Riemann Function

The function obtained by approximating the Riemann Weighted Prime-Power Counting Function $J_2$ in

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

by the Logarithmic Integral $\mathop{\rm Li}\nolimits (x)$. This gives

$$R(n) \equiv 1+\sum_{k=1}^\infty {1\over k\zeta(k+1)} {(\ln n)^k\over k!}$$ (2)

$$= \sum_{m=1}^\infty {\mu(m)\over m} \mathop{\rm Li}\nolimits (n^{1/m}),$$ (3)

where $\zeta(z)$ is the Riemann Zeta Function, $\mu(n)$ is the Möbius Function, and $\mathop{\rm Li}\nolimits (x)$ is the Logarithmic Integral. Then

$$\pi(x)=R(x)-\sum_\rho R(x^p),$$ (4)

where $\pi$ is the Prime Counting Function. Ramanujan independently derived the formula for $R(n)$, but nonrigorously (Berndt 1994, p. 123).

See also Mangoldt Function, Prime Number Theorem, Riemann-Mangoldt Function, Riemann Zeta Function

References

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, 1994.

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 144-145, 1996.

Riesel, H. and Göhl, G. ``Some Calculations Related to Riemann's Prime Number Formula.'' Math. Comput. 24, 969-983, 1970.

Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 28-29 and 362-372, 1991.