  The aim of the present paper is to study the distributions of the length
multiplicities for negatively curved locally symmetric Riemannian manifolds. In
Theorem 2.1, we give upper bounds of the length multiplicities and the square
sums of them for general (not necessarily compact) cases. Furthermore in
Theorem 2.2, we obtain more precise estimates of the length multiplicities and
the power sums of them for arithmetic surfaces whose fundamental groups are
congruence subgroups of the modular group.
