scholarly journals Natural boundary of random Dirichlet series

2006 ◽  
Vol 58 (7) ◽  
pp. 1129-1138 ◽  
Author(s):  
X. Ding ◽  
Y. Xiao
Keyword(s):  
Dirichlet Series ◽  
Natural Boundary ◽  
Random Dirichlet Series

INFINITE PRODUCTS OF CYCLOTOMIC POLYNOMIALS

2015 ◽  
Vol 91 (3) ◽  
pp. 400-411 ◽  
Author(s):  
WILLIAM DUKE ◽  
HA NAM NGUYEN
Keyword(s):  
Asymptotic Behaviour ◽  
Dirichlet Series ◽  
Unit Circle ◽  
Roots Of Unity ◽  
Cyclotomic Polynomials ◽  
Natural Boundary ◽  
Infinite Products

We study analytic properties of certain infinite products of cyclotomic polynomials that generalise some products introduced by Mahler. We characterise those that have the unit circle as a natural boundary and use associated Dirichlet series to obtain their asymptotic behaviour near roots of unity.

Deficient functions of random dirichlet series

2007 ◽  
Vol 27 (2) ◽  
pp. 291-296
Author(s):  
Junying Zhou ◽  
Daochun Sun
Keyword(s):  
Dirichlet Series ◽  
Random Dirichlet Series

scholarly journals Law of the iterated logarithm for a random Dirichlet series

2020 ◽  
Vol 25 (0) ◽  
Author(s):  
Marco Aymone ◽  
Susana Frómeta ◽  
Ricardo Misturini
Keyword(s):  
Dirichlet Series ◽  
Iterated Logarithm ◽  
Random Dirichlet Series

scholarly journals Julia lines of general random dirichlet series

2012 ◽  
Vol 62 (4) ◽  
pp. 919-936 ◽  
Author(s):  
Qiyu Jin ◽  
Guantie Deng ◽  
Daochun Sun
Keyword(s):  
Dirichlet Series ◽  
Random Dirichlet Series

scholarly journals Asymptotics of some generalized Mathieu series

2020 ◽  
Vol 126 (3) ◽  
pp. 424-450
Author(s):  
Stefan Gerhold ◽  
Friedrich Hubalek ◽  
Živorad Tomovski
Keyword(s):  
Dirichlet Series ◽  
Gamma Function ◽  
Mellin Transform ◽  
Asymptotic Estimates ◽  
Precise Asymptotics ◽  
Singular Behavior ◽  
Natural Boundary ◽  
First Order ◽  
Elementary Estimate ◽  
Mathieu Series

We establish asymptotic estimates of Mathieu-type series defined by sequences with power-logarithmic or factorial behavior. By taking the Mellin transform, the problem is mapped to the singular behavior of certain Dirichlet series, which is then translated into asymptotics for the original series. In the case of power-logarithmic sequences, we obtain precise first order asymptotics. For factorial sequences, a natural boundary of the Mellin transform makes the problem more challenging, but a direct elementary estimate gives reasonably precise asymptotics. As a byproduct, we prove an expansion of the functional inverse of the gamma function at infinity.

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