Sum[Sum[DivisorSigma[0, k], {k, 1, n}] - (n Log[n] - (2 EulerGamma - 1) n), {n, 1, x}] \[Proportional] (1/2) x Log[x] + (EulerGamma - 1/4) x + O[x^(3/4)] /; (x -> Infinity)

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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> n </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> x </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <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> </mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> n </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> x </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> x </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 4 </mn> </mrow> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <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> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> x </ci> </uplimit> <apply> <plus /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> DivisorSigma </ci> <cn type='integer'> 0 </cn> <ci> k </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <times /> <ci> n </ci> <apply> <ln /> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <eulergamma /> </apply> <cn type='integer'> -1 </cn> </apply> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <eulergamma /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> <ci> x </ci> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <ci> x </ci> <apply> <ln /> <ci> x </ci> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> x </ci> <cn type='rational'> 3 <sep /> 4 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> x </ci> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02