# What's the limit of this product? $\prod_{n=2}^\infty \left(1-\frac1{n^p}\right)$

Sorry I'm really out of any sort of number theory... is this a known product? \begin{align} P & = \prod_{n=2}^\infty \left(1-\dfrac1{n^p}\right) \end{align} As a special case I would be interested in $p=3$. Thank you very much

• It can be written in terms of the gamma function. – Sangchul Lee Feb 18 '16 at 7:02
• that would work, can you tell me how? – Dac0 Feb 18 '16 at 7:03
• The basic idea is to factor $(1-n^{-p})$ into linear terms and utilize the Weierstrass product for $\Gamma(z)$ to simply the resulting expression. For a technical reason it is easier to work with partial products first. – Sangchul Lee Feb 18 '16 at 7:08
• Of course the valuable fact is an understandable proof, which I hope see from an user to learn it. But here you have a tool for the future (type product (1-1/k^3), k=2 to infinity) in INFINITE PRODUCTS here wolframalpha.com/input/… – user243301 Feb 18 '16 at 7:25
• Just give the infinite product to maple and you will get a closed form! – Mhenni Benghorbal Feb 18 '16 at 7:26

## 1 Answer

From the Weierstrass product for $\Gamma(z)$, we can easily find that

$$\frac{e^{(\gamma-1)z}}{\Gamma(2-z)} = \prod_{n=2}^{\infty} \left( 1 - \frac{z}{n} \right) e^{z/n}.$$

Writing

$$\prod_{n=2}^{\infty} \left(1 - \frac{1}{n^p}\right) = \prod_{n=2}^{\infty} \prod_{\omega : \omega^p = 1} \left( 1 - \frac{\omega}{n} \right) e^{\omega/n}$$

and interchanging the order of summation, we have

$$\prod_{n=2}^{\infty} \left(1 - \frac{1}{n^p}\right) = \prod_{\omega : \omega^p = 1} \frac{1}{\Gamma(2-\omega)}.$$

For some special choice of $p$ (such as $p = 2$ or $p = 3$), this can be simplified further.

• Great Answer!!! – Dac0 Feb 18 '16 at 7:44