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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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