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Re: bug report: MoebiusMu sum

• To: mathgroup at smc.vnet.net
• Subject: [mg93445] Re: bug report: MoebiusMu sum
• From: mark mcclure <mcmcclur at unca.edu>
• Date: Sat, 8 Nov 2008 03:59:56 -0500 (EST)
• References: <200811060906.EAA15853@smc.vnet.net> <gf16vt$ian$1@smc.vnet.net>

> On 6 Nov 2008, at 18:06, Jan Irigi Olsina wrote:
> > Sum[MoebiusMu[k]/k Zeta[0],{k,1,inf}]
> > returns 0 in Mathematica 6.0.0.
>
> > On the other hand
> > NSum[MoebiusMu[k]/k Zeta[0],{k,1,inf}]
> > gives particular numerical result different from 0

and

On Nov 7, 5:58 am, Andrzej Kozlowski <a... at mimuw.edu.pl>
responded:
> Actually, it seems the other way around - the bug is in NSum.

I'm not sure if I would even call this a "bug" in NSum, but
rather a known fact that the algorithms in NSum don't work
so well when the summand is oscillating in an essentially
unpredicatable way.  We can also deduce that the exact Sum
is probably correct, without using any high-powered number
theory (although, I certainly looked it up and found it to
be a very nice theorem).  According to the documentation:
  You should realize that with sufficiently pathological
  summands, the algorithms used by NSum can give wrong
  answers. In most cases, you can test the answer by
  looking at its sensitivity to changes in the setting
  of options for NSum.

In this case, the result varies greatly when the NSumTerms
option is increased; an error message is also returned for
large values of NSumTerms.

We might also check
NSum[MoebiusMu[k]/k, {k, 1, 10000}]
vs
N[Sum[MoebiusMu[k]/k, {k, 1, 10000}]]
and note the difference.

Incidently, the documentation indicates that N[Sum[...]]
calls NSum; this is evidently false.

Mark McClure

• Follow-Ups:
  • Re: Re: bug report: MoebiusMu sum
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>

• References:
  • bug report: MoebiusMu sum
    • From: Jan Irigi Olsina <jan.olsina@gmail.com>