We know it converges for any prime $p$. I just want to know how to compute its exact value: $$\prod_{n=1}^{\infty} (1-p^{-n})$$
$\begingroup$
$\endgroup$
11
-
7$\begingroup$ I could be wrong here, but I doubt this has a closed form evaluation. Or, do you want to ask for computing approximations. In any case this is a particular value of the Euler function (or Q-Pochhammer symbol). It would also help in assesing whether this question is on-topic here, if you could give some context why and what exactly you want to know related to this. (Cf FAQ and How to ask). $\endgroup$– user9072Commented Aug 14, 2012 at 10:57
-
4$\begingroup$ I don't think that's right: when $p=2$ you get $0.288788\dots$. Is there an error in how the question was phrased? $\endgroup$– Henry CohnCommented Aug 14, 2012 at 12:39
-
1$\begingroup$ I'm away from my references, but I'm pretty sure it's not $1/4$ when $p=2$. It's related to the Dedekind eta-function. $\endgroup$– Gerry MyersonCommented Aug 14, 2012 at 12:41
-
3$\begingroup$ I came across this product some years ago, in its role as the asymptotic probability (as $n\to\infty$) that an $n\times n$ matrix over the $p$-element field is non-singular. If I remember correctly, I had some reason then for believing that there isn't a simple expression (like $1/4$) for the product when $p=2$, but I don't now remember what the reason was or how convincing it was. I do remember using an Excel spreadsheet to compute the partial product out to at least 20 terms, but I think my reason was not just "by inspection of the spreadsheet". $\endgroup$– Andreas BlassCommented Aug 14, 2012 at 14:34
-
3$\begingroup$ This product also shows up in the Cohen-Lenstra heuristics for the distribution of p-Sylow subgroups of class groups of imaginary quadratic fields. This is expected to be related to Andreas Blass's comment about the probability that an n by n matrix over F_p is nonsingular, thanks to a paper of Friedman and Washington. $\endgroup$– SimonCommented Aug 14, 2012 at 16:11
|
Show 6 more comments
3 Answers
$\begingroup$
$\endgroup$
3
Wolfram MathWorld gives the expression of this product in terms of the q-Pochhammer symbol and the Jacobi theta function. See formulas (46) and (47) in
answered Aug 14, 2012 at 12:30
-
4$\begingroup$ Well, it is a q-Pochhammer symbol essentially by definition (as I commented). $\endgroup$– user9072Commented Aug 14, 2012 at 12:35
-
$\begingroup$ Yes, it is. But its exact value is still very difficult. $\endgroup$– masonCommented Aug 14, 2012 at 14:18
-
$\begingroup$ Determining its exact value is unlikely to be a well-posed question. $\endgroup$ Commented Aug 14, 2012 at 20:16
$\begingroup$
$\endgroup$
4
Let's fix the issue of giving bounds on the infinite product. The pentagonal number theorem gives, after grouping pairs of consecutive terms with the same sign, an alternating series with terms that are decreasing in modulus. So for instance for $c:=\prod _ {n\ge1} (1-2^{-n})$ one has
$$1-\frac{1}{2}-\frac{1}{4}+\frac{1}{32}+\frac{1}{128} -\frac{1}{4096}-\frac{1}{ 32768 } < c < 1-\frac{1}{2}-\frac{1}{4}+\frac{1}{32}+\frac{1}{128}$$ that is $ 0.288787842 < c < 0.2890625, $ in any case larger than $1/4$.
answered Aug 14, 2012 at 20:07
-
2$\begingroup$ +1 for including this information and thus avoiding the possiblilty of any confusion on this matter. $\endgroup$– user9072Commented Aug 14, 2012 at 20:15
-
$\begingroup$ You can find it in GTM 245, P209 , ex 1. $\endgroup$– masonCommented Aug 19, 2012 at 1:41
-
$\begingroup$ Now even in the notes of my calculus course, why. ;-) $\endgroup$ Commented Aug 21, 2012 at 6:18
-
$\begingroup$ Note that this problem has nothing to do with primes. The product converges if $|p|>1$. $\endgroup$ Commented Aug 24, 2012 at 17:38
$\begingroup$
$\endgroup$
The Dedekind eta function is $$ \eta = q^{1/24}\prod_{k=1}^\infty(1-q^k) = e^{\pi i \tau/12}\prod_{k=1}^\infty(1-e^{2\pi i k \tau}) $$ which converges for complex $q$ with $|q| < 1$. It is often written in terms of a complex argument $\tau$ with $\text{Im}\;\tau > 0$, where $q=e^{2\pi i \tau}$. The factor $q^{1/24}$ in front gives this desirable number-theoretic properties, but clearly it can be evaluated with that factor if and only if it can be evaluated without.
Some exact values are known for the eta function (see the Wikipedia page). But not $q=1/p$, $p$ prime.
answered Jul 3, 2020 at 14:15