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Re: Troubles with HarmonicNumber, empty sums, and Zeta

• To: mathgroup at smc.vnet.net
• Subject: [mg85934] Re: Troubles with HarmonicNumber, empty sums, and Zeta
• From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
• Date: Thu, 28 Feb 2008 02:47:29 -0500 (EST)
• References: <200802130915.EAA21588@smc.vnet.net> <fouma4$qu3$1@smc.vnet.net> <fphbpe$84a$1@smc.vnet.net>

"David W.Cantrell" <DWCantrell at sigmaxi.net> wrote:
[snip]
> Here's another bug, which might be unrelated to the other troubles I had
> mentioned:
>
> In[8]:= Limit[Zeta[z, 9/10] - Zeta[z, 9/10 + z], z -> 1]
>
> Out[8]= Infinity
>
> But it's easy to guess the correct result:
>
> In[16]:= N[Zeta[z, 9/10] - Zeta[z, 9/10 + z] /. z -> 9999/10000, 12]
> Out[16]= 1.11103060226
> In[17]:= N[Zeta[z, 9/10] - Zeta[z, 9/10 + z] /. z -> 10001/10000, 12]
> Out[17]= 1.11119161012
> In[18]:= (% + %%)/2
> Out[18]= 1.11111110619
>
> and so our guess would be 10/9.
>
> Furthermore, we can get the correct exact result by "converting" to
> HarmonicNumber by hand (and then there is no need of Limit):
>
> In[19]:= -HarmonicNumber[-1/10, z] + HarmonicNumber[-1/10 + z, z] /. z ->
> 1 Out[19]= -HarmonicNumber[-1/10] + HarmonicNumber[9/10]
> In[20]:= FullSimplify[%]
> Out[20]= 10/9
>
> Finally, I wonder about things like ZetaClassical and
> ZetaClassicalRegularized. They are used at the Wolfram Functions site.
> For example, the third and fourth entries at
> <http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta2/02/>
> concern ZetaClassical and ZetaClassicalRegularized, resp. And in the
> output from one of the bugs mentioned in my original post in this thread,
> there was mention of ZetaClassical in a system dump. So are ZetaClassical
> and ZetaClassicalRegularized things that users of Mathematica have access
> to, or are they strictly for internal work, to be hidden from the user?

Having just now updated to version 6.0.2, I'm sorry to note that bugs
mentioned earlier in this thread still exist. And here's another bug
related to Zeta (but probably not directly related to bugs previously
mentioned in this thread):

In[27]:= Sum[i*(1/(i - 1/10)^2 - 1/(i + 9/10)^2), {i, 1, Infinity}]

Out[27]= EulerGamma - (2*Sqrt[5/8 + Sqrt[5]/8]*Pi)/(-1 + Sqrt[5]) -
(100/361)*HypergeometricPFQ[{19/10, 19/10, 2}, {29/10, 29/10}, 1] +
2*Log[4] + Log[5] + (1/4)*Log[5/8 - Sqrt[5]/8] - (1/4)*Sqrt[5]*Log[5/8 -
Sqrt[5]/8] + (1/4)*Log[5/8 + Sqrt[5]/8] + (1/4)*Sqrt[5]*Log[5/8 +
Sqrt[5]/8] - (1/2)*Log[-1 + Sqrt[5]] - (1/2)*Sqrt[5]*Log[-1 + Sqrt[5]] -
(1/2)*Log[1 + Sqrt[5]] + (1/2)*Sqrt[5]*Log[1 + Sqrt[5]] +
(1/10)*PolyGamma[1, 9/10]

In[28]:= N[%]

Out[28]= ComplexInfinity

In[29]:= FullSimplify[%%]

Out[29]= Indeterminate

Out[27] is incorrect. (BTW, it seems strange to me that N[Out[27]] is
ComplexInfinity while FullSimplify[Out[27]] is Indeterminate. But that is
not my point.) A correct result from In[27] would have been simply
Zeta[2, 9/10], and that's why I had said this bug is related to Zeta.

In[37]:= N[Zeta[2, 9/10]]

Out[37]= 1.92254

In[38]:= NSum[i*(1/(i - 1/10)^2 - 1/(i + 9/10)^2), {i, 1, Infinity}]

Out[38]= 1.92254

David W. Cantrell

• References:
  • Troubles with HarmonicNumber, empty sums, and Zeta
    • From: "David W.Cantrell" <DWCantrell@sigmaxi.net>