 mersenneforum.org A probabilistic measure arising in approximating pi(x) / Question
 Register FAQ Search Today's Posts Mark Forums Read 2021-06-23, 16:29 #1 mart_r   Dec 2008 you know...around... 23·29 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... 10100110112 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?  Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post Bobby Jacobs Prime Gap Searches 42 2019-02-27 21:54 wildrabbitt Math 57 2015-09-17 18:26 Kathegetes Miscellaneous Math 16 2014-07-13 03:48 only_human Puzzles 0 2010-03-23 18:49 hasan4444 Factoring 17 2009-10-28 14:34

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