Prime zeta function

Imagine you have a magical machine that helps you count all the prime numbers. This machine is called the Prime Zeta function. It works by taking all the prime numbers and adding them up in a special way to get a final answer.

To start, the machine looks at the number 2, which is the first prime number. Then it looks at the number 3, which is the second prime number. It continues in this way, adding up all the prime numbers in a certain pattern.

The pattern is like counting the fractions 1/1, 1/2, 1/3, 1/4, 1/5, and so on all the way up to infinity. The sum of these fractions is called the harmonic series.

But instead of counting fractions, the Prime Zeta function counts prime numbers. And instead of adding up fractions in order, it adds up the reciprocals of the prime numbers raised to a certain power. For example, it might add up 1/2^s, 1/3^s, 1/5^s, and so on, where s is a number that can change.

When the Prime Zeta function is done adding up all the prime numbers in this pattern, it gives you a special number. This number can tell you things about how the prime numbers are distributed and how many there are.

Overall, the Prime Zeta function is a magical machine that helps you count prime numbers and understand more about them.