User:Ljfiedler/sandbox

The Gram series is an important approximation to the prime-counting function. It is equal to the dominant term, R(x), in the explicit formula for the prime-counting function, $\pi$(x), as well as being of great practical usefulness in the calculation of the smaller, so-called "noisy" contributions to π(x) from the nontrivial zeros of the Riemann zeta function owing to their appearance in the explicit formula as R(x&rho;).

The expression for the Gram series G(x) is


 * $$G(x) = \sum_{n=1}^{\infty} \frac{ \mu (n)}{n} \operatorname{li}(x^{1/n}) = 1 + \sum_{k=1}^\infty \frac{(\ln x)^k}{k! k \zeta(k+1)}$$.

History
The Gram series was originally developed by the Danish mathematician Jørgen Pedersen Gram (1850–1916). He published the first collection of the nontrivial Riemann zeros on the critical line in 1903 (the first fifteen zeros) to six decimal places.

Derivation
The Gram series can be quickly derived from the following expression for the logarithmic integral.


 * $$\operatorname{li}(x^{1/n})=\gamma - \ln n + \ln \ln x + \sum_{k=1}^{\infty} \frac {(\ln x)^k} {k!k {n^k}}$$, for x &ne;1,

where &gamma; ≈ 0.5772 is the Euler-Mascheroni constant.

Ramanujan's series
Srinivasa Ramanujan found a closely related series that approximates Riemann's series much more quickly than does Gram's series. In actual computation it provides a much quicker alternative to evaluating R(x). His series is given as


 * $$G_R(x) = \frac {4} {\pi} \sum_{k=1}^{\infty} \frac {(-1)^{k-1} k} {(2k-1) B_{2k}} \left(\frac {\ln x} {2 \pi}\right)^{2k-1} $$

where Bn is the nth Bernoulli number.