AO. Univ. Prof. MAG. Dr. Manfred Kühleitner (1967–2022) An Obituary

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In this paper, we consider the hyperelliptic analogue of the Frobenius endomorphism generator and show that it produces sequences with large linear complexity on the Jacobian of genus 2 curves.
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Linear Complexity of Sequences on Koblitz Curves of Genus 2

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We examine four different notions of variation for real-valued functions defined on the compact ring of integers of a non-Archimedean local field, with an emphasis on regularity properties of functions with finite variation, and on establishing non-Archimedean Koksma inequalities. The first version of variation is due to Taibleson, the second due to Beer, and the remaining two are new. Taibleson variation is the simplest of these, but it is a coarse measure of irregularity and it does not admit a Koksma inequality. Beer variation can be used to prove a Koksma inequality, but it is order-dependent and not translation invariant. We define a new version of variation which may be interpreted as the graph-theoretic variation when a function is naturally extended to a certain subtree of the Berkovich affine line. This variation is order-free and translation invariant, and it admits a Koksma inequality which, for a certain natural family of examples, is always sharper than Beer’s. Finally, we define a Fourier-analytic variation and a corresponding Koksma inequality which is sometimes sharper than the Berkovich-analytic inequality.
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Non-Archimedean Koksma Inequalities, Variation, and Fourier Analysis

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Let (<italic>a</italic><italic>n</italic>)n=0∞ be a second-order linear recurrence sequence with constant coefficient. We study the limit points and asymptotic distribution of the sequence of consecutive ratios <italic>a</italic><italic>n</italic>+1/<italic>a</italic><italic>n</italic>.
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Consecutive Ratios in Second-Order Linear Recurrence Sequences

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For sequences sufficiently close to (<italic>a</italic> log <italic>n</italic>), with an arbitrary real constant <italic>a</italic>, this note describes the precise asymptotics of the associated empirical distributions modulo one, with respect to the Kantorovich metric as well as a discrepancy-style metric. In particular, the note demonstrates how these asymptotics depend on <italic>a</italic> in a delicate, discontinuous way. The results strengthen and complement known facts in the literature.
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A Note on the Distributions of (log <italic>n</italic>)mod 1

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The Minkowski question-mark function ?(<italic>x</italic>) is a continuous monotonous function defined on [0, 1] interval. It is well known fact that the derivative of this function, if exists, can take only two values: 0 and +∞.It isalso known that the value of the derivative ? (<italic>x</italic>)atthe point <italic>x</italic> =[0; <italic>a</italic>1,<italic>a</italic>2,...,<italic>a</italic><italic>t</italic>,...] is connected with the limit behaviour of the arithmetic mean (<italic>a</italic>1 +<italic>a</italic>2 +···+<italic>a</italic><italic>t</italic>)<italic>/t</italic>. Particularly, N. Moshchevitin and A. Dushistova showed that if
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2022-0014_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><msub><mrow><mi>a</mi></mrow><mn>1</mn></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mn>2</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mi>t</mi></msub><mo>&lt;</mo><msub><mrow><mi>κ</mi></mrow><mn>1</mn></msub><mo>,</mo></mrow></math>
<tex-math>{a_1} + {a_2} + \cdots + {a_t} &lt; {\kappa _1},</tex-math>
</alternatives>
</disp-formula>
where <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2022-0014_eq_002.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>κ</mi></mrow><mn>1</mn></msub><mo>=</mo><mn>2</mn><mo>log</mo><mrow><mo>(</mo><mrow><mfrac><mrow><mn>1</mn><mo>+</mo><msqrt><mn>5</mn></msqrt></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><mo>/</mo><mo>log</mo><mn>2</mn><mo>=</mo><mn>1.3884</mn><mo>…</mo></mrow></math>
<tex-math>{\kappa _1} = 2\log \left( {{{1 + \sqrt 5 } \over 2}} \right)/\log 2 = 1.3884 \ldots</tex-math>
</alternatives>
</inline-formula>, then ?′(<italic>x</italic>)=+∞.They also proved that the constant <italic>κ</italic>1 is non-improvable. We consider a dual problem: how small can be the quantity <italic>a</italic>1 + <italic>a</italic>2 + ··· + <italic>a</italic><italic>t</italic> − <italic>κ</italic>1<italic>t</italic> if we know that ? (<italic>x</italic>) = 0? We obtain the non-improvable estimates of this quantity.
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On the Derivative of the Minkowski Question-Mark Function

