The Prime Number Double Product

The paper presents a new formula for the characteristic function of the prime numbers in form of a finite double product using only roots of unity.


Introduction
There exist formulas for the Möbius function [2], the divisor functions or the Ramanujan sums involving only roots of unity.There are also many formulas for the characteristic function of the prime numbers using the floor function and Wilson's theorem among other things [1].
However, to the best of the author's knowledge, there isn't a formula for the characteristic function of the prime numbers involving only roots of unity without using the floor function and Wilson's theorem.In this paper, we present such a formula similar to these for the Möbius function or the divisor functions.This formula has the form of a finite double product of simple terms of roots of unity.The only proof found in the literature uses infinite methods, namely an infinite series, even if the two products are finite and there is no proof based only on finite methods.

Definitions
Let  n  N be a natural number.This formula is interesting because it is a product of  (n 1) 2 1 complex numbers giving as result a real number, moreover a natural number.It is surprisingly that the natural number  n n1 is exactly equal to the double product term if  n is a prime number.Now it is time for the proof.

Proof:
In the proof we distinguish three cases: Let us start with the first.
The prime number double product 59 1.) In this case the formula takes the form: which is already true.

3.)
 n  prime.This is the hardest part of the proof, but also the most beautiful.
In this case we should have:   n is a prime, every term of the double product is not contained in  ,0  , we can take the logarithm on both sides to get: 1 e Then we have: In this case we have to use two well-known facts from algebra.
The prime number double product and for no two"a's" we have the same "b's".


(a 1 ,a 2 and a Now we can compute In total, we get for To finish the proof, we have to show that:  For this, define now The prime number double product In fact, it turns out that this formula is not only valid if  n is a prime, moreover it holds true for all  n  N.
The first few cases of this formula are: The last general formula has an easy proof.  Thus we obtain: Observe that We also have the following formula for

Conclusion
With this, I end my article over the Prime Number Double Product by noting that this formula, even if it is useless for primality testing, is only true, because three different areas of mathematics work perfectly together, namely, number theory, analysis and algebra.For me it was unexpected that a finite region in the complex plane (complex version) or the real line (real version) determines the distribution of the prime numbers in a nontrivial way.
Let  (n) be the characteristic function of 1 and the prime numbers, that is:  0 and 0 * x  0 x  C.
*b for some b  N .
n  prime and a  n *b.We have thatZ nZ  is a field and there is an isomorphism between the sets n *b and by fact 2.), we have  e claimed formula. In the next step, we can use the two Prime Number Double Product formulas to get two formulas for  We must only take the square of the absolute value of the Prime Number Double Product, to get its real version.
2  C and Euler' s theorem e ix  cos(x) i sin(x).