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Re: bug report: MoebiusMu sum

  • To: mathgroup at smc.vnet.net
  • Subject: [mg93414] Re: [mg93401] bug report: MoebiusMu sum
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Fri, 7 Nov 2008 05:59:31 -0500 (EST)
  • References: <200811060906.EAA15853@smc.vnet.net>

Actually, it seems the other way around - the bug is in NSum.  Sum is  
quite correct here. Note that Zeta[0] does not play any role here and  
it is known that

Sum[MoebiusMu[k]/k ,{k,1,Infinity}]==0.

In fact, this statement is equivalent to the Prime Number Theorem (in  
other words, you can deduce it form the Prime Number Theorem and you  
can deduce the Prime Number Theorem from it). For a proof see (for  
example): Apostol, Introduction to Analytic Number Theory, page 97  
(Theorem 4.16).

Andrzej Kozlowski


On 6 Nov 2008, at 18:06, Jan Irigi Olsina wrote:

> I just wanted to report a bug:
>
> 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. Is this a bug in
> Sum?
>
> Thank you!
>
> Jan Olsina
>



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