  We study the distribution of graded degrees of freedom in classically stable
oriented closed string vacua and use the Rankin-Selberg transform to link it to
the finite one-loop vacuum energy. In particular, we find that the spectrum of
physical excitations not only must enjoy asymptotic supersymmetry but actually,
at very large mass, bosonic and fermionic states must follow a universal
oscillating pattern, whose frequencies are related to the zeros of the Riemann
zeta-function. Moreover, the convergence rate of the overall number of the
graded degrees of freedom to the value of the vacuum energy is determined by
the Riemann hypothesis. We discuss also attempts to obtain constraints in the
case of tachyon-free open-string theories.
