Gram Series
where
is the Riemann Zeta Function. This approximation to the Prime Counting Function is 10 timesbetter than 
for 
but has been proven to be worse infinitely often by Littlewood (Ingham 1990). An equivalent formulation due toRamanujan 
is
(Berndt 1994), where
is a Bernoulli Number. The integral analog, also found by Ramanujan, is
(Berndt 1994).
References
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, pp. 124-129, 1994. Gram, J. P. ``Underségelser angaaende Maengden af Primtal under en given Graeense.'' K. Videnskab. Selsk. Skr. 2, 183-308, 1884. Ingham, A. E. Ch. 5 in The Distribution of Prime Numbers. New York: Cambridge, 1990. Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 74, 1991.
- Ramanujan's Tau-Dirichlet Series
- Ramanujan's Tau Function
- Ramanujan Theta Functions
- Ramp Function
- Ramphoid Cusp
- Ramsey Number
- Ramsey's Theorem
- Randelbrot Set
- Random Distribution
- Random Dot Stereogram
- Random Graph
- Random Matrix
- Random Number
- Random Percolation
- Random Polynomial
- Random Variable
- Random Walk
- Random Walk--1-D
- Random Walk--2-D
- Random Walk--3-D
- Range (Image)
- Range (Line Segment)
- Range (Statistics)
- Rank
- Rank (Group)