  We explicitly construct a diffeomorphic pair (p(x),p^{-1}(x)) in terms of an
appropriate quadric spline interpolating the prime series. These continuously
differentiable functions are the smooth analogs of the prime series and the
prime counting function, respectively, and contain the basic information about
the specific behavior of the primes. We employ p^{-1}(x) to find approximate
solutions of Diophantine equations over the primes and discuss how this
function could eventually be used to analyze the von Koch estimate for the
error in the prime number theorem which is known to be equivalent to the
Riemann hypothesis.
