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