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EulerPhi






Mathematica Notation

Traditional Notation









Number Theory Functions > EulerPhi[n] > Summation > Asymptotic infinite summation





http://functions.wolfram.com/13.06.23.0011.01









  


  










Input Form





Sum[UnitStep[x - EulerPhi[k]], {k, 1, Infinity}] \[Proportional] ((Zeta[2] Zeta[3])/Zeta[6]) x + R[x] /; (x -> Infinity) && R[x] \[LessLess] x Exp[(-(1 - \[CurlyEpsilon])) Sqrt[(Log[x] Log[Log[x]])/2]] && \[CurlyEpsilon] > 0










Standard Form





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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> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <semantics> <mi> &#952; </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mrow> <semantics> <mi> &#981; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Phi]&quot;, EulerPhi] </annotation> </semantics> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <mfrac> <mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;2&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;3&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 6 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;6&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mfrac> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mi> R </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <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> <mrow> <mi> R </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#8810; </mo> <mrow> <mi> x </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#949; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> </msup> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#949; </mi> <mo> &gt; </mo> <mn> 0 </mn> </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> <infinity /> </uplimit> <apply> <ci> UnitStep </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> EulerPhi </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <ci> Zeta </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <ci> Zeta </ci> <cn type='integer'> 6 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> x </ci> </apply> <apply> <ci> R </ci> <ci> x </ci> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> x </ci> <infinity /> </apply> <apply> <ci> LessLess </ci> <apply> <ci> R </ci> <ci> x </ci> </apply> <apply> <times /> <ci> x </ci> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#949; </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <ln /> <ci> x </ci> </apply> <apply> <ln /> <apply> <ln /> <ci> x </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <ci> &#949; </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List["UnitStep", "[", RowBox[List["x_", "-", RowBox[List["EulerPhi", "[", "k", "]"]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", "2", "]"]], " ", RowBox[List["Zeta", "[", "3", "]"]]]], ")"]], " ", "x"]], RowBox[List["Zeta", "[", "6", "]"]]], "+", RowBox[List["R", "[", "x", "]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["x", "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List[RowBox[List["R", "[", "x", "]"]], "\[LessLess]", RowBox[List["x", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["1", "-", "\[CurlyEpsilon]"]], ")"]]]], " ", SqrtBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Log", "[", "x", "]"]], " ", RowBox[List["Log", "[", RowBox[List["Log", "[", "x", "]"]], "]"]]]]]]]]]]]], "&&", RowBox[List["\[CurlyEpsilon]", ">", "0"]]]]]]]]]]










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





2007-05-02