Explicit Formula - Applications

Applications

Riemann's original use of the explicit formula was to give an exact formula for the number of primes less than a given number. To do this, take F(log(y)) to be y1/2/log(y) for 0 ≤ y ≤ x and 0 elsewhere. Then the main term of the sum on the right is the number of primes less than x. The main term on the left is Φ(1); which turns out to be the dominant terms of the prime number theorem, and the main correction is the sum over non-trivial zeros of the zeta function. (There is a minor technical problem in using this case, in that the function F does not satisfy the smoothness condition.)