Hint: Define suitable sequence, use Abel's transformation to transform general term of series and Dirichle's theorem about uniform convergence of functional series. (Answer: )
I offer this direct proff. Function is defined for , by Dirichle criteria. Let
Now, for , and
Let . By Abel's transformation, we have
it's only neccessary to prove that series for is uniform convergent for
it's easy to prove that
and it's decreasing.
For we have
and for , so that
And anestly, I know very little about Riman's zeta-function.