On Bohr's theorem for general Dirichlet series

We present quantitative versions of Bohr's theorem on general Dirichlet series D = Sigma a(n)e(-lambda)(ns) assuming different assumptions on the frequency lambda = (lambda(n)), including the conditions introduced by Bohr and Landau. Therefore, using the summation method by typical (first) means invented by M. Riesz, without any condition on lambda, we give upper bounds for the norm of the partial sum operator S-N(D) := Sigma(N)(n=1) a(n)(D)e(-lambda ns) of length N on the space D-infinity(ext)(lambda) of all somewhere convergent lambda-Dirichlet series, which allow a holomorphic and bounded extension to the open right half plane [Re > 0]. As a consequence for some classes of lambda's we obtain a Montel theorem in D-infinity(lambda); the space of all D is an element of D-infinity(ext) (lambda) which converge on [Re > 0]. Moreover, following the ideas of Neder we give a construction of frequencies lambda for which D-infinity(lambda) fails to be complete.