A theorem on Dirichlet series. (Un théorème sur les séries de Dirichlet.) (French) Zbl 0765.30002

Delange, Hubert; Tenenbaum, Gérald

Monatsh. Math. 113, No. 2, 99-105 (1992).

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.

Reviewer: H.Delange

MSC:

30B50	Dirichlet series, exponential series and other series in one complex variable
11M41	Other Dirichlet series and zeta functions

Keywords:

Dirichlet product of series; convergence; uniform convergence; Dirichlet series