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Dirichlet approximation and universal Dirichlet series [article]

Richard Aron, Frédéric Bayart, Paul Gauthier, Manuel Maestre and Vassili Nestoridis
2016 arXiv   pre-print
We also strengthen the notion of universal Dirichlet series.  ...  We characterize the uniform limits of Dirichlet polynomials on a right half plane.  ...  Thus all Dirichlet series of f coincide on C + .  ... 
arXiv:1608.06457v1 fatcat:q3wirkj4wfe3fhgeepq54vhux4

Automatic Dirichlet Series

J.-P Allouche, M Mendès France, J Peyrière
2000 Journal of Number Theory  
Dirichlet series whose coefficients are generated by finite automata define meromorphic functions on the whole complex plane.  ...  For another interesting similar class of Dirichlet series, namely Dirichlet series of (completely) d-multiplicative sequences, see [12] .  ...  DIRICHLET SERIES OF AUTOMATIC SEQUENCES With a sequence (u n ) n 0 taking its values in C we associate two Dirichlet series given by If the sequence (u n ) n 0 takes only finitely many values (this is  ... 
doi:10.1006/jnth.1999.2487 fatcat:25bzv5ehc5eflonj6s7ouuafom

Series de Dirichlet

Emilio Lauret
2008 Revista de Educación Matemática  
Las series de Dirichlet asociadas a caracteres de Dirichlet son llamadas L-series, y se denotan L(s, χ) donde χ es un caracter de Dirichlet.  ...  Sea F (s) = f (n)n −s una serie de Dirichlet.  ...  − L (s, χ) L(s, χ) = − d ds (log L(s, χ)) = d ds Teorema de Dirichlet Usaremos la notación de congruencias. Notemos que n es de la forma qN + r si y sólo si n ≡ r (mód N ) .  ... 
doaj:2e72a710cd9142abbf05abab6503beda fatcat:uolh6v7ruzhqhol4kdlwvbjtve

Admissible Dirichlet Series [article]

Stanley Burris, Karen Yeats
2005 arXiv   pre-print
We propose a definition of "Admissible Dirichlet Series" as the analog of Hayman's 1956 definition of "Admissible Power Series".  ...  In this paper a notion of admissibility for functions that have Dirichlet series expansions is proposed.  ...  It is clear that each F k (s) satisfies (A1)-(A3), and has a Dirichlet series expansion with abscissa of convergence α = 0.  ... 
arXiv:math/0507487v1 fatcat:s6xr4snjgzc53l6sfxs4qfcaze

Note on Dirichlet Series. IV. On the Singularities of Dirichlet Series

Chuji Tanaka
1953 Proceedings of the American Mathematical Society  
If lim»^« log n/X" = 0, then there exists a Dirichlet series X/"-i &» exp (-Xns) having o-= o-, as the natural boundary such that arg (6")=arg (a") (» = 1, 2, ■ ■ ■), and lim«..«, \bn/a"\ =1.Proof.  ... 
doi:10.2307/2031809 fatcat:ujyrnn7h4zdtdhki25f4mhv53q

Note on Dirichlet series. IV. On the singularities of Dirichlet series

Chuji Tanaka
1953 Proceedings of the American Mathematical Society  
If lim»^« log n/X" = 0, then there exists a Dirichlet series X/"-i &» exp (-Xns) having o-= o-, as the natural boundary such that arg (6")=arg (a") (» = 1, 2, ■ ■ ■), and lim«..«, \bn/a"\ =1.Proof.  ... 
doi:10.1090/s0002-9939-1953-0053263-x fatcat:yhcehvo6ajbsjb5u6kxdw537ai

On lacunary Dirichlet series

I. I. Hirschman, J. A. Jenkins
1950 Proceedings of the American Mathematical Society  
Let us take for the constants X*, 0<o<o<l <4<9< from a well known theorem concerning restricted overconvergence of Dirichlet series, see [l, p. 141], that there exists an increasing sequence of integers  ...  License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use on lacunary dirichlet series License or copyright restrictions may apply to redistribution  ... 
doi:10.1090/s0002-9939-1950-0036836-7 fatcat:d3oc6fystrhy7dmne7s2qn4exi

Shifted Multiple Dirichlet Series [article]

Jeff Hoffstein, Min Lee
2014 arXiv   pre-print
We develop certain aspects of the theory of shifted multiple Dirichlet series and study their meromorphic continuations.  ...  This series can be transformed as a Dirichlet series given in (4.10).  ...  Dirichlet series are dealt with as complete objects, and are used to build shifted multiple Dirichlet series in several variables.  ... 
arXiv:1412.5917v1 fatcat:c3q3yasuufewdajfgvtboccjsa

