Is there a "quantum" Riemann zeta function? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T22:26:04Z http://mathoverflow.net/feeds/question/45950 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45950/is-there-a-quantum-riemann-zeta-function Is there a "quantum" Riemann zeta function? Theo Johnson-Freyd 2010-11-13T19:43:19Z 2010-11-23T21:16:33Z <p>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.</p> <p>I recently found myself wishing I could do something similar for the sum of the <em>quantum</em> integers. Recall that at quantum parameter $q = e^{i\hbar}$, <em>quantum $n$</em> is the complex number <code>$$[n]_q = \frac{q^n - q^{-n}}{q - q^{-1}} = q^{n-1} + q^{n-3} + \dots + q^{3-n} + q^{1-n}.$$</code> The point is that <code>$[n]_1 = n$</code>.</p> <blockquote> <p><strong>Question:</strong> Are there established methods to sum the divergent series <code>$\sum_{n=1}^\infty [n]_q$</code> and its cousins? For example, is there some well-behaved function <code>$\zeta_q(s)$</code> for which the series is naturally the $s=-1$ value?</p> </blockquote> <p>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.</p> <p>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:</p> <p><code>$$\sum_{n=1}^\infty [n]_q = \frac1{q-q^{-1}} \sum_{n=1}^\infty (q^n - q^{-n}) = \frac1{q-q^{-1}} \left( \sum_{n=1}^\infty q^n - \sum_{n=1}^\infty q^{-n}\right) =$$</code> <code>$$= \frac1{q-q^{-1}} \left( \frac{q}{1-q} - \frac{q^{-1}}{1-q^{-1}}\right) = \frac{q+1}{(q-q^{-1})(q-1)}$$</code></p> http://mathoverflow.net/questions/45950/is-there-a-quantum-riemann-zeta-function/45958#45958 Answer by mathphysicist for Is there a "quantum" Riemann zeta function? mathphysicist 2010-11-13T20:06:41Z 2010-11-13T20:21:28Z <p>The paper by Cherednik <a href="http://www.springerlink.com/content/a0cw6fvunupf1da4/" rel="nofollow">On q-analogues of Riemann's zeta function</a> 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 <em>quantum zeta function</em> appears to have a somewhat different meaning, see e.g. the paper <a href="http://dx.doi.org/10.1088/0305-4470/29/21/014" rel="nofollow">On the quantum zeta function</a> by R.E. Crandall.</p> http://mathoverflow.net/questions/45950/is-there-a-quantum-riemann-zeta-function/47148#47148 Answer by F. C. for Is there a "quantum" Riemann zeta function? F. C. 2010-11-23T21:16:33Z 2010-11-23T21:16:33Z <p>Here is another article dealing with similar functions:</p> <p><a href="http://linkinghub.elsevier.com/retrieve/pii/0022314X89900784" rel="nofollow">q-analogue of Riemann’s ζ-function and q-Euler numbers.</a> by Junya Satoh.</p> <p>There are also many articles by Taekyun Kim on related functions.</p> <p>One key point is that the value of the function $\zeta_q$ at negative integers is a fraction which has no limit when $q$ goes to $1$. One can obtain a relation to the $q$-Bernoulli numbers introduced by Carlitz in 1948, by taking a difference with the value of a modified $\zeta_q$ function.</p>