ABSTRACT
Trigonometric and Fourier series constitute one of the oldest parts of analysis. The study of convergence of Fourier series is both deep and subtle. It would take us far afield to consider the matter in any detail. Riemann’s theory of the integral, and his accompanying ideas on Fourier series, has made an indelible impression on calculus and real analysis. Bernhard Riemann was shy and modest, with little awareness of his own extraordinary powers; thus, at age nineteen, he went to the University of Goottingen with the aim of pleasing his father by studying theology. Riemann soon tired of the curriculum, and with his father’s acquiescence he turned to mathematics. In a fragmentary note found among his posthumous papers, Riemann wrote that he had a proof of the Riemann hypothesis, and that it followed from a formula for the Riemann zeta function, which he had not simplified enough to publish.
