Calculating a difficult sum
 

What are good ways to numerically calculate integrals? Background (skip, skim, or read depending on your patience):
[INDENT][COLOR="Navy"]I was trying to estimate the value of a difficult arithmetic sum recently and I'm not quite sure how to go about it.

The summands are ~ 1/xlog^2 x and so clearly converge, but a finite sum as far as I was able to manage (a few million terms, though I could reach a billion with a bit more time) aren't enough to get a value I can trust to even two decimal places, and it's surely meaningless beyond four.

The obvious next step was to take the sum as far as I cared to go and use an integral for the rest. Fortunately the terms are just well-behaved enough that I have viable upper and lower bounds. With some effort I managed to generate three integrals that give upper and lower bounds (the lower bound is two piecewise integrals).

So I typed all this up in my programming language of choice and wrote a wrapper function that let me set precision and how far to set the sum/integral cutoff. It was only then that I realized that the integral was apparently rather numerically unstable -- even though the bounding functions were smooth in the domain of consideration. Every time I raised the precision I got a new answer, to the limit of my patience.

My best though so far was to subtract the main term 1/xlog^2 x out of the integral, since it has closed-form integral -1/log x. But this hasn't helped much so far.[/COLOR][/INDENT]
I'm just looking for general strategies, either better ways to calculate the integrals or a different approach to the sum. Because the sum includes primes it's a pain to work with numerically... but Pierre Dusart's bounds are tight enough at large numbers that I think I can make the integral approach work. But first I have to know that what I think is the integral is, in fact, the integral!

Why not evaluate the integrals formulaically using an implementation of Risch's algorithm (although Cherry's algorithmic extension to Rich's result seems more practical; but, as you know, that's not saying much :smile:), and then evaluate the resulting formula numerically at the points you want?

If I interpret Cherry's paper correctly (which you gave me 4 months ago), he does implement his algorithm in Macsyma.

Is Mathematica able to compute this integral (I know, it's a dumb question; but it's worth a shot)?

Neither Mathematica nor Maxima can evaluate the integral in closed form. That's not surprising: most integrals can't be, and I wouldn't have posted if it could have been.

Sometimes a change of variable in the integral helps. The integral convergence may be the same problem as the summation convergence - the integrand decreases so slowly that you need to keep extending the range of integration. The idea is to get a new integration variable that rapidly covers ranges of the old integration variable. Perhaps z = log x.

William