π(x)

In order to know Riemann's zeta function, you need to understand how to count prime numbers. Basically, there is no pattern to find prime numbers, but when you decide the realm of integer such as 100 or 10000, you can pick up prime numbers and count them. Visual basic is useful. The program is based on sieve of Eratosthenes.π(x) means the sum of prime numbers until x. When x is 5, π(x)=3. The graph is like this.

π(10a) means that x are 10,20,30,40・・・・. You count prime numbers infinitely with the same pattern.

If you practiced π(x), you could understand J(x) and the golden key.

J(x)=π(x)+1/2π(√x)+1/3π*root(3, x)+1/4π*root(4, x)+1/5π*root(5, x)+・・・

1/s*logζ(s)=∫[0→∞]J(x)*x^(-s-1)dx

I also count prime numbers which are the same with π(x).