#15846: Incorrect series expansion of zeta(s) at 1
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Reporter: mmezzarobba | Owner:
Type: defect | Status: new
Priority: major | Milestone: sage-6.4
Component: symbolics | Resolution:
Keywords: | Merged in:
Authors: | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: | Stopgaps:
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Old description:
> The series expansion of the zeta function around its singularity at `s=1`
> is incorrect. Within sage, we have
>
> {{{
> sage: zeta(x).series(x==1,2)
> (Infinity) + (zetaderiv(1, 1))*(x - 1) + Order((x - 1)^2)
> }}}
>
> when in fact, we should have
>
> {{{
> sage: zeta(x).series(x==1, 1)
> 1*(-1+x)^(-1)+(log(Pi)+log(2)+euler_gamma+2*zetaderiv(1,0))+Order(-1+x)
> }}}
>
> (see http://dlmf.nist.gov/25.2.E4 for the expansion like it is supposed
> to be.)
New description:
The series expansion of the zeta function around its singularity at `s=1`
is incorrect. Within Sage, we have
{{{
sage: zeta(x).series(x==1,2)
(Infinity) + (zetaderiv(1, 1))*(x - 1) + Order((x - 1)^2)
}}}
when in fact, we should have
{{{
sage: zeta(x).series(x==1, 1)
1*(-1+x)^(-1)+(log(pi)+log(2)+euler_gamma+2*zetaderiv(1,0))+Order(-1+x)
}}}
(see http://dlmf.nist.gov/25.2.E4 for the expansion like it is supposed to
be.)
--
Comment (by behackl):
I managed to find a fix for the problem in GiNaC, which should -- more or
less -- directly translate into a fix for Pynac, which Sage uses as a
substitute for GiNaC.
When using zeta's functional equation http://dlmf.nist.gov/25.4.E2 for the
series expansion around `s=1`, then within GiNaC, I obtain the expression
now stated in the description.
For a follow-up ticket, it would probably be nice to tell sage that
`zetaderiv(0,1)` is equal to `-log(sqrt(2*pi)`, as it simplifies the
expansion to the more likeable form
{{{
sage: zeta(x).series(x==1, 1)
1*(-1+x)^(-1)+(euler_gamma)+Order(-1+x)
}}}
Maxima probably has a similar problem like !SymPy, GiNaC et al.; I guess
we should report the issue there as well?
--
Ticket URL: <http://trac.sagemath.org/ticket/15846#comment:7>
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