Distribution of Prime Numbers and Recent Advances Toward the Riemann Hypothesis

Abstract

The distribution of prime numbers lies at the heart of analytic number theory. The Prime Number Theorem (PNT) gives the first-order asymptotic behaviour of the prime-counting function (\pi(x)), while the finer structure of the primes appears to be governed by the zeros of the Riemann zeta function (\zeta(s)). The Riemann Hypothesis (RH), asserting that all nontrivial zeros of (\zeta(s)) lie on the critical line (\Re(s)=\tfrac12), would yield near–optimal error terms in the PNT and far-reaching consequences for the distribution of primes in short intervals, arithmetic progressions, and many other problems.

This paper reviews the classical framework connecting primes and zeta zeros via explicit formulas and discusses the role of zero-free regions and zero-density estimates. It then surveys selected recent advances connected—directly or indirectly—to RH: improved explicit zero-free regions, refinements in the proportion of zeros known to lie on the critical line, results on the pair correlation of zeros and its implications, developments around the de Bruijn–Newman constant, and progress on bounded gaps between primes. While a proof of RH remains elusive, these developments significantly sharpen our understanding of both prime distribution and the landscape of zeta zeros, suggesting promising directions for future research.