On lindelöfs hypothesis concerning the riemann zeta-function

Abstract 1.1 The "hypothesis of Lindelöf" concerning the Zeta-function ζ(s) = ζ ( σ + it) is expressed most simply by the equation ζ(1/2 + it) = 0 (|t|) (A) for every positive e. It may also be expressed by the assertions (B1) that ζ( σ + it) = 0 (|t|), (B) for σ 1/2, or (B2) that this equations is true for σ ≧ 1/2, or (B3) that it is true uniformly for σ ≧ 1/2. The equivalence of these various assertions is a corollary of well known theorems in the general theory of Dirichlet's series.