PrimePi[x] == Sum[KroneckerDelta[g[k], Subscript[n, g[k]], 1], {k, 1, Floor[x]}] /; FactorInteger[k] == {{Subscript[p, 1], Subscript[n, 1]}, {Subscript[p, 2], Subscript[n, 2]}, \[Ellipsis], {Subscript[p, g[k]], Subscript[n, g[k]]}} && Element[Subscript[p, j], Primes] && Element[Subscript[n, j], Integers] && Subscript[n, j] > 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PrimePi", "[", "x", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["Floor", "[", "x", "]"]]], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["g", "[", "k", "]"]], ",", SubscriptBox["n", RowBox[List["g", "[", "k", "]"]]], ",", "1"]], "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["FactorInteger", "[", "k", "]"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["p", "1"], ",", SubscriptBox["n", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["p", "2"], ",", SubscriptBox["n", "2"]]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["p", RowBox[List["g", "[", "k", "]"]]], ",", SubscriptBox["n", RowBox[List["g", "[", "k", "]"]]]]], "}"]]]], "}"]]]], "\[And]", RowBox[List[SubscriptBox["p", "j"], "\[Element]", "Primes"]], "\[And]", RowBox[List[SubscriptBox["n", "j"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["n", "j"], ">", "0"]]]]]]]]

<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> &#960; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Pi]&quot;, PrimePi] </annotation> </semantics> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mi> x </mi> <mo> &#8971; </mo> </mrow> </munderover> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mo> , </mo> <msub> <mi> n </mi> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </msub> <mo> , </mo> <mn> 1 </mn> </mrow> </msub> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> factors </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mo> { </mo> <mrow> <msub> <mi> p </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> n </mi> <mn> 1 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> p </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> n </mi> <mn> 2 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mrow> <mo> { </mo> <mrow> <msub> <mi> p </mi> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </msub> <mo> , </mo> <msub> <mi> n </mi> <mrow> <mi> g </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </msub> </mrow> <mo> } </mo> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> p </mi> <mi> j </mi> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8473; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalP]&quot;, Function[Primes]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> n </mi> <mi> j </mi> </msub> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </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'> 1 </cn> </lowlimit> <uplimit> <apply> <floor /> <ci> x </ci> </apply> </uplimit> <apply> <ci> KroneckerDelta </ci> <apply> <ci> g </ci> <ci> k </ci> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> g </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> factors </ci> <ci> k </ci> </apply> <list> <list> <apply> <ci> Subscript </ci> <ci> p </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> p </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </list> <ci> &#8230; </ci> <list> <apply> <ci> Subscript </ci> <ci> p </ci> <apply> <ci> g </ci> <ci> k </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <apply> <ci> g </ci> <ci> k </ci> </apply> </apply> </list> </list> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> <primes /> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> j </ci> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PrimePi", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["Floor", "[", "x", "]"]]], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["g", "[", "k", "]"]], ",", SubscriptBox["n", RowBox[List["g", "[", "k", "]"]]], ",", "1"]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["FactorInteger", "[", "k", "]"]], "\[Equal]", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["p", "1"], ",", SubscriptBox["n", "1"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["p", "2"], ",", SubscriptBox["n", "2"]]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["p", RowBox[List["g", "[", "k", "]"]]], ",", SubscriptBox["n", RowBox[List["g", "[", "k", "]"]]]]], "}"]]]], "}"]]]], "&&", RowBox[List[SubscriptBox["p", "j"], "\[Element]", "Primes"]], "&&", RowBox[List[SubscriptBox["n", "j"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["n", "j"], ">", "0"]]]]]]]]]]

2001-10-29