More thoughts, regarding the time complexity of statistical sums Assuming the Riemann Hypothesis, and neglecting constant and logarithmic factors, the error of the logarithmic integral in approximating prime_pi(x) is sqrt(x). (sqrt(x))
There is a sum with terms asymptotic to sqrt(x) * sin(t * log(x)) / t, along with a simple prime power correction that gives better accuracy than the logarithmic integral. "t" sums over Riemann zeta zeros. Increasing the number of Riemann zeta zeros by a factor of 10 decreases the error by a factor of approximately 2.3 (this is an observed constant, presumably irrational), this beneficial effect can be neglected for the current analysis. In the NTFP formula, we deal with areas under or between curves and staircase segments. As the heights of the excluded waves decrease, the frequencies increase. When we have a sinusoid, and perform a horizontal squash by some factor, a definite integral will be smaller by the factor (more sophisticated statistical or otherwise analysis may be useful). A random walk with each term decreased by the factor appears. (t) If we multiply the interval size by n, the standard error decreases by a factor of sqrt(n). (sqrt(t)) C = sqrt(x) / (t * sqrt(t)) t = cbrt(x) On Tuesday, March 21, 2017 at 2:30:40 AM UTC-7, kstueve wrote: > > A reiteration of my Winter 2010 Number Theory Final Project, at UW in > Seattle. > > This represents a continuous integration around x of Riemann's explicit > formula, combined with a full sieving in that interval, to effectively > produce the average of many sample errors. Multiply the number of zeros by > 10 in Riemann's explicit formula, and the error decreases by a factor of > 2.3. The effective number of samples depends on the height of zeros > included in the sum. Statistics tells us that if we multiply the number of > samples by n, the standard error will decrease by a factor of sqrt(n). > > Kevin Marshall Stueve > -- You received this message because you are subscribed to the Google Groups "sage-nt" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send an email to [email protected]. Visit this group at https://groups.google.com/group/sage-nt. For more options, visit https://groups.google.com/d/optout.
