For a tuple of integers $\underline{s}:=(s_1,\dots, s_d)$, the multiple zeta value (MZV) at $\underline{s}$ is defined as: $$ \zeta(s_1,\dots, s_d):= \sum_{n_1>\dots>n_d>0}\frac 1 {n_1^{s_1}\dots n_d^{s_d}}.$$

In all the references I have seen, people restrict to tuples $\underline{s}$ such that $s_1 > 1$ and $s_i \geq 1$ for all $i\geq 2$, mentionning convergence of the series.

However, it seems to me that this series converges as soon as $s_1>1$, $s_1+s_2>2$, \dots, $s_1 + \dots s_d>d$ which allows more MZV at integers. For example, $\zeta(5,-1)$ converges.

My question is twofold:

1) Have these MZV at "negative integers" been studied and are they "number theoretically interesting"?

2)I also understand that there should be a reason to exclude these values and that they don't fit in the general philosophy about MZV. For example, we would have a non trivial relation such as $\zeta(s,0)=\zeta(s-1) -\zeta(s)$ between MZV of different weights. (The weight of $\underline{s}$ is by definition $s_1+\dots s_d$. As far as I understood, the general belief is that all relations between MZV are generated by relations between MZV of the same weight but I don't think that the one above can.)

Is there an a priori good reason to restrict to positive integers?

  • $\begingroup$ I was under the impression that negative input values are also sometimes considered by using some kind of regularization. Unfortunately, I don't have a reference. $\endgroup$ – S. Carnahan May 22 '12 at 8:40
  • $\begingroup$ I think Goncharov's article on periods and mixed motives doesn't restrict to positive integers. $\endgroup$ – Gjergji Zaimi May 22 '12 at 8:57
  • $\begingroup$ @S. Carnahan: as far as I understand, regularization has to do with giving a meaning to diverging series sur as $\zeta(1,2)$. But the series I refer to do converge. $\endgroup$ – Oblomov May 22 '12 at 12:29

In response to 1: Yes, these have been studied, for example, in Multiple zeta values at non-positive integers (2001) and (2014), see also The first derivative multiple zeta values at non-positive integers (2010) and Analytic continuation of multiple zeta-functions and the asymptotic behavior at non-positive integers (2013).

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