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Re: Troubles with HarmonicNumber, empty sums, and Zeta

• To: mathgroup at smc.vnet.net
• Subject: [mg85715] Re: Troubles with HarmonicNumber, empty sums, and Zeta
• From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
• Date: Wed, 20 Feb 2008 06:51:35 -0500 (EST)
• References: <200802130915.EAA21588@smc.vnet.net> <fouma4$qu3$1@smc.vnet.net>

Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
> On 13 Feb 2008, at 10:15, David W.Cantrell wrote:
[snip]
> > In[27]:= N[Zeta[-1/2] - Zeta[-1/2, -2]]
> > Out[27]= -2.41421
> >
> > [BTW, is there some reasonably direct way to simplify
> > Zeta[-1/2] - Zeta[-1/2, -2]  to  -1 - Sqrt[2]  using Mathematica?]
> >
> > David W. Cantrell
> >
>
> It depends on the meaning of "direct". But there certainly is a
> "short" way:
>
>   RootApproximant[Zeta[-2^(-1)] - Zeta[-2^(-1), -2]]
>   -1 - Sqrt[2]

Thanks, Andrzej, for mentioning RootApproximant. It's a new function of
which I had been unaware.

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?

David W. Cantrell

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