  The microscopic formula for the degeneracies of 1/8 BPS black holes in type
II string theory compactified on a six dimensional torus can be expressed as a
sum of several terms. One of the terms is a function of the Cremmer-Julia
invariant and gives the leading contribution to the entropy in the large charge
limit. The other terms, which give exponentially subleading contribution,
depend not only on the Cremmer-Julia invariant, but also on the arithmetic
properties of the charges, and in fact exist only when the charges satisfy
special arithmetic properties. We identify the origin of these terms in the
macroscopic formula for the black hole entropy, based on quantum entropy
function, as the contribution from non-trivial saddle point(s) in the path
integral of string theory over the near horizon geometry. These saddle points
exist only when the charge vectors satisfy the arithmetic properties required
for the corresponding term in the microscopic formula to exist. Furthermore the
leading contribution from these saddle points in the large charge limit agrees
with the leading asymptotic behaviour of the corresponding term in the
degeneracy formula.
