Gram Series
The Gram series is an approximation to the prime counting function given by
 |
(1)
|
where 
is the Riemann zeta function (Hardy 1999, p. 24). This approximation is 10 times better than 
for 
but has been proven to be worse infinitely often by Littlewood (Ingham 1990).
The Gram series is equivalent to the Riemann prime counting function (Hardy 1999, pp. 24-25)
 |
(2)
|
where 
is the logarithmic integral and 
is the Möbius function (Hardy 1999, pp. 16 and 23; Borwein et al. 2000), but is much more tractable for numeric computations. For example, the plots above show the difference 
where 
is computed using the Wolfram Language's built-in NSum command (black) and approximated using the first 
(blue), 
(green), 
(yellow), 
(orange), and 
(red) points.
A related series due to Ramanujan is
(Berndt 1994, p. 124; Hardy 1999, p. 23), where 
is a Bernoulli number. The integral analog, also found by Ramanujan, is
 |
(7)
|
(Berndt 1994, p. 129; Hardy 1999, p. 23).
REFERENCES:
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, 1994.
Borwein, J. M.; Bradley, D. M.; and Crandall, R. E. "Computational Strategies for the Riemann Zeta Function." J. Comput. Appl. Math. 121, 247-296, 2000.
Gram, J. P. "Undersøgelser angaaende Maengden af Primtal under en given Graeense." K. Videnskab. Selsk. Skr. 2, 183-308, 1884.
Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.
Ingham, A. E. Ch. 5 in The Distribution of Prime Numbers. New York: Cambridge University Press, 1990.
Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, p. 225, 1996.
Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 74, 1991.
