The translation \(\frac 12\) in the theory of Dirichlet series (Q1852451)

Many results in the theory of Dirichlet series lead to a conclusion on the behavior of this or related series at a point shifted by \(1/2.\) It is worth giving a couple of (known) examples from the paper. First, if \(\sum|a_n|^2<\infty,\) then \(\sum a_n n^{-1/2+it}\) converges for almost all \(t.\) Next, if \(\sum a_n n^{-s}\) and \(\sum b_n n^{-s}\) converge at \(s=0,\) then the product of these series converges at \(s=1/2.\) Both results are sharp. In the paper this phenomenon is studied in detail. Besides considering results related to above examples, the authors show the non-existence of Rudin-Shapiro type Dirichlet polynomials.