Riemann Weighted Prime-Power Counting Function

The Riemann weighted prime-power counting function is defined by

$$J_2(x) \equiv \left\{\begin{array}{ll}\pi(x)+{\textstyle{1\over 2}}\pi(x^{1/2})... ...(x^{1/3})+\ldots & \mbox{}\\ \quad {\rm otherwise} & \mbox{}\end{array}\right.$$ (1)

$$= \lim_{t\to\infty} {1\over 2\pi i} \int_{2-iT}^{2+iT} {x^s\over s} \ln \zeta(s)\,ds.$$ (2)

The Prime Counting Function is given in terms of $J_2(x)$ by

$$\pi(x)=\sum_{n=1}^\infty {\mu(n)\over n} J_2(x^{1/n}).$$ (3)

The function also satisfies the identity

$${\ln \zeta(s)\over s} = \int_1^\infty J_2(x)x^{-s-1}\,dx.$$ (4)

See also Mangoldt Function, Prime Counting Function, Riemann's Formula