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 $[n]\_1 = n$.



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> **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?
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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:


$$ \sum\_{n=1}^\infty [n]\_q = \frac1{q-q^{-1}} \sum\_{n=1}^\infty (q^n - q^{-n}) = \frac1{q-q^{-1}} \left( \sum\_{n=1}^\infty q^n - \sum\_{n=1}^\infty q^{-n}\right) = $$
$$ = \frac1{q-q^{-1}} \left( \frac{q}{1-q} - \frac{q^{-1}}{1-q^{-1}}\right) = \frac{q+1}{(q-q^{-1})(q-1)}$$



Is there a "quantum" Riemann zeta function?
The paper by Cherednik [On q-analogues of Riemann's zeta function](https://doi.org/10.1007/s00029-001-8095-6 "Sel. math., New ser. 7, 447–491 (2001). zbMATH review at https://zbmath.org/?q=an:1001.11033") ([arXiv:math/9804099](https://doi.org/10.48550/arXiv.math/9804099)) 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](https://doi.org/10.1088/0305-4470/29/21/014 "J. Phys. A, Math. Gen. 29, No. 21, 6795–6816 (1996). zbMATH review at https://zbmath.org/?q=an:0905.58040") by R.E. Crandall.


