Sums involving Farey fractions
M. Newman, J. Lehner (1969)
Acta Arithmetica
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M. Newman, J. Lehner (1969)
Acta Arithmetica
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Jaroslav Hančl (2002)
Journal de théorie des nombres de Bordeaux
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The main result of this paper is a criterion for linear independence of continued fractions over the rational numbers. The proof is based on their special properties.
H. Kesten (1964)
Acta Arithmetica
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Michael Drmota, Johannes Gajdosik (1998)
Journal de théorie des nombres de Bordeaux
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By using a generating function approach it is shown that the sum-of-digits function (related to specific finite and infinite linear recurrences) satisfies a central limit theorem. Additionally a local limit theorem is derived.
Nobushige Kurokawa, Masato Wakayama (2005)
Rendiconti del Seminario Matematico della Università di Padova
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Franciszek Czekała (1996)
Applicationes Mathematicae
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The paper is concerned with the asymptotic normality of a certain statistic based on the logarithms of disjoint m-spacings. The exact and asymptotic mean and variance are computed in the case of uniform distribution on the interval [0,1]. This result is generalized to the case when the sample is drawn from a distribution with positive step density on [0,1].
J. Mikusiński (1953)
Studia Mathematica
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C. Hooley (1963)
Acta Arithmetica
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Catalina Calderón García, María José Zárate Azcuna (1990)
Extracta Mathematicae
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J.-L. Nicolas, A. Sárközy (2000)
Journal de théorie des nombres de Bordeaux
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Let denote the number of partitions of into parts, each of which is at least . By applying the saddle point method to the generating series, an asymptotic estimate is given for , which holds for , and .