²+1,且实二次数域K ₂=Q(Öq)的类数为1.     

Some Results Connected with the Class Number Problem in Real Quadratic Fields

We investigate arithmetic properties of certain subsets of square-free positive integers and obtain in this way some results concerning the class number h（d） of the real quadratic field Q（√d）. In particular, we give a new proof of the result of Hasse, asserting that in this case h（d） = 1 is possible only if d is of the form p, 2q or qr. where p.q. r are primes and q≡r≡3（mod 4）.     

Bounds of the Ideal Class Numbers of Real Quadratic Function Fields

The theory of continued fractions of functions is used to give a lower bound for class numbers h(D) of general real quadratic function fields K=k(D)over k=Fq(T)．For five series of real quadratic function fields K,the bounds of h(D)are given more explicitly,e．g.,if D=F^2 C．then h(D)≥degF／degP;if D=(SG)^2 cS．then h(D)≥degS／degP;if D=(A^m a)^2 A,then h(D)≥degA／degP,where P is an irreducible polynomial splitting in K,c∈Fq．In addition,three types of quadratic function fields K are found to have ideal class numbers bigger than one．     

Dihedral Congruence Primes and Class Fields of Real Quadratic Fields

We show that for a real quadratic field F the dihedral congruence primes with respect to F for cusp forms of weight k and quadratic nebentypus are essentially the primes dividing expressions of the form ε^k−1₊±1 where ε₊ is a totally positive fundamental unit of F. This extends work of Hida. Our results allows us to identify a family of (ray) class fields of F which are generated by torsion points on modular abelian varieties.     

The Problem on Class Numbers of Quadratic Number Fields

It is a survey of the problem on class numbers of quadratic number fields.     

Prime Producing Quadratic Polynomials and Class Number One or Two

Let d≡ 5 mod 8 be a positive square-free integer and let h(d) be the class number of the real quadratic field ℚ(√d). Let p be a divisor of d = pq and let . Assume that is prime or equal to 1 for all integers x with 0≤x<W. Under the assumption that the Riemann hypothesis is true, we prove that if , then h(d) < 2. Furthermore we show that h(d)< 2 implies d < 4245. In the case when there exists at least one split prime less than W, we prove the following results without any assumptions on the Riemann hypothesis. If then h< 2 or h = 4. If , then h≤ 2, h = 4 or h = 2^t−2, where t is the number of primes dividing d. In the case when h = 2^t−2 we have , where φ = 2 or 4. 2000 Mathematics Subject Classification: Primary–11R29     

实二次域上理想类的Zeta-函数在-3处的值

用简单连分数给出了实二次域理想类的Zeta-函数在-3处值的一个具体的计算公式.