A trick to calculate Pi(a) for any a?..

by eljose
Tags: trick
 Sci Advisor HW Helper P: 9,396 A trick to calculate Pi(a) for any a?.. Jose, I know many things about the zeta function, and pi(x). And I'm pointing out that you are not helping yourself by continually making obfuscatory posts like this. Imagine someone wanders in and sees that post without knowing your history? What are they to make of it? Would you care to domonstrate how effective you "numerical method" is? By, say, doing something with it? For instance pi(x) satisfies http://mathworld.wolfram.com/LegendresFormula.html http://mathworld.wolfram.com/LehmersFormula.html http://mathworld.wolfram.com/MeisselsFormula.html and this lovely formula $$\pi(x) = -1 + \sum_{r=3}^{n} [(r-2)! - r \lfloor \frac{(r-2)!}{r} \rfloor ]$$ Here is another Wolfram page that shows that using Mellin transforms in calculating pi(x) is well known http://mathworld.wolfram.com/Riemann...gFunction.html here is something that shows how to use the zeroes of the zeta function to calculate pi(x) http://www.maths.ex.ac.uk/~mwatkins/zeta/encoding2.htm remember you wanted an integral where the zeroes of the zeta function were the poles? We are not saying what you have is wrong. But I'd ask you to re-evaluate your claims. Once you claimed to have solved pi(x), remember? And try and write it in a clearer format so that others might be able to decipher already quite tricky things (you are in this thread presuming that people inuitively know what pi, R, Ln, gaussian integration, legendre polynomials are without explicit reference to them, never mind not explicitly saying what the indexing sets of the sums are). And finally, what is the effectiveness of your algorithm, what is its cost?
 P: 501 Here you are matt the math in latex,let be the integral equation for pi(x): $$\frac{lnR(s)}{s}=\int_0^{\infty}\frac{pi(x)}{x({x^s}-1)}$$ we use an approximate formula of integration related to PI(a) where a is the point we want to calculate PI(x) function in we have $$\frac{lnR(s)}{s}=K(s,a)\pi(a)+\sum_jK(x_j,s)\pi(x_j)$$ now choosing several s_j we have a system we only have to solve this system to obtain pi(a) with K the kernel $$\frac{1}{x({x^s}-1)}$$