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What is known about the Dirichlet series given by

$$\zeta(s)=\sum_{n=1}^{\infty} \frac{\#\mathrm{groups}(n)}{n^{s}}$$

where $\#\mathrm{groups}(n)$ is the number of isomorphism classes of finite groups of order $n$. Specifically: does it converge? If so, where? Do the residues at any of its poles have interesting values? Can it be expresed in terms of the classical Riemann zeta function? Is this even an interesting object to think about?

Mathematica has a list of $\#\mathrm{groups}(n)$ for $1 \le n \le 2047$. Plotting the partial sum seems to indicate that it does converge and has a pole at $s=1$.

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up vote 4 down vote accepted

According to a sci.math posting by Avinoam Mann which I found at the best upper bound is #groups(n) $\le n^{c(\log n)^2}$ for some constant $c$. That would indicate that your Dirichlet series diverges for all $s$, having arbitrarily large terms.

See also (the very first entry in the OEIS), which is where I got the link above.

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Wouldn't one need a lower bound to prove divergence? The problem with an upper bound is that you have to bound $\#groups(2^{n})$ for all $n$, when really many (density $\approx \frac{1}{3}$) of terms in the sequence are 1! – Jackson Dec 10 '12 at 14:35
Right, see my answer which has a lower bound (as does the link in Robert's answer). Also, note that Robert isn't just saying that the a_n get large, but instead that a_n/n^s gets large no matter what s is. – Noah Snyder Dec 10 '12 at 14:39
Perfect, thanks to both of you. – Jackson Dec 10 '12 at 15:57

To add a little more detail Robert's answer, Higman and Sims proved that $$\#\text{groups}(p^n) = p^{2n^3 / 27+O(n^8/3)}.$$ Since this grows faster than polynomially in $p^n$, the $p^n$th term of your series grows without bound as Robert says. For more information see this blog post and these MO answers. I myself learned of Higman-Sims results from a beautiful paper of Poonen's, where he gives a similar formula for the dimension of the space of rank n commutative algebras.

The original papers are:

  1. Graham Higman, Enumerating p-groups. I. Inequalities, Proc. London Math. Soc. (3) 10 (1960), 24–30. MR 0113948
  2. Charles C. Sims, Enumerating p-groups, Proc. London Math. Soc. (3) 15 (1965), 151–166. MR 0169921
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