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DivisorSigma






Mathematica Notation

Traditional Notation









Number Theory Functions > DivisorSigma[k,n] > Summation > Asymptotic finite summation





http://functions.wolfram.com/13.05.23.0012.01









  


  










Input Form





Sum[DivisorSigma[k, j] DivisorSigma[l, j + \[Delta]], {j, 1, n}] \[Proportional] ((Zeta[k + 1] Zeta[l + 1])/((k + l + 1) Zeta[k + l + 2])) DivisorSigma[-k - l - 1, \[Delta]] n^(k + l + 1) + O[n^(k + 1)] /; (n -> Infinity) && Element[k, Integers] && k >= 1 && Element[l, Integers] && l >= 1 && Element[\[Delta], Integers] && \[Delta] >= 0










Standard Form





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MathML Form







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</mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> l </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;l&quot;, &quot;+&quot;, &quot;1&quot;]], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> l </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> l </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;l&quot;, &quot;+&quot;, &quot;2&quot;]], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> &#963; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Sigma]&quot;, DivisorSigma] </annotation> </semantics> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> l </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> &#948; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> n </mi> <mrow> <mi> k </mi> <mo> + </mo> <mi> l </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> n </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </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> k </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> l </mi> <mo> &#8712; </mo> <mi> k </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#948; </mi> <mo> &#8712; </mo> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <ci> DivisorSigma </ci> <ci> k </ci> <ci> j </ci> </apply> <apply> <ci> DivisorSigma </ci> <ci> l </ci> <apply> <plus /> <ci> j </ci> <ci> &#948; </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> l </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> k </ci> <ci> l </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> k </ci> <ci> l </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> DivisorSigma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> l </ci> </apply> <cn type='integer'> -1 </cn> </apply> <ci> &#948; </ci> </apply> <apply> <power /> <ci> n </ci> <apply> <plus /> <ci> k </ci> <ci> l </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> n </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> n </ci> <infinity /> </apply> <apply> <in /> <ci> k </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <ci> l </ci> <ci> k </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <in /> <ci> &#948; </ci> <ci> k </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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Date Added to functions.wolfram.com (modification date)





2007-05-02





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