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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, π(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^ρ).

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 ≠1,

where γ ≈ 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 B_n is the n^th Bernoulli number.

See also

• Prime-counting function in number theory
• Jørgen Pedersen Gram, the eponymous Danish mathematician of the Gram series

External Links

• Wolfram Mathworld – Gram Series

Category:Approximation theory Category:Analytic number theory