我们开发了新的技术，使用抽象算子理论，以获得渐近公式的晶格计数问题的无限体积双曲流形，误差项是统一的晶格移动通过“同余”子群。我们给出仿射线性筛理论的下列应用。根据费马的思想，考虑两个平方和中的素数问题，f（c，d）=c^2+d^2，但将（c，d）限制在轨道O =（0，1）.Gamma，其中Gamma是SL（2，Z）的无限指数非初等有限生成子群。假设Reimann曲面Gamma \H^2在无穷远处有一个尖点。我们证明了f（O）的值集包含无穷多个整数，对于任意R>4/（delta-theta），至多有R个素因子，其中theta>1/2是谱隙，delta为<1 is the Hausdorff dimension of the limit set of Gamma. If delta>149/150，那么我们可以取theta=5/6，给出R=25。对于delta-theta>4/9，该方法的极限为R=9。这是相同数量的素因子达到布伦的原始攻击的孪生素数猜想。
英文摘要：
We develop novel techniques using abstract operator theory to obtain asymptotic formulae for lattice counting problems on infinite-volume hyperbolic manifolds, with error terms which are uniform as the lattice moves through "congruence" subgroups.
We give the following application to the theory of affine linear sieves. In the spirit of Fermat, consider the problem of primes in the sum of two squares, f(c,d)=c^2+d^2, but restrict (c,d) to the orbit O = (0,1).Gamma, where Gamma is an infinite-index non-elementary finitely-generated subgroup of SL(2,Z). Assume that the Reimann surface Gamma\H^2 has a cusp at infinity. We show that the set of values f(O) contains infinitely many integers having at most R prime factors for any R>4/(delta-theta), where theta>1/2 is the spectral gap and delta<1 is the Hausdorff dimension of the limit set of Gamma. If delta>149/150, then we can take theta=5/6, giving R=25. The limit of this method is R=9 for delta-theta>4/9. This is the same number of prime factors as attained in Brun's original attack on the twin prime conjecture.
筛被构造用于第一类孪生素数，其形式为2 m +/-D，D>=3奇数。它们的特征在于它们的双D-I秩m。他们没有平价问题。使用奇素数p>=5来识别和计数非秩数。Twin-D-I秩和非秩构成正整数的集合。非秩的正则性允许获得双D-I秩的数量。它涉及相当多的取消，使渐近形式的主要条款崩溃的预期形式，但其系数取决于D。
构造了一个距离为4的孪生素数的筛子，它们的形式为3（2 m +1）+/-2，其特征在于它们的孪生-4秩为2 m +1。它没有平价问题。非秩被标识为所有其他奇数，并使用奇素数p>=5计数。双4秩和非秩构成奇数的集合。非等级的规则允许收集关于它们的信息以获得双4等级的数量的Legendre类型的总和。由于其中有相当多的消去项，其主项的渐近律具有其系数的期望形式和大小。
Weighted enumeration of number-fields and counting points that take bounded squarefree values along certain polynomials using Pseudo and Sudo maximal orders
