  Letting the truncation point for the Logarithmic Integral's asymptotic
expansion be the variable to solve for, though not to be minimized as with the
usual Stieltjes truncation. Instead solved with respect to the prime
truncation, producing exactly the prime count function $\pi(n)$ by controlling
the number of terms in the asymptotic expansion. This prime truncation varies
about an average function, whose derivation and application is the focus of
this paper. An approximate closed form solution for the average prime
truncation is constructed and its properties examined. From this, a new bound
is obtained on the summation $\sum_{\rho} li(n^{\rho})$ over the non-trivial
zeros $\rho$, of the Riemann zeta function.
