Arithmetical aspects of certain functional equations

Lutz G. Lucht

Displaying similar documents to “Arithmetical aspects of certain functional equations”

An integral involving the remainder term in the Piltz divisor problem

R. Sitaramachandrarao (1987)

Acta Arithmetica

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Linear independence of values of a certain generalisation of the exponential function – a new proof of a theorem of Carlson

Rolf Wallisser (2005)

Journal de Théorie des Nombres de Bordeaux

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Let $Q$ be a nonconstant polynomial with integer coefficients and without zeros at the non–negative integers. Essentially with the method of Hermite, a new proof is given on linear independence of values at rational points of the function $\begin{matrix} G (x) = \sum_{n = 0}^{\infty} \frac{x^{n}}{Q (1) Q (2) \dots Q (n)} . \end{matrix}$

On mean values of Dirichlet polynomials with real characters

Matti Jutila (1975)

Acta Arithmetica

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Actions of sets of integers on irrationals

Daniel Berend (1987)

Acta Arithmetica

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On Ramanujan expansions of certain arithmetical functions

Hubert Delange (1976)

Acta Arithmetica

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On multiplication of infinite series

A. Alexiewicz (1948)

Studia Mathematica

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On two analytic functions

K. Mahler (1987)

Acta Arithmetica

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On normal lattice configurations and simultaneously normal numbers

Mordechay B. Levin (2001)

Journal de théorie des nombres de Bordeaux

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Let $q, q_{1}, \dots, q_{s} \geq 2$ be integers, and let $α_{1}, α_{2}, \dots$ be a sequence of real numbers. In this paper we prove that the lower bound of the discrepancy of the double sequence ${(\{α_{m} q^{n}\}, \dots, \{α_{m + s - 1} q^{n}\})}_{m, n = 1}^{M N}$ coincides (up to a logarithmic factor) with the lower bound of the discrepancy of ordinary sequences ${(x n)}_{n = 1}^{M N}$ in $s$ -dimensional unit cube $(s, M, N = 1, 2, \dots)$ . We also find a lower bound of the discrepancy (up to a logarithmic factor) of the sequence ${(\{α_{1} q_{1}^{n}\}, \dots, \{α_{s} q_{s}^{n}\})}_{n = 1}^{N}$ (Korobov’s problem).

Integrability theorems for trigonometric series

Bruce Aubertin, John Fournier (1993)

Studia Mathematica

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We show that, if the coefficients (an) in a series $a_{0} / 2 + \sum_{n = 1}^{\infty} a_{n} c o s (n t)$ tend to 0 as n → ∞ and satisfy the regularity condition that $\sum_{m = 0}^{\infty} {\sum_{j = 1}^{\infty} [\sum_{n = j 2^{m}}^{(j + 1) 2^{m} - 1} | a_{n} - a_{n + 1} |] ²}^{1 / 2} < \infty$ , then the cosine series represents an integrable function on the interval [-π,π]. We also show that, if the coefficients (bn) in a series $\sum_{n = 1}^{\infty} b_{n} s i n (n t)$ tend to 0 and satisfy the corresponding regularity condition, then the sine series represents an integrable function on [-π,π] if and only if $\sum_{n = 1}^{\infty} | b_{n} | / n < \infty$ . These conclusions were previously known to hold under stronger restrictions on the sizes...