<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we study the sequence (<italic>f</italic> (<italic>p</italic>n))n≥1,where <italic>p</italic>n is the <italic>n</italic>th prime number and <italic>f</italic> is a function of a class of slowly increasing functions including <italic>f</italic> (<italic>x</italic>)=log<italic>b x</italic><italic>r</italic> and <italic>f</italic> (<italic>x</italic>)=log<italic>b</italic>(<italic>x</italic> log <italic>x</italic>)<italic>r</italic>,where <italic>b</italic> ≥ 2 is an integer and <italic>r></italic> 0 is a real number. We give upper bounds of the discrepancy <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0003_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mi>i</mi></msub></mrow><mo>*</mo></msubsup><mrow><mo>(</mo><mrow><mi>f</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>p</mi></mrow><mi>n</mi></msub></mrow><mo>)</mo></mrow><mo>,</mo><mi>g</mi></mrow><mo>)</mo></mrow></mrow></math><tex-math>D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right)</tex-math></alternatives></inline-formula> for a distribution function <italic>g</italic> and a sub-sequence (<italic>N</italic><italic>i</italic>)<italic>i</italic>≥1 of the natural numbers. Especially for <italic>f</italic> (<italic>x</italic>)= log<italic>b x</italic><italic>r</italic>, we obtain the effective results for an upper bound of <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0003_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><msub><mrow><mi>N</mi></mrow><mi>i</mi></msub></mrow><mo>*</mo></msubsup><mrow><mo>(</mo><mrow><mi>f</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>p</mi></mrow><mi>n</mi></msub></mrow><mo>)</mo></mrow><mo></mo><mi>g</mi></mrow><mo>)</mo></mrow></mrow></math><tex-math>D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right)</tex-math></alternatives></inline-formula>.</abstract>

In this paper, we study the sequence (f (pn))n≥1,where pn is the nth prime number and f is a function of a class of slowly increasing functions including f (x)=logb xr and f (x)=logb(x log x)r,where b ≥ 2 is an integer and r> 0 is a real number. We give upper bounds of the discrepancy DNi*(f(pn),g)D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right) for a distribution function g and a sub-sequence (Ni)i≥1 of the natural numbers. Especially for f (x)= logb xr, we obtain the effective results for an upper bound of DNi*(f(pn)g)D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right).

Distribuion of Leading Digits of Numbers II

The International University of Kagoshima

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Distribuion of Leading Digits of Numbers II"}

In this paper, we study the sequence (f (pn))n≥1,where pn is the nth prime number and f is a function of a class of slowly increasing functions including f...

Distribuion of Leading Digits of Numbers II

Publié en ligne: 27 mars 2020

Pages: 19 - 42

Reçu: 06 nov. 2017

Accepté: 16 mai 2018

DOI: https://doi.org/10.2478/udt-2019-0003

Mots clés
prime numbers, Benford’s law, distribution function, discrepancy, regularly varying function

© 2019 Yukio Ohkubo et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Distribuion of Leading Digits of Numbers II

Publié en ligne: 27 mars 2020

Pages: 19 - 42

Reçu: 06 nov. 2017

Accepté: 16 mai 2018

DOI: https://doi.org/10.2478/udt-2019-0003

Mots clésprime numbers, Benford’s law, distribution function, discrepancy, regularly varying function

© 2019 Yukio Ohkubo et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Mots clés
prime numbers, Benford’s law, distribution function, discrepancy, regularly varying function