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We examine the uniform distribution theory of H. Weyl when there is a periodic perturbation present. As opposed to the classical setting, in this case the conditions for (mod 1) density and (mod 1) uniform distribution turn out to be different.
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Weyl’s Uniform Distribution Under Periodic Perturbation

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In this paper we study the distribution of leading digits of imaginary parts of Riemann zeta zeros in the <italic>b</italic>-adic expansion.
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The International University of Kagoshima

Distribution of Leading Digits of Imaginary Parts of Riemann Zeta Zeros

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Let <italic>K</italic> be a number field which is multiquadratic or quartic cyclic. We prove several results about the Kummer extensions of <italic>K</italic>, namely concerning the intersection between the Kummer extensions and the cyclotomic extensions of <italic>K</italic>. For <italic>G</italic> a finitely generated subgroup of <italic>K</italic>×, we consider the cyclotomic-Kummer extensions <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2022-0017_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>K</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>ζ</mi></mrow><mrow><mi>n</mi><mi>t</mi></mrow></msub><mo>,</mo><mroot><mi>G</mi><mi>n</mi></mroot></mrow><mo>)</mo></mrow><mo>/</mo><mi>K</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>ζ</mi></mrow><mrow><mi>n</mi><mi>t</mi></mrow></msub></mrow><mo>)</mo></mrow></mrow></math>
<tex-math>K\left( {{\zeta _{nt}},\root n \of G } \right)/K\left( {{\zeta _{nt}}} \right)</tex-math>
</alternatives>
</inline-formula> for all positive integers <italic>n</italic> and <italic>t</italic>, and we describe an explicit finite procedure to compute at once the degree of all these extensions.
</abstract>

Kummer Theory for Multiquadratic or Quartic Cyclic Number Fields

AHEAD OF PRINT

Volume 18 (2023): Issue 1 (July 2023)

Volume 17 (2022): Issue 2 (December 2022)

Volume 17 (2022): Issue 1 (May 2022)

Volume 16 (2021): Issue 2 (December 2021)

Volume 16 (2021): Issue 1 (June 2021)

Volume 15 (2020): Issue 2 (December 2020)

Volume 15 (2020): Issue 1 (June 2020)

Volume 14 (2019): Issue 2 (December 2019)

Volume 14 (2019): Issue 1 (June 2019) The sixth International Conference on Uniform Distribution Theory (UDT 2018) CIRM, Luminy, Marseilles, France, October 1–5, 2018

Volume 13 (2018): Issue 2 (December 2018)

Volume 13 (2018): Issue 1 (June 2018)

Volume 12 (2017): Issue 2 (December 2017)

Volume 12 (2017): Issue 1 (June 2017)

Volume 11 (2016): Issue 2 (December 2016)

Volume 11 (2016): Issue 1 (June 2016)

Uniform distribution theory

 The journal is published by:The BOKU-University of Natural Resources and Applied Life Sciences (Vienna, Austria) and Institute of Mathematics of the Slovak Academy of Sciences, (Bratislava, Slovakia)  The aim of this journal is to publish original research papers on various aspects of uniform distribution theory, especially theoretical and computational aspects of combinatorial, diophantine and probabilistic number theory. In particular we welcome submissions on the following topics:   Distribution of one dimensional and multidimensional sequences.  Distribution of binary sequences.  Distribution of integer sequences and sequences from groups and  Distribution problems in finite abstract sets.  Continuously uniform distributions.  Discrepancies.  Pseudo-random number generators.  Quasi-Monte Carlo integration.  Quasi-Monte Carlo methods in financial mathematics.  Theory of densities.  Theory of distribution functions of sequences.  Distribution of integer points in large domains.  Spectral properties of sequences.  Dynamics emerging from sequences.  Number theoretic ciphers and codes.  Combinatorial number theory.  Trigonometric sums.  Diophantine approximations and diophantine equations.  Continued fractions.  Quantum chaos.  In terms of the AMS subject classification, the main subject areas supported are:11K, 11J, 11B and 11L.  Company Registration Number: 00166791; ISSN 1336–913X; EV 4556/12; math.boku.ac.at/udt  Archiving  Sciendo archives the contents of this journal in Portico - digital long-term preservation service of scholarly books, journals and collections.  Plagiarism Policy  The editorial board is participating in a growing community of Similarity Check System's users in order to ensure that the content published is original and trustworthy. Similarity Check is a medium that allows for comprehensive manuscripts screening, aimed to eliminate plagiarism and provide a high standard and quality peer-review process. 