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Gaurav Thakur
12/27/02 1:30pm


I have been having some trouble with the results given by the Zeta, Sum and Product functions. In particular, there seems to be some issues with derivatives of Zeta and a quirky convergence bug with the Sum and Product. I also had a couple of general questions which I will get to later on. I am using Mathematica student version 4.2 on a windows system to obtain these results.

1: The zeta function problem has to do with the derivatives of Zeta at the pole. Here are some input/output messages from Mathematica; they should give a general idea of what is going on.

In[1]:= Zeta[1]
Out[1]:= ComplexInfinity

In[2]:= Series[Zeta[x],{x,0,3}]
Out[2]:= 1/(x-1) + EulerGamma - StieltjesGamma[1](x-1) + StieltjesGamma[2](x-2)/2 + O[x-1]^2

So far everything looks good. However, it starts giving strange answers for the derivatives:

In[3]:= Zeta'[1]
Out[3]:= StieltjesGamma[1]

In[4]:= Zeta'[2]
Out[4]:= StieltjesGamma[2]

In[5]:= Limit[D[Zeta[x] + 1/(1-x),x], x->Infinity]
Out[5]:= -Infinity

In[6]:= Series[Zeta'[x],{x,0,3}]
Out[6]:= StieltjesGamma[1] + StieltjesGamma[2](x-1) - 1/2 StieltjesGamma[3](x-1)^2 + 1/6 StieltjesGamma[4](x-1)^3 + O[x-1]^4

In the last one, notice that the 1/(x-1) term from the previous series disappeared altogether, whereas it should have become -1/(x-1)^2. The problem seems to lie somewhere in the symbolic simplifier, since the numerical results are more accurate:

In[7]:= ND[Zeta[x],x,1]
Out[7]:= Indeterminate

In[8]:= ND[Zeta'[x]-1/(1-x),x,1.0000001]
Out[8]:= 0.07281616283461904

However, plotting Zeta'[x] generates some unusual behavior around x=1, and the plot actually appears to have multiple poles in that area. (it looks sort of like Gamma[1/x] around x=0 ) I am not quite sure but I don't think that this is correct either; the zeta function moves to the pole pretty much smoothly and uniformly, so the derivative should reflect that.

2: There is also another (possibly related) problem I am having. Check this out:

In[9]:= Product[k^(1/k),{k,0,Infinity}] (* this does not converge *)
Out[9]:= Indeterminate

In[10]:= Product[Exp[Log[k]/k)],{k,0,Infinity}] (* same thing as above in a messier form *)
Out[10]:= Exp[StieltjesGamma[1]]

The last of these seems to somehow mess up the kernel completely, since after performing the calculation it gives lots of erroneous results with Sum and Product. The kernel must be closed and restarted to fix the problem, and some other combinations of input also appear to have similar effects. Here is some of the stuff I got out of it:

In[11]:= Sum[k,{k,0,Infinity}]
Out[11]:= -1/12

In[12]:= Sum[k!,{k,0,Infinity}]
Out[12]:= (-E+I Pi+ExpIntegralEi[1]) / E

In[13]:= Sum[Log[k],{k,0,Infinity}]
Out[13]:= 1/2 Log[2Pi]

All of those obviously diverge, but they are specific cases of more general functions that have analytic continuations to the divergent areas (for example, the In[11] summand is 1/(k^-1), or Zeta[-1]) and it is indeed giving the right answers for those functions. Therefore, I suspect that it is not doing the convergence tests properly for some reason. Once I restarted the kernel though, it gave the correct answers (ComplexInfinity) for all them. Any ideas as to what is happening here?

3: Is it possible to add new rules and stuff to FunctionExpand, Integrate and the like? (I want to add in a couple of new functions and transformation rules for them)

4: There seems to be a function called FDPowerConstant that is occasionally used on the website. I could not find any information on this in the helpfiles and it does not appear to do anything when used as input in Mathematica; anyone know what this does?

5: Can the Nest function accept an index-like parameter that drops in a different value for each nesting? For example, suppose I want to get a numerical value of Log[1Log[2Log[3Log[4...]]]] and so on to 2000, can I do this without actually typing in the whole thing manually?

That's about it; thanks in advance for any responses.


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