  We prove an extension of the Landau-Gonek formula. As an application we
recover unconditionally some of the consequences of a pair correlation estimate
that previously was known under the Riemann hypothesis. As one corollary we
prove that at least two-thirds of the zeros of the zeta function are simple
under a zero density hypothesis, which is weaker than the Riemann hypothesis.
The results in this paper can be viewed as pair correlation estimates
independent of the Riemann hypothesis.
