  We propose a new basis in Witten's open string field theory, in which the
star product simplifies considerably. For a convenient choice of gauge the
classical string field equation of motion yields straightforwardly an exact
analytic solution that represents the nonperturbative tachyon vacuum. The
solution is given in terms of Bernoulli numbers and the equation of motion can
be viewed as novel Euler--Ramanujan-type identity. It turns out that the
solution is the Euler--Maclaurin asymptotic expansion of a sum over wedge
states with certain insertions. This new form is fully regular from the point
of view of level truncation. By computing the energy difference between the
perturbative and nonperturbative vacua, we prove analytically Sen's first
conjecture.
