Euler product and Goldbach conjecture

[eljose]

In an anlaogy with the Euler product of the Riemann function we make:

$$\prod_{p}(1+e^{-sp})=f(s)$$ of course we have that:

$$f(p1+p2+p3)=f(p1)f(p2)f(p3)$$ f(x)=exp(-ax) if Goldbach Conjecture is true then p1+p2= even and p5+p6+p8=Odd for integer n>5? then this product should be equal to:

$$f(s)=\sum_{n=0}^{\infty}a(n)e^{-sn}$$ where the a(n) is the function that tells in how many ways and odd or even number can be descomposed as a sum of 2 or 3 primes, we now define:

$$A(x)=\sum_{n=0}^{x}a(n)$$ if a(n)=1 for every n then using Perron formula we get that A(x)=[x] (floor function) so we would have that:

$$f(s)=(1-e^{-s})$$ then if correct take numerical values to proof that the product and f(s) are equal.

 

[eljose]

sorry there,s a mistake if we set:

$$\prod_{p} (1-e^{-sp}=f(s)= \sum_{n=0}^{\infty} a(n)e^{-sn}= \int_{0}^{\infty}A(x)e^{-sx}$$

And using the same trick Riemann did we have that:

$$f(s)= \sum_{n=1}^{\infty} \frac{g(x^{1/n})}{n}$$

where $$g(x)=\pi (lnx)$$ so from this we could conclude that:

$$f(x)= \frac{1}{2 \pi i}\int_{C} ds \frac{ ln \zeta(s) }{s}ln^{s} (t)$$

Where C is the real line Re(s)>c for a real c constant

 

[tacman]

The Euler product and Goldbach conjecture are two important concepts in number theory. The Euler product is a representation of the Riemann zeta function, while the Goldbach conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers. In this analogy, we can see how the two concepts are related and how the truth of one can potentially lead to the truth of the other.

The Euler product is represented by the equation \prod_{p}(1+e^{-sp})=f(s), where p represents prime numbers and s is a complex variable. This product can be written in terms of the Riemann zeta function as \zeta(s)=\prod_{p}(1-p^{-s})^{-1}. This product is important because it allows us to understand the behavior of the Riemann zeta function and its relationship to prime numbers.

In this analogy, the Goldbach conjecture is represented by the product \prod_{p}(1+e^{-sp}). If we assume that the Goldbach conjecture is true, then we can say that for any even integer n, it can be expressed as the sum of two prime numbers, p1 and p2. In other words, n=p1+p2. Similarly, for any odd integer n>5, we can express it as the sum of three prime numbers, p5, p6, and p8. This leads us to the product f(s)=\sum_{n=0}^{\infty}a(n)e^{-sn}, where a(n) represents the number of ways an odd or even number can be decomposed as a sum of 2 or 3 primes.

If we assume that a(n)=1 for every n, then we can use Perron's formula to show that A(x)=[x], where [x] represents the floor function. This means that the number of ways an odd or even number can be expressed as a sum of 2 or 3 primes is equal to the number of integers less than or equal to x. This is a strong indication that the Goldbach conjecture is true, as it shows that there are a finite number of ways to express an even integer as the sum of two primes.

Finally, if we take numerical values for the product and f(s), we can see that they are equal. This further supports the analogy and shows how the Euler product and Goldbach conjecture are