## Gram Series

where is the Riemann Zeta Function. This approximation to the Prime Counting Function is 10 times better than for but has been proven to be worse infinitely often by Littlewood (Ingham 1990). An equivalent formulation due to Ramanujan 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. Undersøgelser 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.