Weighted Prime Powers Truncation of the Asymptotic Expansion for the Logarithmic Integral: Properties and Applications

Abstract

Given the asymptotic expansion for the logarithmic integral $\int_0^n \frac{dt}{\ln(t)}$, obtained from repeated integration by parts until the expansion terms reach a minimum; approaching zero. Which determines a cut-off for the number of terms in the expansion and this truncation is a function of $n$. By dropping the minimization constraint and introducing a new variable $x$ for the number of expansion terms, consider the question: Where to truncate the asymptotic expansion for the logarithmic integral to equal the prime count function $\pi(n)$?. Although constructing this new truncation function requires using the non-trivial zeros $\rho$ of the Riemann zeta function. There exists a closed form approximation that does not utilize the zeros at all. From this, a new bound is obtained on the summation $\sum_{\rho} li(n^{\rho})$. Which is then compared to an equivalent form of the Schoenfeld bound derived for this same summation. Resulting in a proof that the Riemann Hypothesis is true.