<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>In this article, we are going to look at the requirements regarding a monotone function f ∈ ℝ →ℝ <sub>≥0</sub>, and regarding the sets of natural numbers <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2021-0016_eq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mrow><mrow><mo>(</mo><mrow><msub><mrow><mi>A</mi></mrow><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></msubsup><mo>⊆</mo><mi>d</mi><mi>m</mi><mi>n</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow></mrow></math>
<tex-math>\left( {{A_i}} \right)_{i = 1}^\infty \subseteq dmn\left( f \right)</tex-math>
</alternatives>
</inline-formula>, which requirements are sufficient for the asymptotic<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2021-0016_eq_002.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><munder><mo>∑</mo><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>n</mi><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mi>N</mi></msub></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>P</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>≤</mo><msup><mrow><mi>N</mi></mrow><mi>θ</mi></msup></mrow></mtd></mtr></mtable></mrow></munder><mrow><mi>f</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mo>∼</mo><mi>ρ</mi><mrow><mo>(</mo><mrow><mn>1</mn><mo>/</mo><mi>θ</mi></mrow><mo>)</mo></mrow><munder><mo>∑</mo><mrow><mi>n</mi><mo>∈</mo><msub><mrow><mi>A</mi></mrow><mi>N</mi></msub></mrow></munder><mrow><mi>f</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></mrow></mrow></math>
<tex-math>\sum\limits_{\matrix{{n \in {A_N}} \hfill \cr {P\left( n \right) \le {N^\theta }} \hfill \cr } } {f\left( n \right) \sim \rho \left( {1/\theta } \right)\sum\limits_{n \in {A_N}} {f\left( n \right)} }</tex-math>
</alternatives>
</disp-formula> to hold, where N is a positive integer, θ ∈ (0, 1) is a constant, P(n) denotes the largest prime factor of n, and ρ is the Dickman function.</p>
</abstract>

In this article, we are going to look at the requirements regarding a monotone function f ∈ ℝ →ℝ ≥0, and regarding the sets of natural numbers 


(Ai)i=1∞⊆dmn(f)
\left( {{A_i}} \right)_{i = 1}^\infty \subseteq dmn\left( f \right)

, which requirements are sufficient for the asymptotic


∑n∈ANP(n)≤Nθf(n)∼ρ(1/θ)∑n∈ANf(n)
\sum\limits_{\matrix{{n \in {A_N}} \hfill \cr {P\left( n \right) \le {N^\theta }} \hfill \cr } } {f\left( n \right) \sim \rho \left( {1/\theta } \right)\sum\limits_{n \in {A_N}} {f\left( n \right)} }

 to hold, where N is a positive integer, θ ∈ (0, 1) is a constant, P(n) denotes the largest prime factor of n, and ρ is the Dickman function.

On sums of monotone functions over smooth numbers

Acta Universitatis Sapientiae, Mathematica

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

{"article-title":"On sums of monotone functions over smooth numbers"}

In this article, we are going to look at the requirements regarding a monotone function f ∈ ℝ →ℝ ≥0, and regarding the sets of natural numbers...

On sums of monotone functions over smooth numbers

Published Online: Aug 26, 2021

Page range: 273 - 280

Received: Jun 28, 2020

DOI: https://doi.org/10.2478/ausm-2021-0016

Keywords
smooth number, monotone function, Dickman function, Abel’s identity

© 2021 Gábor Román, published by Sciendo

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

On sums of monotone functions over smooth numbers

Published Online: Aug 26, 2021

Page range: 273 - 280

Received: Jun 28, 2020

DOI: https://doi.org/10.2478/ausm-2021-0016

Keywordssmooth number, monotone function, Dickman function, Abel’s identity

© 2021 Gábor Román, published by Sciendo

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

Keywords
smooth number, monotone function, Dickman function, Abel’s identity