My post today will be discussing two ideas: the primorial and the paper “The product over all primes is “ (2003).
The primorial is certain generalization of the factorial, but running on prime numbers. While the factorial is defined as
the primorial is defined as follows
The first primorial numbers for are
We can also extend the notion of primorial to integer numbers
where is the prime counting function. The first primorial numbers for integer numbers are
In fact, if you take the limit
since the Chebyshev function provides
By the other hand, the factorial of infinity can be regularized
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
If you have an increasing sequence of numbers , then we can define the regularized products thanks to the Riemann zeta function (this technique is called zeta function regularization procedure):
and where the is some sequence of positive numbers. The Artin-Hasse exponential can be defined in the following way:
and there is the Möbius function. From this exponential, we can easily get that
Using the prime zeta function
Now, if we remember that
and from this we get
And similarly, it can be proved the beautiful formula
Moreover, using the Riemann zeta function
we also have
In particular, e.g., we get
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
and it converges . But then, the series
converges . Then, if , and , with we have a limit
Therefore, the meromorphic extension of this formula to the whole complex plane provides that
as we wanted to prove (Q.E.D.).
Let the prime numbers and the primorial be with you!
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