AN EFFECTIVE BOUND FOR THE HUBER CONSTANT FOR COFINITE FUCHSIAN GROUPS

作者：Friedman J S^*; Jorgenson J; Kramer J

来源：Mathematics of Computation, 2011, 80(274): 1163-1196.

DOI：10.1090/S0025-5718-2010-02430-5

Let Gamma be a cofinite Fuchsian group acting on hyperbolic two-space H. Let M = Gamma\H be the corresponding quotient space. For gamma, a closed geodesic of M, let l(gamma) denote its length. The prime geodesic counting function pi(M)(u) is defined as the number of Gamma-inconjugate, primitive, closed geodesics gamma such that e(l(gamma)) <= u. The prime geodesic theorem states that:
mu(M)(u) = Sigma(0 <=lambda M,j <= 1/4) li(u(sM,j)) +O(M) (u(3/4)/log u),
where 0 = lambda(M,0) < lambda(M,1) < ... are the eigenvalues of the hyperbolic Laplacian acting on the space of smooth functions on M and s(M,j) = 1/2 + root 1/4 - lambda(M,j). Let C(M) be the smallest implied constant so that
|pi(M)(u) - Sigma(0 <=lambda M,j <= 1/4) li(u(sM,j))| <= C(M) u(3/4)/log u for all u > 1.
We call the (absolute) constant CM the Huber constant.
The objective of this paper is to give an effectively computable upper bound of C(M) for an arbitrary cofinite Fuchsian group. As a corollary we bound the Huber constant for PSL(2,Z), showing that C(M) <= 16,607,349,020,658 approximate to exp(30.44086643).

• 出版日期2011-4