<abstract xmlns="http://www.w3.org/1999/xhtml">Applying the theory of distribution functions of sequences we find the relative densities of the first digits also for sequences <italic>xn</italic> not satisfying Benford’s law. Especially for sequence <italic>xn</italic> = <italic>nr</italic>, <italic>n</italic> = 1, 2, . . . and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0003_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>n</mi></msub><mo>=</mo><msubsup><mi>p</mi><mi>n</mi><mi>r</mi></msubsup></math><tex-math>$x_n = p_n^r $</tex-math></alternatives></inline-formula>, <italic>n</italic> = 1, 2, . . ., where <italic>pn</italic> is the increasing sequence of all primes and <italic>r</italic> > 0 is an arbitrary real. We also add rate of convergence to such densities.</abstract>

Applying the theory of distribution functions of sequences we find the relative densities of the first digits also for sequences xn not satisfying Benford’s law. Especially for sequence xn = nr, n = 1, 2, . . . and xn=pnr$x_n = p_n^r $, n = 1, 2, . . ., where pn is the increasing sequence of all primes and r > 0 is an arbitrary real. We also add rate of convergence to such densities.

Distribution of Leading Digits of Numbers

Department of Business Administration, The International University of Kagoshima, 8-34-1 Sakanoue, Kagoshima-shi, 891-0197,

Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava,

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Distribution of Leading Digits of Numbers"}

Applying the theory of distribution functions of sequences we find the relative densities of the first digits also for sequences xn not satisfying Benford’s...

Distribution of Leading Digits of Numbers

Published Online: Jan 13, 2017

Page range: 23 - 45

Received: Sep 23, 2015

Accepted: Nov 14, 2015

DOI: https://doi.org/10.1515/udt-2016-0003

Keywords
Benford’s law, distribution function, prime number

© 2016 Yukio Ohkubo et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Distribution of Leading Digits of Numbers

Published Online: Jan 13, 2017

Page range: 23 - 45

Received: Sep 23, 2015

Accepted: Nov 14, 2015

DOI: https://doi.org/10.1515/udt-2016-0003

KeywordsBenford’s law, distribution function, prime number

© 2016 Yukio Ohkubo et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Benford’s law, distribution function, prime number