A theorem on Dirichlet series (Q5905658)

We consider \(k\) Dirichlet series \(\sum a_ j(n)n^{-s}\) (\(1\leq j\leq k\)), \(k\geq 2\). We suppose that for each \(j\) the series \(\sum a_ j(n)n^{-s}\) converges for \(s=s_ j=\sigma_ j+it_ j\), and that \(\text{Max }\sigma_ j-\text{Min }\sigma_ j<1/(k-1)\). We prove that the (Dirichlet) product of these series converges uniformly on every bounded segment of the line \(\text{Re s}=(\sigma_ 1+\dots+\sigma_ k)/k+1-1/k\) and we give an information on the rapidity of convergence. The number \(1-1/k\) cannot be replaced by a smaller one.