Prime Numbers and the Reimann Hypothesis

Abstract

Prime numbers are beautiful, mysterious, and beguiling mathematical objects. The mathematician Bernhard Riemann made a celebrated conjecture about primes in 1859, the so-called Riemann hypothesis, which remains one of the most important unsolved problems in mathematics. Through the deep insights of the authors, this book introduces primes and explains the Riemann hypothesis. Students with a minimal mathematical background and scholars alike will enjoy this comprehensive discussion of primes. The first part of the book will inspire the curiosity of a general reader with an accessible explanation of the key ideas. The exposition of these ideas is generously illuminated by computational graphics that exhibit the key concepts and phenomena in enticing detail. Readers with more mathematical experience will then go deeper into the structure of primes and see how the Riemann hypothesis relates to Fourier analysis using the vocabulary of spectra. Readers with a strong mathematical background will be able to connect these ideas to historical formulations of the Riemann hypothesis. The Riemann Hypothesis. Thoughts about numbers ; What are prime numbers? ; "Named" prime numbers ; Sieves ; Questions about primes ; Further questions about primes ; How many primes are there? ; Prime numbers viewed from a distance ; Pure and applied mathematics ; A probabilistic first guess ; What is a "good approximation" ; Square root error and random walks ; What is Riemann's Hypothesis ; The mystery moves to the error term ; Cesàro smoothing ; A view of Li(X) -- [pi](X) ; The prime number theorem ; The staircase of primes ; Tinkering with the staircase of primes ; Computer music files and prime numbers ; The word "spectrum" ; Spectra and trigonometric sums ; The spectrum and the staircase of primes ; To our readers of Part I -- Distributions. Slopes of graphs that have no slopes ; Distributions ; Fourier Transforms : second visit ; Fourier Transform of delta ; Trigonometric series ; A sneak preview of Part III --- The Riemann Spectrum of prime numbers. On losing no information ; From primes to the Riemann Spectrum ; How many [theta][subscript i]'s are there? ; Further questions about the Riemann Spectrum ; From the Riemann Spectrum to primes -- Back to Riemann. Building [pi](X) from the Spectrum ; As Riemann envisioned it ; Companions to the zeta function.