The distribution of prime numbers, one of the central questions in number theory, is elegantly described by the Prime Number Theorem, which states that the number of primes up to a given value x grows asymptotically like x/log(x). We present a complete proof of this theorem following the approach of Newman and Zagier, using only basic tools from complex analysis.

A deeper refinement is provided by the von Mangoldt exact formula, which expresses the Chebyshev weighted prime-counting function in terms of the zeros of the Riemann zeta function. This formula makes explicit the connection between prime numbers and the non-trivial zeros of the zeta function, showing how each zero contributes to the oscillations in the distribution of primes and to deviations from smooth, regular growth.

The Riemann explicit formula offers an even more direct relationship between the number of primes up to x and the zeros of the zeta function, capturing the deviations from the Prime Number Theorem as a sum over these zeros. Together, these formulas deepen our understanding of prime distribution and highlight the profound connection between analytic and arithmetic structures.

We outline the derivation of both exact formulas, omitting some of the most technical steps.