Ramanujan, Zeta Functions, and the Hidden Spectrum of the Primes

More than a century after Srinivasa Ramanujan's discoveries, his extraordinary identities continue to illuminate some of the deepest structures in mathematics.This book explores the remarkable connections between Ramanujan's infinite series, modular forms, theta functions, automorphic representations, Hilbert-Pólya theory, random matrix statistics, adelic harmonic analysis, Langlands correspondences, and noncommutative geometry.Moving from classical analytic number theory to modern arithmetic geometry, the book develops a unified perspective in which Ramanujan's formulae appear as analytic shadows of a deeper arithmetic spectral structure.While making no claim to resolve the Riemann Hypothesis, it argues that any successful theory of the zeta function must explain the profound dualities and spectral phenomena already visible in Ramanujan's work.Suitable for advanced students, researchers, and mathematicians interested in zeta functions, modular forms, arithmetic geometry, and the hidden structure of the prime numbers. Read more