Occasionally I find myself in a situation where a naive, non-rigorous computation leads me to a divergent sum, like $\sum_{n=1}^\infty n$. In times like these, a standard approach is to guess the right answer by assuming that secretly my non-rigorous manipulations were really manipulating the Riemann zeta function $\zeta(s) = \sum_{n=1}^\infty n^{-s}$ and its cousins. Then it's reasonable to guess that the "correct" answer is, for example, $\sum_{n=1}^\infty n = \zeta(-1) = -\frac1{12}$. Thus the zeta function and its cousins are a valuable tool for other non-number-theoretic problem solving: it's always easier to rigorously prove that your guess is correct (or discover, in trying to prove it, that it's wrong) than it is to rigorously derive an answer from scratch.
I recently found myself wishing I could do something similar for the sum of the quantum integers. Recall that at quantum parameter $q = e^{i\hbar}$, quantum $n$ is the complex number $$[n]_q = \frac{q^n – q^{-n}}{q – q^{-1}} = q^{n-1} + q^{n-3} + \dots + q^{3-n} + q^{1-n}.$$ The point is that $[n]_1 = n$.
Question: Are there established methods to sum the divergent series $\sum_{n=1}^\infty [n]_q $ and its cousins? For example, is there some well-behaved function $\zeta_q(s)$ for which the series is naturally the $s=-1$ value?
Note that when $n$ is a root of unity, the series truncates, and it would be nice (but maybe too much too hope for) if the regularized series agreed with the truncated series at these values.
I should mention also that I consider the following answer tempting but inaccurate, as it definitely doesn't work at roots of unity, which I do care about:
$$ \sum_{n=1}^\infty [n]_q = \frac1{q-q^{-1}} \sum_{n=1}^\infty (q^n – q^{-n}) = \frac1{q-q^{-1}} \left( \sum_{n=1}^\infty q^n – \sum_{n=1}^\infty q^{-n}\right) = $$
$$ = \frac1{q-q^{-1}} \left( \frac{q}{1-q} – \frac{q^{-1}}{1-q^{-1}}\right) = \frac{q+1}{(q-q^{-1})(q-1)}$$
Best Answer
The paper by Cherednik On q-analogues of Riemann's zeta function gives precisely the definition you're after: $$ \zeta_q(s)=\sum\limits_{n=1}^\infty q^{sn}/[n]_q^s $$ His paper also contains a brief discussion of the properties of this $q$-zeta function. On the other hand, the term quantum zeta function appears to have a somewhat different meaning, see e.g. the paper On the quantum zeta function by R.E. Crandall.