# Computation similar to the (real) zeta function

1. Jul 10, 2007

### CRGreathouse

I'm trying to get a numerical approximation for a complicated sum, and I'm currently trying to estimate the size of the infinite right tail. Actually I'm doing the sum in three parts: an 'exact' (well, 100 digit floating point) calculation for the first portion, a calculation involving an algebraic approximation of the natural log* for a portion 3-4 times as large, and then an estimate for the tail.

In the limit, the function I'm summing varies as x^-8, so my natural thought was to use some variation on $\zeta(8)$ since there are good algorithms written for this -- and since I'm using Pari which has one built in. The trouble is that I need to sum not from 1 but from a hundred thousand or so. Subtracting out everything less than it will sap the precision too low to be useful (I would think).

Are there any analytic tools I can use here?

* The natural log is the most expensive part of the calculation, and the quantity I'm taking the log of varies as 1-1/x^2 so I'm using the approximation
$$\ln(1+\alpha)\approx\frac{\alpha}{1+\alpha/2}+\frac{\alpha^3}{12(\alpha+1)}$$
which is within $\mathcal{O}(\alpha^4)$ of the true answer as $\alpha\to0$.

Edit: Oops, stupid me. I'm trying the obvious solution now, we'll see if it works.

Last edited: Jul 10, 2007