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Ordre, convergence et sommabilité de produits de séries de Dirichlet. (Order, convergence and summability of products of Dirichlet series.). (French) Zbl 0977.11037
Summary: Optimal or almost optimal answers are given to the following questions, going back to Stieltjes, Landau and Bohr, about Dirichlet series $$A_j= \sum^{\infty}_{ n=1}a(j,n)n^{-s}$$ $$(j=1,2,\cdots ,k)$$ and their product $$C= \sum^{\infty}_{n=1}c(n)n^{-s}.$$
1) Assuming that the $$A_j$$ converge at points $$\rho_ j$$ and converge absolutely at points $$\rho_j+\tau_j,$$ at which points $$s$$ does it follow that $$C$$ converges?
2) Assuming that the $$A_j$$ converge at points $$\rho_j,$$ at which points $$s$$ does it follow that $$C$$ converges?
3) Assuming that the $$A_j$$ are $$\alpha_j$$-summable at points $$\rho_j,$$ at which points $$s$$ does it follow that $$C$$ is $$\beta$$-summable ?
The answers involve convex functions which enjoy another extremal property: they are the largest order (= Lindelöf) functions compatible with the data.

MSC:
 11M41 Other Dirichlet series and zeta functions 30B50 Dirichlet series, exponential series and other series in one complex variable 40G10 Abel, Borel and power series methods
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References:
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