Sum[Boole[Element[DivisorSigma[0, k]/p, Integers]], {k, 1, x}] \[Proportional] (1 - Zeta[p]/Zeta[p - 1]) x + O[x^(1/(p - 1))] /; (x -> Infinity) && Element[p, Primes]

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> x </mi> </munderover> <mrow> <mi> boole </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <msub> <semantics> <mi> &#963; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Sigma]&quot;, DivisorSigma] </annotation> </semantics> <mn> 0 </mn> </msub> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mi> p </mi> </mfrac> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> p </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;p&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[RowBox[List[&quot;p&quot;, &quot;-&quot;, &quot;1&quot;]], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> x </mi> <mfrac> <mn> 1 </mn> <mrow> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> p </mi> <mo> &#8712; </mo> <mi> &#8473; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> x </ci> </uplimit> <apply> <ci> boole </ci> <apply> <in /> <apply> <times /> <apply> <ci> DivisorSigma </ci> <cn type='integer'> 0 </cn> <ci> k </ci> </apply> <apply> <power /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8469; </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> x </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Zeta </ci> <ci> p </ci> </apply> <apply> <power /> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> x </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> p </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> x </ci> <infinity /> </apply> <apply> <in /> <ci> p </ci> <ci> &#8473; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02