# LOG#104. Primorial objects.

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

(1)

the primorial is defined as follows

The first primorial numbers for are

We can also extend the notion of primorial to integer numbers

(2)

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

(3)

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

(4)

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):

(5)

(6)

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

(7)

Using the prime zeta function

(8)

we obtain

(9)

(10)

Therefore

(11)

Now, if we remember that

(12)

and that

(13)

(14)

and from this we get

(15)

Q.E.D.

And similarly, it can be proved the beautiful formula

(16)

Moreover, using the Riemann zeta function

(17)

we also have

(18)

In particular, e.g., we get

(19)

and

(20)

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

(21)

and it converges . But then, the series

(22)

converges . Then, if , and , with we have a limit

(23)

and thus

(24)

Therefore, the meromorphic extension of this formula to the whole complex plane provides that

(25)

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

Let the prime numbers and the primorial be with you!

View ratings
Rate this article

This site uses Akismet to reduce spam. Learn how your comment data is processed.