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Re: Re: bug report: MoebiusMu sum
On 8 Nov 2008, at 17:59, mark mcclure wrote:
>> 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
>
I suppose the "bug" issue reduces to the question whether NSum should
not start by "asking" Sum if the exact answer is known, at least in
the cases " when the summand is oscillating in an essentially
unpredicatable way", or perhaps issue some warning message rather than
taking the simple and fast route and returning a wrong answer.
Andrzej Kozlowski
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