Book

Description

The Riemann Hypothesis stands as one of the greatest unsolved problems in Number Theory, forming the cornerstone of analytic number theory and profoundly influencing our understanding of the distribution of primes. Since Bernhard Riemann's seminal memoir of 1859, generations of mathematicians have sought to unravel the deep connections between the non-trivial zeros of the Riemann zeta function and the hidden order governing the primes. This book presents a comprehensive analytical study of the Riemann Hypothesis while extending its discussion to several contemporary problems lying at the core of Hilbert's Eighth Problem. Beginning with the historical evolution of PNT, the text develops the theory of the Riemann zeta function, the PNT, explicit formulae, and the role of the critical strip in understanding the asymptotic distribution of prime numbers. A distinguishing feature of this monograph is its integration of original computational and theoretical investigations into prime number research. Beyond the classical theory, the book explores the behaviour of bounded prime gaps (with particular emphasis on gaps less than or equal to 6), and examines their relationship with the distribution of primes. It further investigates the additive structure of primes through Goldbach's Binary Conjecture, introducing strengthened formulations involving twin primes, isolated primes, and (Ap + B)-type in additive decompositions. These studies combine rigorous mathematical reasoning with large-scale computational verification (up to 7,50,000), offering new perspectives on several longstanding questions in additive and analytic number theory. Designed for graduate students, researchers, and enthusiasts of number theory, this volume bridges classical mathematics with modern computational approaches. By bringing together the Riemann Hypothesis, prime distribution, bounded prime gaps, Goldbach-type problems, and original research contributions within a unified analytical framework, the book provides both a comprehensive reference and a fresh perspective on some of the most profound and enduring mysteries surrounding the prime numbers.