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Uniform Distribution Modulo One
2014The theory of Uniform Distribution Modulo One is a branch of Number Theory which goes back to the seminal work of H. Weyl from 1916. For us the main motivation to study this topic lies in its application for numerical integration based on QMC rules.
Gunther Leobacher +1 more
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On the Distribution of αp; Modulo One
Journal of the London Mathematical Society, 1983It was shown by Vinogradov (see Theorem 7, Chapter 4 of [18]) that, for e > 0, there are infinitely many solutions in primes p of the inequalitywhere {x} denotes the fractional part of x and y = 0.1. The value of γ was improved toby Kaufman [9]. On the Riemann Hypothesis he showed that one can take γ =¼ The method used actually shows that, for any real
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On the distribution of αp modulo one (II)
Science in China Series A: Mathematics, 2000Suppose that α is an irrational number and β is a real number. It is proved that there are infinitely many prime numbers p such that ‖ αp-β ‖＜ p -9/22 .
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CHARACTERIZATIONS OF SHIFT‐INVARIANT DISTRIBUTIONS BASED ON SUMMATION MODULO ONE
Australian Journal of Statistics, 1994SummaryLet X1Y1,…, Yn be independent random variables. We characterize the distributions of X and Yj satisfying the equation {X+Y1++Yn}=dX, where {Z} denotes the fractional part of a random variable Z. In the case of full generality, either X is uniformly distributed on [0,1), or Yj has.a shifted lattice distribution and X is shift‐invariant.
Wilms, R.J.G., Thiemann, J.G.F.
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On the distribution of the a-points of a Selberg class L-function modulo one
Archiv der Mathematik, 2015Let \(F(s)=\sum_{n=1}^{+\infty}\frac{a(n)}{n^s}\), \(s=\sigma+it\), be the Dirichlet series defined in the half-plane \(\sigma>1\). The Selberg class \(\mathcal{S}\) consists of the functions \(F(s)\) satisfying following properties: the Ramanujan hypothesis for the coefficients \(a(n)\), existence the Euler product representation over prime numbers ...
Jakhlouti, Med-Taïb +2 more
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Distribution Modulo One and Diophantine Approximation
2012This book presents state-of-the-art research on the distribution modulo one of sequences of integral powers of real numbers and related topics. Most of the results have never before appeared in one book and many of them were proved only during the last decade.
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On the distribution of ap k modulo one
Mathematika, 1991The problem concerning the distribution of the fractional parts of the sequence an k ( k an integer exceeding one) was first considered by Hardy and Littlewood [6] and Weyl [20] earlier this century. This work was developed, with the focus on small fractional parts of the sequence, by Vinogradov [17], Heilbronn [13] and Danicic [2] (see [1 ...
R. C. Baker, G. Harman
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The distribution of αp modulo one
Mathematical Proceedings of the Cambridge Philosophical Society, 2009AbstractWe prove that, for any irrational number α, there are infinitely many primes p such that ∥αp∥ < p−1/3+ε. Here ∥y∥ denotes the distance from y to the nearest integer. The proof uses Harman's sieve method with arithmetical information coming from bounds for averages of Kloosterman sums.
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