General Dirichlet series and Bohr’s equivalence theorem

Abstract

This chapter treats a class of series, called general Dirichlet series, which includes both power series and ordinary Dirichlet series as special cases. Most of the chapter is devoted to a method developed by Harald Bohr [6] in 1919 for studying the set of values taken by Dirichlet series in a half-plane. Bohr introduced an equivalence relation among Dirichlet series and showed that equivalent Dirichlet series take the same set of values in certain halfplanes. The theory uses Kronecker’s approximation theorem discussed in the previous chapter. At the end of the chapter applications are given to the Riemann zeta function and to Dirichlet L-functions.