TITLE: Does any method of summing divergent series work on the harmonic series?
QUESTION [53 upvotes]: It's sort of folklore (as exemplified by this old post at The Everything Seminar) that none of the common techniques for summing divergent series work to give a meaningful value to the harmonic series, and it's also sort of folklore (although I can't remember where I heard this) that the harmonic series is more or less the only important series with this property.
What other methods besides analytic continuation and zeta regularization exist for summing divergent series?  Do they work on the harmonic series?  And are there other well-known series which also don't have obvious regularizations?

REPLY [6 votes]: As other answers have mentioned, the most "natural" value for the harmonic series seems to be the Euler-Mascheroni constant $\gamma$. The article Euler constant as a renormalized value of Riemann zeta function at its pole by Andrei Vieru says

We believe that Euler constant is not just the "renormalized" value of
  the Riemann zeta function in 1. In a sense that we shall clarify it is
  in fact the normal and natural value of zeta of 1. In this paper we
  first propose a limit definition of a function whose values coincide
  everywhere with those of the Riemann zeta function, save in 1, where
  our limit definition yields the Euler constant. Since in the
  literature one can find more than one way to regularize the value of
  the zeta function at s=1, we give asymptotic expansions where, by dint
  of some extended analogies, Euler constant appears to be the true
  "renormalized" value. As a striking example of such analogies, we
  propose an expansion of the logarithm function based on Euler constant
  and on all values of the zeta function at odd positive integers, in
  which all these presumably irrational numbers are accompanied by
  Harmonic numbers of corresponding orders. The other aim of this paper
  is to show how sequences of rationals, often the same, arise in
  computations related to Dirichlet L-functions. Here, a connection with
  the Liouville lambda function appears to have been found. Thus we
  raise the question about the possible usefulness of an extension of
  the Liouville lambda function to rationals.

One example given in the article (on page 4, equation 11) is
$$\ln \Gamma(x) = -\ln x - \gamma x + \sum_{k \geq 2} \frac{\zeta(k)}{k}(-x)^k$$
where $\Gamma$ is the gamma function and $\zeta$ is the zeta function. If we let $\zeta(1) = \gamma$, this can be simplified to
$$\sum_{k \geq 1} \frac{\zeta(k)}{k} x^k = \ln (-x)!$$
which is, I think, supremely elegant. More generally, we seem to have
$$\sum_{k \geq 1} \zeta(k) k^n x^k = \sum_{k=0}^n \left\{{n+1 \atop k+1}\right\} (-x)^{k+1} \psi^{(k)}(1-x)$$
where $\psi^{(k)}$ is the polygamma function and the brackets indicate Stirling numbers of the second kind.