Euler totient function: Representations through equivalent functions (formula 13.06.27.0003)

EulerPhi

$Mathematica Notation$

Number Theory Functions

EulerPhi[n]

Representations through equivalent functions

With related functions

http://functions.wolfram.com/13.06.27.0003.01

Input Form

PrimePi[x] == Sum[Floor[EulerPhi[k]/(k - 1)], {k, 2, Floor[x]}] /; x >= 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PrimePi", "[", "x", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], RowBox[List["Floor", "[", "x", "]"]]], RowBox[List["Floor", "[", FractionBox[RowBox[List["EulerPhi", "[", "k", "]"]], RowBox[List["k", "-", "1"]]], "]"]]]]]], "/;", RowBox[List["x", "\[GreaterEqual]", "0"]]]]]]

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> <semantics> <mi> π </mi> <annotation encoding='Mathematica'> TagBox["\[Pi]", PrimePi] </annotation> </semantics> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mrow> <mo> ⌊ </mo> <mi> x </mi> <mo> ⌋ </mo> </mrow> </munderover> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <semantics> <mi> ϕ </mi> <annotation encoding='Mathematica'> TagBox["\[Phi]", EulerPhi] </annotation> </semantics> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> x </mi> <mo> ≥ </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> PrimePi </ci> <ci> x </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 2 </cn> </lowlimit> <uplimit> <apply> <floor /> <ci> x </ci> </apply> </uplimit> <apply> <floor /> <apply> <times /> <apply> <ci> EulerPhi </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <geq /> <ci> x </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PrimePi", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], RowBox[List["Floor", "[", "x", "]"]]], RowBox[List["Floor", "[", FractionBox[RowBox[List["EulerPhi", "[", "k", "]"]], RowBox[List["k", "-", "1"]]], "]"]]]], "/;", RowBox[List["x", "\[GreaterEqual]", "0"]]]]]]]]

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

2001-10-29