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.
- RSA Encryption
- RSA Number
- Rubber-Sheet Geometry
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