Sulle serie di dirichlet

Pia Nalli
1915 Rendiconti del circolo matematico di Palermo  
di DIRICHLET cir.  ...  In particolare, il prodotto di n serie di DIRICHLET convergenti in tutto il piano, n-I delle quali hanno un semipiano di convergenza assoluta, ~ in tutto il piano sommabile del prim'ordine.  ... 
doi:10.1007/bf03014842 fatcat:hufezkacnrectkavicanhtfwdq

Dirichlet and Poincaré series

A. Good
1985 Glasgow Mathematical Journal  
Let L je (s) denote the Dirichlet series attached to F= F} and £ as in (6) and (7) .  ...  The associated Dirichlet series oo L £ (s) = £ a«(n)n" s , s = a + if, ffl S (^) k S k -s ) , (9) where F(s) denotes the gamma function.  ...  In Theorem 2 certain combinations of the Dirichlet series attached to cusp forms are expressed by series which, in contrast to the Dirichlet series, converge absolutely inside the critical strip.  ... 
doi:10.1017/s0017089500006066 fatcat:rdek6yseofaw5bcgvtfsqxxrtu

On Lacunary Dirichlet Series

I. I. Hirschman, J. A. Jenkins
1950 Proceedings of the American Mathematical Society  
Let us take for the constants X*, 0<o<o<l <4<9< from a well known theorem concerning restricted overconvergence of Dirichlet series, see [l, p. 141], that there exists an increasing sequence of integers  ...  License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use on lacunary dirichlet series License or copyright restrictions may apply to redistribution  ... 
doi:10.2307/2032321 fatcat:pzkmfxjhfzhdvfwsoytj2aia2i

Introduction: Multiple Dirichlet Series [chapter]

Daniel Bump
2012 Multiple Dirichlet Series, L-functions and Automorphic Forms  
In the theory of multiple Dirichlet series it might be useful to substituteW w for the p-part at a finite number of places.  ...  Despite the similarity of this multiple Dirichlet series to that of [36] , this series does not fit the same way in the theory of Weyl group multiple Dirichlet series.  ...  If the a n are themselves L-functions or other Dirichlet series, this is then a multiple Dirichlet series. One may try to study moments of L-functions this way.  ... 
doi:10.1007/978-0-8176-8334-4_1 fatcat:tkgfrjuvejbbradofp3l6chqbe

On Kubota's Dirichlet series

Ben Brubaker, Daniel Bump
2006 Journal für die Reine und Angewandte Mathematik  
The basic building blocks in the Weyl group multiple Dirichlet series are the Kubota Dirichlet series.  ...  The Kubota Dirichlet series are the entry point to a fascinating universe.  ...  • We exhibit concrete finite-dimensional families of Dirichlet series that are closed under the functional equations. The description of these Dirichlet series is as simple as we can imagine.  ... 
doi:10.1515/crelle.2006.073 fatcat:x7uamuykorgjbpdbtmeu4efxua

Dirichlet series and series with Stirling numbers [article]

Khristo N. Boyadzhiev
2021 arXiv   pre-print
This paper presents a number of identities for Dirichlet series and series with Stirling numbers of the first kind.  ...  As coefficients for the Dirichlet series we use Cauchy numbers of the first and second kinds, hyperharmonic numbers, derangement numbers, binomial coefficients, central binomial coefficients, and Catalan  ...  They are defined by the Example 5 . 5 Dirichlet series with derangement numbers.  ... 
arXiv:2109.09167v2 fatcat:flu2xxtdl5gmhe3vmyoctmqyia

The multiple Dirichlet product and the multiple Dirichlet series

Tomokazu Onozuka
2017 International Journal of Number Theory  
u^{*}(1)=1\neq 0$ $u^{*}\in$ $U$ $u^{*}$ 2.2 $u_{EZ}$ $u_{EZ}(1)=0$ $u_{EZ}\not\in U$ 2.2 Dirichlet $F(s_{1}, \ldots, s_{k};f)$ $f\in U$ Euler-Zagier 3 Dirichlet 2 Dirichlet Dirichlet 1 Dirichlet Theorem3.1  ...  .$ Dirichlet $*$ $(f*g)(n)= \sum_{a\cdot b--n,a_{)}b\in \mathbb{N}^{k}}f(a)g(b)$ . $k=1$ Dirichlet Dirichlet $k$ $I$ $I(n):=\{\begin{array}{ll}1 (n=1) ,0 (otherwise).\end{array}$ Theorem 2.2.  ... 
doi:10.1142/s1793042117501184 fatcat:2alb6fulvnbkbc7y5p6tmkhspq
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