#15846: Incorrect series expansion of zeta(s) at 1
-------------------------------+------------------------
       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:
-------------------------------+------------------------

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>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica, 
and MATLAB

-- 
You received this message because you are subscribed to the Google Groups 
"sage-trac" group.
To unsubscribe from this group and stop receiving emails from it, send an email 
to [email protected].
To post to this group, send email to [email protected].
Visit this group at http://groups.google.com/group/sage-trac.
For more options, visit https://groups.google.com/d/optout.

Reply via email to