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 2021-06-23, 16:29 #1 mart_r     Dec 2008 you know...around... 29B16 Posts A probabilistic measure arising in approximating pi(x) / Question Given the logarithmic integral $Li(x) = \int_2^x \frac{dt}{\log t}$ and the smooth part of Riemann's prime counting formula as the equivalent of the Gram series $R(x) = 1+\sum_{n=1}^\infty \frac{\log^n x}{n\cdot n!\cdot\zeta(n+1)},$ is there a constant c such that $c = \lim_{m\rightarrow\infty} \frac{1}{m} \sum_{x=2}^m \log(\lvert\frac{Li(x)-\pi(x)}{R(x)-\pi(x)}\rvert)$ ?
 2021-07-19, 18:48 #2 mart_r     Dec 2008 you know...around... 66710 Posts Let me admit at this point that I still don't seem to understand how to approximate pi(x) by employing the nontrivial zeta zeroes, specifically how I achieve sufficient convergence of x^(1/2 ± t i) as t gets larger. I'd like to see how well the above mentioned value c fares when x is large, but for this I need to have a pi(x) approximation that makes use of at least a couple of those zeroes. Are there any freely available programs for this?

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