Using the Turán–Kubilius inequality for y-friable integers proved by De la Brèteche and Tenenbaum. We provide, the friable mean value of the additive function f where f(n) is either <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44426_2025_11_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$ \tilde{\omega }(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>ω</mi> <mo stretchy="false">~</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44426_2025_11_Article_IEq2.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$\tilde{\Omega }(n),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">~</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> the number of distinct prime factors, the total number of prime factors p of a positive integer n such that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44426_2025_11_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="121" /> </InlineMediaObject> <EquationSource Format="TEX">$s_q(p)\equiv a (\mathrm{mod\,}b) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>s</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>≡</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="normal">mod</mi> <mspace width="0.166667em" /> </mrow> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44426_2025_11_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="115" /> </InlineMediaObject> <EquationSource Format="TEX">$(a,b\in {\mathbb {Z}}, b\geqslant 2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi mathvariant="double-struck">Z</mi> <mo>,</mo> <mi>b</mi> <mo>⩾</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44426_2025_11_Article_IEq5.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">$s_q(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>s</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the sum of the digits in base <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="44426_2025_11_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$q\geqslant 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>⩾</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> of the positive integer n.

On the friable mean value of additive arithmetic functions

期刊论文
发表日期：

2025年7月21日

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Walid Wannes,
Hichem Zouari

摘要

Using the Turán–Kubilius inequality for y-friable integers proved by De la Brèteche and Tenenbaum. We provide, the friable mean value of the additive function f where f(n) is either $ \tilde{\omega }(n)$ $\tilde{ω} (n)$ or $\tilde{\Omega }(n),$ $\tilde{Ω} (n),$ the number of distinct prime factors, the total number of prime factors p of a positive integer n such that $s_q(p)\equiv a (\mathrm{mod\,}b) $ $s_{q} (p) \equiv a (\mod b)$ $(a,b\in {\mathbb {Z}}, b\geqslant 2)$ $(a, b \in Z, b ⩾ 2)$ and $s_q(n)$ $s_{q} (n)$ denote the sum of the digits in base $q\geqslant 2$ $q ⩾ 2$ of the positive integer n.