Re: Re: bug report: MoebiusMu sum

*To*: mathgroup at smc.vnet.net*Subject*: [mg93460] Re: [mg93445] Re: bug report: MoebiusMu sum*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Sun, 9 Nov 2008 05:25:48 -0500 (EST)*References*: <200811060906.EAA15853@smc.vnet.net> <gf16vt$ian$1@smc.vnet.net> <200811080859.DAA14264@smc.vnet.net>

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

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

**Re: bug report: MoebiusMu sum***From:*mark mcclure <mcmcclur@unca.edu>