Book

Ramanujan’s mathematics has never belonged only to the past. His formulas continue to open unexpected paths between infinite series, special functions, prime numbers and the Riemann zeta function. Investigations into Ramanujan’s Methods in Analytic Number Theory brings together a collection of studies shaped by that remarkable legacy. The book opens with 1729, the number made famous by Ramanujan’s conversation with G. H. Hardy, and then moves into a series of deeper investigations: a refined expansion for harmonic numbers, Ramanujan’s formula for odd values of the Riemann zeta function, critical values of the zeta function associated with the tau function, and a polynomial expansion for the Hurwitz zeta function. The later chapters turn to generalized Ramanujan primes, divisor sums, Robin’s inequality, positivity arguments, and Hilbert-space formulations connected with the Riemann Hypothesis. Throughout the book, careful attention is given to an essential mathematical distinction; what a formula suggests, what it proves, and where the limits of a method begin. Rather than treating Ramanujan’s identities as isolated curiosities, this book looks beneath them at the symmetry, analytic continuation, contour methods, arithmetic structure, and computational ideas that make them work. Written for readers interested in analytic number theory, special functions, and Ramanujan’s continuing influence, it offers a thoughtful exploration of both the power of these methods and the questions that remain open.