User group for PFGW & PrimeForm programs Yahoo Group need help about 2 constants =============================================== Pierre CAMI Message 1 of 3 Mar 10, 2006 ----------------------------------------------- I try to calculate the constant sum from 1 to infinity of n/(p(n)^2) with p(n) = n-th prime as i d'nt found this value searching on the net. It seems to me that this need to converge as for high n it is like n/(p(n)^2) ~ 1/n*(log(n)^2) and n*(log(n)^2) > n^1 I found that it is > 1.1 with a very low convergence , so is it a way to found this value with let say 5 or 10 digits after the 1 ? Same question about sum of 1/n*(log(n)^2) Thanks for any help Pierre =============================================== David Broadhurst Message 2 of 3 Mar 10, 2006 ----------------------------------------------- --- In primeform@yahoogroups.com, "Pierre CAMI" <pierrecami@...> wrote: > Same question about sum of 1/n*(log(n)^2) I assume that you mean the sum of 1/(n*log(n)^2) over all integers n>1. Here you can use an Euler-MacLaurin method since you are have a smooth (infinitely differentiable) function summed over integers and the integral of 1/(x*log(x)^2) is -1/log(x), which vanishes at infinity. Using only a single derivative I got 2.10974280123689221 2.10974280123689138 2.10974280123688166 by truncating at n = 10^3, 10^4, 10^5. This can easily be improved by taking the third derivative: 2.1097428012368919744790 2.1097428012368919744792 2.1097428012368919744792 BUT, for the sum of n/prime(n)^2 you have no hope of using the Euler-MacLaurin method, since the fluctutations of primality destroy the assumption of smoothness So you asked one very hard question and one very easy question. David =============================================== David Broadhurst Message 3 of 3 Mar 10, 2006 ----------------------------------------------- --- In primeform@yahoogroups.com, "David Broadhurst" <d.broadhurst@...> wrote: > > Same question about sum of 1/n*(log(n)^2) .. > 2.1097428012368919744790 > 2.1097428012368919744792 > 2.1097428012368919744792 PS: If you Google 2.10974280123689 you will see that this constant appears in a college mathematics journal to which I do not have access. >> MR1478271 Kreminski, Rick Using Simpson's rule to approximate sums of infinite series. College Math. J. 28 (1997), no. 5, 368--376. 65B10 [There will be no review of this item.] << Perhaps college students know about Bart Simpson, but not about Leonhard Euler :-? Here is a more accurate value, courtesy of Leonhard: 2.1097428012368919744792571976165513263855319843947420226499156 As you can see analysis is quite easy; it is /arithmetic/ that is truly hard. David =============================================== Cached by Georg Fischer at Nov 14 2019 12:47 with clean_yahoo.pl V1.4