Gram Series

$\begin{displaymath} R(x)=1+\sum_{k=1}^\infty {(\ln x)^k\over k k! \zeta(k+1)}, \end{displaymath}$

where $\zeta(z)$ is the Riemann Zeta Function. This approximation to the Prime Counting Function is 10 times better than $\mathop{\rm Li}\nolimits (x)$ for

but has been proven to be worse infinitely often by Littlewood (Ingham 1990). An equivalent formulation due to Ramanujan

$\begin{displaymath} G(x)\equiv {4\over \pi}\sum_{k=1}^\infty {(-1)^{k-1}k\over B_{2k}(2k-1)} \left({\ln x\over 2\pi}\right)^{2k-1}\sim\pi(x) \end{displaymath}$

(Berndt 1994), where $B_{2k}$ is a Bernoulli Number. The integral analog, also found by Ramanujan, is

$\begin{displaymath} J(x)\equiv \int_0^\infty {(\ln x)^t\,dt\over t\Gamma(t+1)\zeta(t+1)}\sim\pi(x) \end{displaymath}$

(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.