## Riemann Function

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

 (1)

by the Logarithmic Integral . This gives
 (2) (3)

where is the Riemann Zeta Function, is the Möbius Function, and is the Logarithmic Integral. Then
 (4)

where is the Prime Counting Function. Ramanujan independently derived the formula for , 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.