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Bogomolny-Leyvraz-Schmit (1996) and Peter (2002) proposed an asymptoticformula for the correlation of the multiplicities in length spectrum on themodular surface, and Lukianov (2007) extended its asymptotic formula to theRiemann surfaces derived from the congruence subgroup $\Gam_0(n)$ and thequaternion type co-compact arithmetic groups. The coefficients of the leadingterms in these asymptotic formulas are described in terms of Euler productsover prime numbers, and they appear in eigenvalue statistic formulas found byRudnick (2005) and Lukianov (2007) for the Laplace-Beltrami operators on thecorresponding Riemann surfaces. In the present paper, we further extend theirasymptotic formulas to the higher level correlations of the multiplicities forany congruence subgroup of the modular group.

Correlations of multiplicities in length spectra for congruence subgroups