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

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