Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations

John Shackell

Displaying similar documents to “Growth orders occurring in expansions of Hardy-field solutions of algebraic differential equations”

A generalized formula of Hardy.

Campbell, Geoffrey B. (1994)

International Journal of Mathematics and Mathematical Sciences

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Goldbach numbers in sparse sequences

Jörg Brüdern, Alberto Perelli (1998)

Annales de l'institut Fourier

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We show that for almost all $n \in N$ , the inequality $| p_{1} + p_{2} - \exp ((\log n)^{γ}) | < 1$ has solutions with odd prime numbers $p_{1}$ and $p_{2}$ , provided $1 < γ < \frac{3}{2}$ . Moreover, we give a rather sharp bound for the exceptional set. This result provides almost-all results for Goldbach numbers in sequences rather thinner than the values taken by any polynomial.

On the simultaneous approximation of $a$ , $b$ and $a^{b}$

A. Bijlsma (1977)

Compositio Mathematica

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On the new convergence criteria for Fourier series of Hardy and Littlewood

Gilbert Walter Morgan (1935)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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A “Hardy-Littlewood” approach to the $S$ -unit equation

G. R. Everest (1989)

Compositio Mathematica

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Asymptotic values and the growth of analytic functions in spiral domains.

James E. Brennan, Alexander L. Volberg (1993)

Publicacions Matemàtiques

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In this note we present a simple proof of a theorem of Hornblower which characterizes those functions analytic in the open unit disk having asymptotic values at a dense set in the boundary. Our method is based on a kind of ∂-mollification and may be of use in other problems as well.

The complex sum of digits function and primes

Jörg M. Thuswaldner (2000)

Journal de théorie des nombres de Bordeaux

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Canonical number systems in the ring of gaussian integers $ℤ [i]$ are the natural generalization of ordinary $q$ -adic number systems to $ℤ [i]$ . It turns out, that each gaussian integer has a unique representation with respect to the powers of a certain base number $b$ . In this paper we investigate the sum of digits function $ν_{b}$ of such number systems. First we prove a theorem on the sum of digits of numbers, that are not divisible by the $f$ -th power of a prime. Furthermore, we establish an Erdös-Kac type...

On the $b$ -ary expansion of an algebraic number

Yann Bugeaud (2007)

Rendiconti del Seminario Matematico della Università di Padova

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On an estimate of Walfisz and Saltykov for an error term related to the Euler function

Y.-F. S. Pétermann (1998)

Journal de théorie des nombres de Bordeaux

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The technique developed by A. Walfisz in order to prove (in 1962) the estimate $H (x) ≪ {(log x)}^{2 / 3} {(log log x)}^{4 / 3}$ for the error term $H (x) = \sum_{n \leq x} \frac{φ (n)}{n} - \frac{6}{π^{2}} x$ related to the Euler function is extended. Moreover, the argument is simplified by exploiting works of A.I. Saltykov and of A.A. Karatsuba. It is noted in passing that the proof proposed by Saltykov in 1960 of $H (x) ≪ {(log x)}^{2 / 3} {(log log x)}^{1 + ϵ}$ is erroneous and once corrected “only” yields Walfisz’ result. The generalizations obtained can be applied to error terms related to various classical - and less classical -...