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On the fractional parts of x / n and related sequences. I

Bahman Saffari, R. C. Vaughan (1976)

Annales de l'institut Fourier

This paper and its sequels deal with a new concept of distributions modulo one which is connected with the Dirichlet divisor and similar problems. Each of the theorems has some independent interest, and in addition some of the techniques developed lead to improvements in certain applications of the hyperbola method.

On the fractional parts of x / n and related sequences. II

Bahman Saffari, R. C. Vaughan (1977)

Annales de l'institut Fourier

As promised in the first paper of this series (Ann. Inst. Fourier, 26-4 (1976), 115-131), these two articles deal with the asymptotic distribution of the fractional parts of x h ( x ) where h is an arithmetical function (namely h ( n ) = 1 / n , h ( n ) = log n , h ( n ) = 1 / log n ) and n is an integer (or a prime order) running over the interval [ y ( x ) , x ) ] . The results obtained are rather sharp, although one can improve on some of them at the cost of increased technicality. Number-theoretic applications will be given later on.

On the Frobenius number of a modular Diophantine inequality

José Carlos Rosales, P. Vasco (2008)

Mathematica Bohemica

We present an algorithm for computing the greatest integer that is not a solution of the modular Diophantine inequality a x mod b x , with complexity similar to the complexity of the Euclid algorithm for computing the greatest common divisor of two integers.

On the functional properties of Bessel zeta-functions

Takumi Noda (2015)

Acta Arithmetica

Zeta-functions associated with modified Bessel functions are introduced as ordinary Dirichlet series whose coefficients are J-Bessel and K-Bessel functions. Integral representations, transformation formulas, a power series expansion involving the Riemann zeta-function and a recurrence formula are given. The inverse Laplace transform of Weber's first exponential integral is the basic tool to derive the integral representations. As an application, we give a new proof of the Fourier series expansion...

On the Fundamental Group of self-affine plane Tiles

Jun Luo, Jörg M. Thuswaldner (2006)

Annales de l’institut Fourier

Let A 2 × 2 be an expanding matrix, 𝒟 2 a set with | det ( A ) | elements and define 𝒯 via the set equation A 𝒯 = 𝒯 + 𝒟 . If the two-dimensional Lebesgue measure of 𝒯 is positive we call 𝒯 a self-affine plane tile. In the present paper we are concerned with topological properties of 𝒯 . We show that the fundamental group π 1 ( 𝒯 ) of 𝒯 is either trivial or uncountable and provide criteria for the triviality as well as the uncountability of π 1 ( 𝒯 ) . Furthermore, we give a short proof of the fact that the closure of each component of int ( 𝒯 ) is a locally...

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