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HarmonicNumber bug and related Zeta disappointments

• To: mathgroup at smc.vnet.net
• Subject: [mg75606] HarmonicNumber bug and related Zeta disappointments
• From: "David W. Cantrell" <DWCantrell at sigmaxi.net>
• Date: Sun, 6 May 2007 01:54:23 -0400 (EDT)

I'm fairly certainly that what I've found is a bug in HarmonicNumber. But
I'd like the opinions of others. Furthermore, I'm still using version 5.2
and so would be interested to know if the same behavior is also found in
version 6.

In[2]:= HarmonicNumber[Infinity, r]
Out[2]= Zeta[r]

This is certainly correct for r > 1, but it is incorrect for 0 < r < 1, in
which case the result should instead have been Infinity. Note that the
following limit is correct:

In[6]:= Assuming[0 < r < 1, Limit[HarmonicNumber[z, r], z -> Infinity]]
Out[6]= Infinity

Below, I list related items which are disappointing (but are not bugs IMO).

1. In light of

<http://functions.wolfram.com/GammaBetaErf/HarmonicNumber2/02/0001/>

it would have been nice if the following simplification had given 0,
instead of just returning essentially the original expression:

In[26]:= FullSimplify[HarmonicNumber[z, r] - (Zeta[r] - Zeta[r, z + 1])]
Out[26]= HarmonicNumber[z, r] - Zeta[r] + Zeta[r, 1 + z]

2.  Unfortunately  FullSimplify[Zeta[r, Infinity], r > 1]
and  FullSimplify[Zeta[r, Infinity], 0 < r < 1] both return just
Zeta[r, Infinity], while the desired results are 0 and -Infinity, resp.

Similarly, related limits, such as Limit[Zeta[3/2, x], x -> Infinity]  and
Limit[Zeta[1/2, x], x -> Infinity], are returned unevaluated.

David W. Cantrell