Riemann-Mangoldt Function

$$f(x) = \sum_{n>1}^\infty {\pi_0(x^{1/n})\over n}$$

$$= \mathop{\rm Li}\nolimits (x)-\sum_{{\rm nontrivial\ }\rho\atop \z... ...\rho)=0}\mathop{\rm ei}(\rho\ln x)-\ln 2+\int_x^\infty {dt\over t(t^2-1)\ln t},$$ (1)

where $\zeta(z)$ is the Riemann Zeta Function, $\mathop{\rm Li}\nolimits (x)$ is the Logarithmic Integral and $\mathop{\rm ei}\nolimits (x)$ is the Exponential Integral. The Mangoldt Function is given by

$$\Lambda(n)=\cases{ \ln p & if $n=p^m$\ for $(m\geq 1)$\ and $p$\ prime\cr 0 & otherwise\cr}$$ (2)

$$-{\zeta'(x)\over\zeta(s)} = \sum_{n=1}^\infty {\Lambda(n)\over n^s}$$ (3)

for $\Re[s]>1$.

$$J(x)=\sum_{n\leq x} {\Lambda(n)\over \ln n}.$$ (4)

The Summatory Riemann-Mangoldt function is defined by

$$\psi(x)=\sum_{n\leq x} \Lambda(n) = \theta(x)+\theta(x^{1/2})+\ldots.$$ (5)

See also Prime Number Theorem, Riemann Function

References

Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 364-365, 1991.