2021-06-23, 16:29 | #1 |
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 |
Dec 2008
you know...around...
1010011011_{2} 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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