The Dirichlet-Bohr radius

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Denote by Ω(n) the number of prime divisors of n ∈ ℕ (counted with multiplicities). For x ∈ ℕ define the Dirichlet-Bohr radius L(x) to be the best r > 0 such that for every finite Dirichlet polynomial Σn≤xann-s we have ∑n ≤ x |an| rΩ(n) ≤ supt ∈ ℝ | ∑n ≤ x ann-it|. We prove that the asymptoti...

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Autor principal: Carando, D.
Otros Autores: Defant, A., García, D., Maestre, M., Sevilla-Peris, P.
Formato: Capítulo de libro
Lenguaje:Inglés
Publicado: Instytut Matematyczny 2015
Acceso en línea:Registro en Scopus
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100 1 |a Carando, D. 
245 1 4 |a The Dirichlet-Bohr radius 
260 |b Instytut Matematyczny  |c 2015 
506 |2 openaire  |e Política editorial 
504 |a Aizenberg, L., Generalization of Carathéodory's inequality and the Bohr radius for multidimensional power series (2005) Selected Topics in Complex Analysis, Oper. Theory Adv. Appl., 158, pp. 87-94. , Birkhäuser, Basel 
504 |a Balasubramanian, R., Calado, B., Queffélec, H., The Bohr inequality for ordinary Dirichlet series (2006) Studia Math., 175, pp. 285-304 
504 |a Bayart, F., Defant, A., Frerick, L., Maestre, M., Sevilla-Peris, P., (2014) Multipliers of Dirichlet Series and Monomial Series Expansions of Holomorphic Functions in Infinitely Many Variables, , arXiv:1405.7205 
504 |a Bayart, F., Pellegrino, D., Seoane-Sepúlveda, J.B., The Bohr radius of the n-dimensional polydisk is equivalent to √(log n)/n (2014) Adv. Math., 264, pp. 726-746 
504 |a Boas, H.P., Khavinson, D., Bohr's power series theorem in several variables (1997) Proc. Amer. Math. Soc., 125, pp. 2975-2979 
504 |a Bohr, H., Über die Bedeutung der Potenzreihen unendlich vieler Variabeln in der Theorie der Dirichletschen Reihen ∑an/n8 (1913) Nachr. Ges. Wiss. Göttingen Math.-Phys. Kl., pp. 441-488 
504 |a Bohr, H., Über die gleichmäßige Konvergenz Dirichletscher Reihen (1913) J. Reine Angew. Math., 143, pp. 203-211 
504 |a Bohr, H., A theorem concerning power series (1914) Proc. London Math. Soc., 13 (2), pp. 1-5 
504 |a De La-Bretèche, R., Sur l'ordre de grandeur des polynômes de Dirichlet (2008) Acta Arith., 134, pp. 141-148 
504 |a Defant, A., Frerick, L., Ortega-Cerdà, J., Ounaïes, M., Seip, K., The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive (2011) Ann. of Math., 174 (2), pp. 485-497 
504 |a Defant, A., García, D., Maestre, M., Bohr's power series theorem and local Banach space theory (2003) J. Reine Angew. Math., 557, pp. 173-197 
504 |a Defant, A., Schwarting, U., Sevilla-Peris, P., Estimates for vector valued Dirichlet polynomials (2014) Monatsh. Math., 175, pp. 89-116 
504 |a Dineen, S., Timoney, R.M., Absolute bases, tensor products and a theorem of Bohr (1989) Studia Math., 94, pp. 227-234 
504 |a Hedenmalm, H., Lindqvist, P., Seip, K., A Hilbert space of Dirichlet series and systems of dilated functions in L2(0,1) (1997) Duke Math. J., 86, pp. 1-37 
504 |a Konyagin, S.V., Queffélec, H., The translation 1/2 in the theory of Dirichlet series (2001) Real Anal. Exchange, 27, pp. 155-175 
504 |a Prachar, K., (1957) Primzahlverteilung, , Springer, Berlin 
504 |a Queffélec, H., H. Bohr's vision of ordinary Dirichlet series; Old and new results (1995) J. Anal., 3, pp. 43-60 
504 |a Queffélec, H., Queffélec, M., (2013) Diophantine Approximation and Dirichlet Series, , Harish-Chandra Research Inst. Lecture Notes 2, Hindustan Book Agency, New Delhi 
520 3 |a Denote by Ω(n) the number of prime divisors of n ∈ ℕ (counted with multiplicities). For x ∈ ℕ define the Dirichlet-Bohr radius L(x) to be the best r > 0 such that for every finite Dirichlet polynomial Σn≤xann-s we have ∑n ≤ x |an| rΩ(n) ≤ supt ∈ ℝ | ∑n ≤ x ann-it|. We prove that the asymptotically correct order of L(x) is (log x)1/4x-1/8. Following Bohr's vision our proof links the estimation of L(x) with classical Bohr radii for holomorphic functions in several variables. Moreover, we suggest a general setting which allows translating various results on Bohr radii in a systematic way into results on Dirichlet-Bohr radii, and vice versa. Copyright © 2007-2014 by IMPAN. All rights reserved.  |l eng 
593 |a Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pab I, Ciudad Universitaria, Buenos Aires, 1428, Argentina 
593 |a IMAS, CONICET, Argentina 
593 |a Institut für Mathematik, Universität Oldenburg, Oldenburg, D26111, Germany 
593 |a Departamento de Análisis Matemático, Universidad de Valencia, Doctor Moliner 50, Burjasot (Valencia), 46100, Spain 
593 |a Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Valencia, Spain 
690 1 0 |a BOHR RADIUS 
690 1 0 |a DIRICHLET SERIES 
690 1 0 |a HOLOMORPHIC FUNCTIONS 
700 1 |a Defant, A. 
700 1 |a García, D. 
700 1 |a Maestre, M. 
700 1 |a Sevilla-Peris, P. 
773 0 |d Instytut Matematyczny, 2015  |g v. 171  |h pp. 23-37  |k n. 1  |p Acta Arith.  |x 00651036  |w (AR-BaUEN)CENRE-1163  |t Acta Arithmetica 
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