  We study spin-1/2 Heisenberg XXX antiferromagnet. The spectrum of the
Hamiltonian was found by Hans Bethe in 1931. We study the probability of
formation of ferromagnetic string in the antiferromagnetic ground state, which
we call emptiness formation probability P(n). This is the most fundamental
correlation function. We prove that for the short strings it can be expressed
in terms of the Riemann zeta function with odd arguments, logarithm ln 2 and
rational coefficients. This adds yet another link between statistical mechanics
and number theory. We have obtained an analytical formula for P(5) for the
first time. We have also calculated P(n) numerically by the Density Matrix
Renormalization Group. The results agree quite well with the analytical ones.
Furthermore we study asymptotic behavior of P(n) at finite temperature by
Quantum Monte-Carlo simulation. It also agrees with our previous analytical
results.
