scholarly journals The power series coefficients of functions defined Dirichlet series

1961 ◽  
Vol 5 (1) ◽  
pp. 43-44 ◽  
Author(s):  
W. E. Briggs ◽  
R. G. Buschman
Keyword(s):  
Power Series ◽  
Dirichlet Series

A Multiplicative Analogue of Schur's Tauberian Theorem

2003 ◽  
Vol 46 (3) ◽  
pp. 473-480 ◽  
Author(s):  
Karen Yeats
Keyword(s):  
Power Series ◽  
Asymptotic Behaviour ◽  
Dirichlet Series ◽  
Tauberian Theorem ◽  
Partial Sums ◽  
Regularly Varying Functions ◽  
Regularly Varying

AbstractA theorem concerning the asymptotic behaviour of partial sums of the coefficients of products of Dirichlet series is proved using properties of regularly varying functions. This theorem is a multiplicative analogue of Schur's Tauberian theorem for power series.

Random Dirichlet Functions: Multipliers and Smoothness

1993 ◽  
Vol 45 (2) ◽  
pp. 255-268 ◽  
Author(s):  
W. George Cochran ◽  
Joel H. Shapiro ◽  
David C. Ullrich
Keyword(s):  
Power Series ◽  
Holomorphic Function ◽  
Dirichlet Series ◽  
Dirichlet Space ◽  
General Setting ◽  
Sufficient Condition ◽  
Dirichlet Spaces ◽  
Random Series ◽  
Weighted Dirichlet Spaces ◽  
Almost All

AbstractWe show that if is a holomorphic function in the Dirichlet space of the unit disk, then almost all of its randomizations are multipliers of that space. This parallels a known result for lacunary power series, which also has a version for smoothness classes: every lacunary Dirichlet series lies in the Lipschitz class Lip1/2 of functions obeying a Lipschitz condition with exponent 1/2. However, unlike the lacunary situation, no corresponding “almost sure” Lipschitz result is possible for random series: we exhibit a Dirichlet function with norandomization in Lip1/2. We complement this result with a “best possible” sufficient condition for randomizations to belong almost surely to Lip1/2. Versions of our results hold for weighted Dirichlet spaces, and much of our work is carried out in this more general setting.

scholarly journals Hochschild cohomology of reduced incidence algebras

2016 ◽  
Vol 16 (09) ◽  
pp. 1750168
Author(s):  
Müge Kanuni ◽  
Atabey Kaygun ◽  
Serkan Sütlü
Keyword(s):  
Power Series ◽  
Dirichlet Series ◽  
Formal Power Series ◽  
Hochschild Cohomology ◽  
Incidence Algebras ◽  
Exponential Power ◽  
Formal Power

We compute the continuous Hochschild cohomology of four reduced incidence algebras: the algebra of formal power series, the algebra of exponential power series, the algebra of Eulerian power series, and the algebra of formal Dirichlet series. We achieve the result by carrying out the computation for the coalgebra Cotor-groups of their pre-dual coalgebras.

A connection between power series and Dirichlet series

2021 ◽  
Vol 493 (2) ◽  
pp. 124541
Author(s):  
Luis M. Navas ◽  
Francisco J. Ruiz ◽  
Juan L. Varona
Keyword(s):  
Power Series ◽  
Dirichlet Series

Schlicht Dirichlet Series

1958 ◽  
Vol 10 ◽  
pp. 161-176 ◽  
Author(s):  
M. S. Robertson
Keyword(s):  
Power Series ◽  
Dirichlet Series ◽  
Proper Choice ◽  
Image Position

For power series (1.1) for which (1.2) , it has been known for four decades (1) that ƒ(z) is regular and univalent or schlicht in |z| < 1. This theorem, due to J. W. Alexander, has more recently been studied by Remak (5) who has shown that w = ƒ(z), under the hypothesis (1.2), maps |z| < 1 onto a star-like region, and if (1.2) is not satisfied=(z) need not be univalent in |z| < 1 for a proper choice of the amplitudes of the coefficients an.

scholarly journals Note on the region of overconvergence of Dirichlet series with Ostrowski gaps

1966 ◽  
Vol 7 (4) ◽  
pp. 169-173 ◽  
Author(s):  
J. P. Earl ◽  
J. R. Shackell
Keyword(s):  
Power Series ◽  
Dirichlet Series ◽  
Partial Sums ◽  
Direct Treatment

The main object of this note is to show that a proof given by A. J. Macintyre [2] of a result on the overconvergence of partial sums of power series works more easily in the context of Dirichlet series. Applying this observation to the particular Dirichlet series Σane−ns, we can remove certain restrictions which Macintyre finds necessary in the direct treatment of power series.

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