Classical and modern treatments of Riemann zeta function

Abstract

Tate’s doctoral thesis, Fourier Analysis in Number Fields and Hecke’s Zeta- Functions, on the analytic properties of the class of L-functions introduced by Erich Hecke, is one of the relatively few such dissertations that have become a byword. In it the methods, novel for that time, of Fourier analysis on groups of adeles, were worked out to recover Hecke’s results. In this M.S. thesis, after a brief chapter on the classical treatment of Riemann zeta function, we discuss the local theory, restricted direct products, and the global theory following Tate’s thesis. We compute prime divisors of quadratic fields and quasi-characters of p-adic fields.