zbMATH — the first resource for mathematics

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.

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
Full Text: DOI Numdam EuDML
[1] H. BOHR, On the convergence problem for Dirichlet series, Dan. Mat. Fys. Medd., 25, 6 (1946), 1-18 (I A 17 in Collected Math. Works).
[2] H. BOHR, On multiplication of summable Dirichlet series, Mat. Tidsskr., (1950), 71-75 (I A 19 in Collected Math. Works). · Zbl 0039.08203
[3] H. BOHR, On the summability function and the order function of Dirichlet series, Dan. Mat. Fys. Medd., 27, no. 4 (1952), 3-38 (manuscript prepared by E. Følner) (I A 22 in Collected Math. Works). · Zbl 0046.30203
[4] N. BOURBAKI, Utilisation des nombres réels en topologie générale, Chapitre 9, Paris, Hermann, 1958. · Zbl 0085.37103
[5] H. DELANGE & G. TENENBAUM, Un théorème sur LES séries de Dirichlet, Monatshefte Mat., 113 (1992), 99-105. · Zbl 0765.30002
[6] G. H. HARDY & M. RIESZ, The general theory of Dirichlet series, Cambridge tracts in Math. and Math. Physics, 18 (1915). · JFM 45.0387.03
[7] J.-P. KAHANE, The last problem of harald Bohr, J. Austr. Math. Soc., A 47 (1989), 133-152. · Zbl 0734.30003
[8] J.-P. KAHANE, Sur trois notes de Stieltjes relatives aux séries de Dirichlet. Numéro spécial “100 ans après Stieltjes” des Annales de la Faculté des Sciences de Toulouse, (1996), 33-56. · Zbl 0877.01023
[9] E. LANDAU, Über die multiplication Dirichlet’schen reihen, Rendiconti di Palermo, 24 (1907), 81-160. · JFM 38.0322.01
[10] E. LANDAU, Über das konvergenzproblem der Dirichlet’schen reihen, Rendiconti di Palermo, 28 (1909), 113-151. · JFM 40.0311.03
[11] H. QUEFFELEC, Propriétés presque-sûres et quasi-sûres des séries de Dirichlet et des produits d’Euler, Canad. J. Math., 32 (1980), 531-558. · Zbl 0475.30006
[12] M. RIESZ, Sur l’équivalence de certaines méthodes de sommation, Proc. London Math. Soc., (2) 22 (1924), 412-419. · JFM 50.0154.01
[13] W. RUDIN, Real and complex analysis, 3rd ed. Mc-Graw Hill, 1987, p. 118. · Zbl 0925.00005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.