The correlation between multiplicities of closed geodesics on the modular surface (Q5961018)

The aim of the paper under review is to make rigorous part of an investigation by \textit{E. Bogomolny, F. Leyvraz} and \textit{C. Schmit} [Commun. Math. Phys. 176, 577-617 (1996; Zbl 0861.11034)] on the correlation of the eigenvalues of the Laplacian on the modular surface. The author proves an explicit product representation of the following type for the correlation function of the multiplicities of closed geodesics on the modular surface: For \(n>2\) let \(g(n)\) denote the number of closed geodesics on \(\text{SL}_2 (\mathbb{Z})\setminus\mathbb{H}\) which correspond to primitive conjugacy classes with trace \(n\) and let \(\beta(n):= g(n)(\log n)/n-1\). Then for every integer \(r\geq 0\) the limit \[ \gamma(r):= \lim_{N\to\infty} \sum_{n=2}^N \beta(n) \beta(n+r) \] exists, and its value is given by a complicated product of local factors which extends over all rational primes \(p\). This result was already stated and made plausible on heuristic grounds in the aforementioned paper by Bogomolny et al. The present approach is based on the connection between the numbers \(g(n)\) and class numbers of indefinite primitive binary quadratic forms and on Dirichlet's class number formula. The main step is to show that \(\beta\) is almost periodic and to compute its Fourier coefficients; then the result follows from Parseval's formula.