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Computation similar to the (real) zeta function

  1. Jul 10, 2007 #1

    CRGreathouse

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    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 [itex]\zeta(8)[/itex] 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
    [tex]\ln(1+\alpha)\approx\frac{\alpha}{1+\alpha/2}+\frac{\alpha^3}{12(\alpha+1)}[/tex]
    which is within [itex]\mathcal{O}(\alpha^4)[/itex] of the true answer as [itex]\alpha\to0[/itex].

    Edit: Oops, stupid me. I'm trying the obvious solution now, we'll see if it works.
     
    Last edited: Jul 10, 2007
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