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Based on Raymond Manzoni's answer https://math.stackexchange.com/a/2822928 we know that Gram series in sum over non-trivial zeros of zeta-function has a slow converging because of large x. And we know that general converging series for Ei[1] (exponential integral) looks very similar to Gram series (except the zeta function term in the denominator in Gram series) and has the same problem for large x as the Gram series. But it exists a very rapidly converging continued fractions for Ei[1] (for large x): https://www.boost.org/doc/libs/1_68_0/libs/math/doc/html/math_toolkit/expint/expint_n.html#math_toolkit.expint.expint_n.implementation

Maybe there is a way to find similar continued fractions for Gram series?

Vadim
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  • If you have the hope that, say , $\pi(10^{300})$ can be determined exactly with such an approach, I must disappoint you. $li(x)$ is already an excellent approximation and I do not think we can improve this significantly without a lot of effort. – Peter Dec 11 '21 at 16:08
  • No, I'm about x<10^9 where Gram series is better – Vadim Dec 11 '21 at 16:10
  • In this range , of course. – Peter Dec 11 '21 at 16:11

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