LOG#104. Primorial objects. My post today will be discussing two ideas: the primorial and the paper “The product over all primes is $4\pi^2$ (2003).

The primorial is certain generalization of the factorial, but running on prime numbers. While the factorial is defined as $n!=n\cdot (n-1)\cdots 3\cdot \cdot 2\cdot 1$

the primorial is defined as follows $\boxed{p_n\#=\prod_{k=1}^np_k}$

The first primorial numbers for $n=1,2,\ldots$ are $2,6,30,210,2310,30030,510510,\ldots$

We can also extend the notion of primorial to integer numbers $\boxed{n\#=\prod_{k=1}^{\pi (n)}p_k}$

where $\pi (n)$ is the prime counting function. The first primorial numbers for integer numbers are $1,2,6,6,30,30,210,210,210,210,2310,\ldots$

In fact, if you take the limit $\displaystyle{\lim_{n\rightarrow \infty}(p_n)^{1/p_n}=e}$

since the Chebyshev function provides $\displaystyle{\lim_{x\rightarrow \infty}\dfrac{x}{\theta (x)}=1}$

By the other hand, the factorial of infinity can be regularized $\infty ! =1\cdot 2\cdot 3\cdots =\sqrt{2\pi}$

The paper mentioned above provides a set of cool formulae related to infinite products of prime numbers powered to some number! The main result of the paper is that $\boxed{\prod_p p=4\pi^2}$

If you have an increasing sequence of numbers $0<\lambda_1\leq \lambda_2\leq \ldots$, then we can define the regularized products thanks to the Riemann zeta function (this technique is called zeta function regularization procedure): $\boxed{\prod_{n=1}^\infty \lambda_n =e^{-\zeta'_\lambda (0)}}$ $\boxed{\zeta_\lambda (s)=\sum_{n=1}^\infty \lambda_n^{-s}}$

and where the $\lambda_n$ is some sequence of positive numbers. The Artin-Hasse exponential can be defined in the following way: $\boxed{\exp (X)=\prod_{n=1}^\infty (1-X^n)^{-\mu (n)/n}}$

and there $\mu (n)$ is the Möbius function. From this exponential, we can easily get that $\boxed{e^{p^{-s}}=\prod_{n=1}^\infty (1-p^{-ns})^{-\mu (n)/n}}$

Using the prime zeta function $\displaystyle{\mathcal{P}(s)= \sum_{p}\dfrac{1}{p^s}}$

we obtain $\displaystyle{e^{\mathcal{P}(s)}=\prod_p e^{p^{-s}}=\prod_p \prod_{n=1}^\infty (1-p^{-ns})^{-\mu (n)/n}}$ $\displaystyle{e^{\mathcal{P}(s)}=\prod_{n=1}^\infty \prod_p (1-p^{-ns})^{-\mu (n)/n}=\prod_{n=1}^\infty \zeta (ns)^{\mu (n)/n}}$

Therefore $\boxed{e^{\mathcal{P}(s)}=\prod_{n=1}^\infty \zeta (ns)^{\mu (n)/n}}$

Now, if we remember that $\displaystyle{\sum_{n=1}^ \infty\dfrac{\mu (n)}{n^{s}}=\dfrac{1}{\zeta (s)}}$

and that $\displaystyle{\mathcal{P}'(s)=\sum_{n=1}^\infty \dfrac{\mu (n)}{n}\dfrac{n\zeta' (ns)}{\zeta (ns)}=\sum_{n=1}^\infty \mu (n)\dfrac{\zeta' (ns)}{\zeta (ns)}}$ $\displaystyle{\mathcal{P}'(0)=\left(\sum_{n=1}^\infty \mu (n)\right) \dfrac{\zeta' (0)}{\zeta (0)}=\dfrac{\zeta' (0)}{\zeta (0)^2}=-2\log (2\pi)}$

and from this we get $\displaystyle{\prod_p p=e^{\mathcal{P}'(0)}=(2\pi)^2=4\pi^2}$

Q.E.D.

And similarly, it can be proved the beautiful formula $\boxed{\displaystyle{\prod_p p^s=(2\pi)^{2s}}}$

Moreover, using the Riemann zeta function $\displaystyle{\zeta (s)=\prod_p (1-p^{-s})^{-1}=\dfrac{\displaystyle{\prod_p p^s}}{\displaystyle{\prod_p (p^s-1)}}}$

we also have $\displaystyle{\boxed{\prod_p(p^s-1)=\dfrac{(2\pi)^{2s}}{\zeta (s)}}}$

In particular, e.g., we get $\displaystyle{\prod_p (p-1)=0}$

and $\displaystyle{\prod_p (p^2-1)=48\pi^2}$

The final part of the paper is a proof using a “more convergent” series of the previous “prime products”/products of prime numbers. It uses the series $\displaystyle{e^{\mathcal{P}(s,t)}=\prod_{n=1}^\infty \zeta (ns)^{\mu (n)/n^t}}$

and it converges $\forall Re (s)>1, Re (t)>1$. But then, the series $\displaystyle{\dfrac{\partial \mathcal{P}}{\partial s}(s,t)=\sum_{n=1}^\infty \dfrac{\mu (n)}{n^{t-1}}\dfrac{\zeta' (ns)}{\zeta (ns)}}$

converges $\forall Re (t)>1, Re (s)> 0$. Then, if $t\in \mathbb{C}$, and $Re (t)>2$, with $s\longrightarrow 0$ we have a limit $\displaystyle{L=\lim_{Re s>0}\lim _{s\rightarrow 0}\dfrac{\partial \mathcal{P}}{\partial s}(s,t)=\left(\sum_{n=1}^\infty \dfrac{\mu (n)}{n^{t-1}}\right)\dfrac{\zeta' (0)}{\zeta (0)}}$

and thus $L=\dfrac{\zeta ' (0)}{\zeta (t-1) \zeta (0)}$

Therefore, the meromorphic extension of this formula to the whole complex plane provides that $\displaystyle{\lim_{t\rightarrow 1} L=\dfrac{\zeta '(0)}{\zeta (0)^2}}$

as we wanted to prove (Q.E.D.).

Let the prime numbers and the primorial be with you!  