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Applying results of the infinitary Ramsey theory, namely the dichotomy principle of Galvin-Prikry, we show that for every sequence ${(α_{j})}_{j = 1}^{\infty}$ of scalars, there exists a subsequence ${(α_{k_{j}})}_{j = 1}^{\infty}$ such that either every subsequence of ${(α_{k_{j}})}_{j = 1}^{\infty}$ defines a universal series, or no subsequence of ${(α_{k_{j}})}_{j = 1}^{\infty}$ defines a universal series. In particular examples we decide which of the two cases holds.

A Disproof of a Conjecture of Robertson and Generalizations

P. G. Todorov (1991)

Publications de l'Institut Mathématique

A gap series with growth conditions and its applications.

Shinji Yamashita (1987)

Mathematica Scandinavica

A generalization of a result of André-Jeannin concerning summation of reciprocals.

Melham, R.S. (2000)

Portugaliae Mathematica

A hybrid of Darboux's method and singularity analysis in combinatorial asymptotics.

Flajolet, Philippe, Fusy, Eric, Gourdon, Xavier, Panario, Daniel, Pouyanne, Nicolas (2006)

The Electronic Journal of Combinatorics [electronic only]

A majorant problem.

Peretz, Ronen (1992)

International Journal of Mathematics and Mathematical Sciences

A note on matrix transformations of holomorphic Dirichlet series.

Lê Hai Khôi (1999)

Portugaliae Mathematica

A note on pluripolar extensions of univalent functions.

Sičiak, Józef (2006)

Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Universitatis Iagellonicae Acta Mathematica

A note on the construction of the proximate type of an entire Dirichlet series with index-pair $(p, q)$

H. M. Srivastava, H. S. Kasana (1987)

Rendiconti del Seminario Matematico della Università di Padova

A note on the number of zeros of polynomials in an annulus

Xiangdong Yang, Caifeng Yi, Jin Tu (2011)

Annales Polonici Mathematici

Let p(z) be a polynomial of the form $p (z) = \sum_{j = 0}^{n} a_{j} z^{j}$ , $a_{j} \in - 1, 1$ . We discuss a sufficient condition for the existence of zeros of p(z) in an annulus z ∈ ℂ: 1 - c < |z| < 1 + c, where c > 0 is an absolute constant. This condition is a combination of Carleman’s formula and Jensen’s formula, which is a new approach in the study of zeros of polynomials.

A probabilistic description of the three theorems of Hardy.

Goonatilake, Rohitha (2002)

Southwest Journal of Pure and Applied Mathematics [electronic only]

A recovery of Brouncker's proof for the quadrature continued fraction.

Sergey Khrushchev (2006)

Publicacions Matemàtiques

350 years ago in Spring of 1655 Sir William Brouncker on a request by John Wallis obtained a beautiful continued fraction for 4/π. Brouncker never published his proof. Many sources on the history of Mathematics claim that this proof was lost forever. In this paper we recover the original proof from Wallis' remarks presented in his Arithmetica Infinitorum. We show that Brouncker's and Wallis' formulas can be extended to MacLaurin's sinusoidal spirals via related Euler's products. We derive Ramanujan's...

A remark on the means of entire Dirichlet series

Gupta, V.P. (1979)

Portugaliae mathematica

A ridiculously simple and explicit implicit function theorem.

Sokal, Alan D. (2009)

Séminaire Lotharingien de Combinatoire [electronic only]

A study of various results for a class of entire Dirichlet series with complex frequencies

Niraj Kumar, Garima Manocha (2018)

Mathematica Bohemica

Let $F$ be a class of entire functions represented by Dirichlet series with complex frequencies $\sum a_{k} e^{〈 λ^{k}, z 〉}$ for which $(| λ^{k} {| / e)}^{| λ^{k} |} k! | a_{k} |$ is bounded. Then $F$ is proved to be a commutative Banach algebra with identity and it fails to become a division algebra. $F$ is also proved to be a total set. Conditions for the existence of inverse, topological zero divisor and continuous linear functional for any element belonging to $F$ have also been established.

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