Analytical Proof of the Prime Number Theorem

Introduction

The Prime Number Theorem (PNT) provides an asymptotic description of the distribution of prime numbers, stating that the number of primes less than a given number $ x $, denoted $ \pi(x) $, is approximately $ \frac{x}{\log x} $ as $ x \to \infty $. This theorem, first conjectured by Carl Friedrich Gauss and later rigorously proved by Hadamard and de la Vallée Poussin in 1896, remains a cornerstone of analytic number theory. The proof relies on complex analysis, particularly the study of the Riemann zeta function $ \zeta(s) $, and its functional equation. This article presents an analytical proof of the PNT, emphasizing the interplay between the zeta function, its functional equation, and the estimation of prime counts.

The Dirichlet Series and the Riemann Zeta Function

The PNT is closely tied to the Dirichlet series representation of the Riemann zeta function:
$$ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}, \quad \text{for } \Re(s) > 1. $$
This series converges for real $ s > 1 $, and its analytic continuation to the entire complex plane is achieved via the functional equation. The zeta function's behavior at $ s = 1 $ is critical, as it diverges logarithmically, reflecting the density of primes. The proof of the PNT hinges on estimating $ \zeta(s) $ for $ \Re(s) > 1 $ and connecting it to the distribution of primes.

The Functional Equation of the Riemann Zeta Function

The functional equation of the Riemann zeta function relates its values at $ s $ and $ 1 - s $:
$$ \zeta(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \frac{\Gamma(1 - s)}{\Gamma(s)}, $$
where $ \Gamma $ is the gamma function. This equation allows the evaluation of $ \zeta(s) $ at non-integer values, providing a bridge between the analytic continuation and the behavior of the zeta function at the critical line $ \Re(s) = 1 $. The functional equation also enables the derivation of asymptotic estimates for $ \zeta(s) $, which are essential for proving the PNT.

The Explicit Formula and Prime Counting

The explicit formula for $ \pi(x) $, derived from the Dirichlet series and the functional equation, provides a precise count of primes up to $ x $:
$$ \pi(x) = \sum_{\substack{p \leq x \ p \text{ prime}}} 1 + \sum_{k=1}^{\infty} \frac{1}{k} \sum_{\substack{m=1 \ (m,k)=1}}^{\infty} \mu(m) \frac{1}{m^k} \left( \frac{1}{(1 - \frac{1}{m})^k} \right), $$
where $ \mu $ is the Möbius function. This formula connects the prime counting function to the Dirichlet series of the Möbius function, enabling the estimation of $ \pi(x) $ through analytic methods. The explicit formula is pivotal in the proof, as it allows the evaluation of $ \zeta(s) $ at $ s = 1 $, which is directly related to the density of primes.

Estimation of the Zeta Function and the Prime Number Theorem

The proof of the PNT involves estimating $ \zeta(s) $ for $ \Re(s) > 1 $ and connecting it to the prime counting function. Using the functional equation, one can show that $ \zeta(s) $ behaves asymptotically like $ \frac{1}{1 - s} $ as $ s \to 1 $. This leads to the estimation:
$$ \pi(x) \