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Gram Series


GramSeries

The Gram series is an approximation to the prime counting function given by

 G(x)=1+sum_(k=1)^infty((lnx)^k)/(kk!zeta(k+1)),
(1)

where zeta(z) is the Riemann zeta function (Hardy 1999, p. 24). This approximation is 10 times better than Li(x) for x<10^9 but has been proven to be worse infinitely often by Littlewood (Ingham 1990).

GramSeriesRiemannComparison

The Gram series is equivalent to the Riemann prime counting function (Hardy 1999, pp. 24-25)

 R(x)=sum_(n=1)^infty(mu(n))/nli(x^(1/n))
(2)

where li(x) is the logarithmic integral and mu(n) 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 G(x)-R(x) where R(x) is computed using the Wolfram Language's built-in NSum command (black) and approximated using the first 10^1 (blue), 10^2 (green), 10^3 (yellow), 10^4 (orange), and 10^5 (red) points.

A related series due to Ramanujan is

G^*(x)=4/pisum_(k=1)^(infty)((-1)^(k-1)k)/(B_(2k)(2k-1))((lnx)/(2pi))^(2k-1)
(3)
=sum_(k=1)^(infty)((lnx)^k)/(kk!zeta(k+1))
(4)
=2sum_(k=1)^(infty)(ln^(2k-1)x)/((2k-1)(2k-1)!zeta(2k))
(5)
=8sum_(k=1)^(infty)((-1)^(k-1)kln^(2k-1)x)/((2k-1)(2pi)^(2k)B_(2k))
(6)

(Berndt 1994, p. 124; Hardy 1999, p. 23), where B_(2k) is a Bernoulli number. The integral analog, also found by Ramanujan, is

 J(x)=int_0^infty((lnx)^tdt)/(tGamma(t+1)zeta(t+1))
(7)

(Berndt 1994, p. 129; Hardy 1999, p. 23).


See also

Riemann Prime Counting Function

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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.

Referenced on Wolfram|Alpha

Gram Series

Cite this as:

Weisstein, Eric W. "Gram Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GramSeries.html

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