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DivisorSigma




Mathematica Notation

Traditional Notation









Number Theory Functions > DivisorSigma[k,n] > Inequalities





http://functions.wolfram.com/13.05.29.0001.01









  


  








Input Form





DivisorSigma[1, n] <= E^EulerGamma n Log[Log[n]] + (n (2 (1 - Sqrt[2]) + EulerGamma - Log[4 Pi]))/Sqrt[Log[n]] /; n > 18








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["DivisorSigma", "[", RowBox[List["1", ",", "n"]], "]"]], "\[LessEqual]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", "EulerGamma"], " ", "n", " ", RowBox[List["Log", "[", RowBox[List["Log", "[", "n", "]"]], "]"]]]], "+", FractionBox[RowBox[List["n", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["2"]]], ")"]]]], "+", "EulerGamma", "-", RowBox[List["Log", "[", RowBox[List["4", " ", "\[Pi]"]], "]"]]]], ")"]]]], SqrtBox[RowBox[List["Log", "[", "n", "]"]]]]]]]], "/;", RowBox[List["n", ">", "18"]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <semantics> <mi> &#963; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Sigma]&quot;, DivisorSigma] </annotation> </semantics> <mn> 1 </mn> </msub> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#8804; </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msqrt> <mn> 2 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mi> log </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msup> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mfrac> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> </msup> <mo> &#8290; </mo> <mi> n </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &gt; </mo> <mn> 18 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <leq /> <apply> <ci> DivisorSigma </ci> <cn type='integer'> 1 </cn> <ci> n </ci> </apply> <apply> <plus /> <apply> <times /> <ci> n </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> </apply> </apply> </apply> <eulergamma /> </apply> <apply> <power /> <apply> <power /> <apply> <ln /> <ci> n </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <eulergamma /> </apply> <ci> n </ci> <apply> <ln /> <apply> <ln /> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <ci> n </ci> <cn type='integer'> 18 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["DivisorSigma", "[", RowBox[List["1", ",", "n"]], "]"]], "\[LessEqual]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", "EulerGamma"], " ", "n", " ", RowBox[List["Log", "[", RowBox[List["Log", "[", "n", "]"]], "]"]]]], "+", FractionBox[RowBox[List["n", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["2"]]], ")"]]]], "+", "EulerGamma", "-", RowBox[List["Log", "[", RowBox[List["4", " ", "\[Pi]"]], "]"]]]], ")"]]]], SqrtBox[RowBox[List["Log", "[", "n", "]"]]]]]]]], "/;", RowBox[List["n", ">", "18"]]]]]]


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





2001-10-29