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http://functions.wolfram.com/13.04.17.0001.01
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PrimePi[x] == PrimePi[Sqrt[x]] + DivisorSum[MoebiusMu[Subscript[d, j]]
Floor[x/Subscript[d, j]], {Subscript[d, j], n}] - 1 /;
n == Product[Subscript[p, k], {k, 1, m}] && Element[Subscript[p, k],
Primes] && Subscript[p, k] <= Sqrt[x]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PrimePi", "[", "x", "]"]], "\[Equal]", RowBox[List[RowBox[List["PrimePi", "[", SqrtBox["x"], "]"]], "+", RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List[RowBox[List["MoebiusMu", "[", SubscriptBox["d", "j"], "]"]], RowBox[List["Floor", "[", FractionBox["x", SubscriptBox["d", "j"]], "]"]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["d", "j"], ",", "n"]], "}"]]]], "]"]], "-", "1"]]]], "/;", RowBox[List[RowBox[List["n", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "m"], SubscriptBox["p", "k"]]]]], "\[And]", RowBox[List[SubscriptBox["p", "k"], "\[Element]", "Primes"]], "\[And]", RowBox[List[SubscriptBox["p", "k"], "\[LessEqual]", SqrtBox["x"]]]]]]]]]
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<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> <mrow> <munder> <mo> ∑ </mo> <mrow> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> | </mo> <mi> n </mi> </mrow> </munder> <mrow> <mrow> <semantics> <mi> μ </mi> <annotation encoding='Mathematica'> TagBox["\[Mu]", MoebiusMu] </annotation> </semantics> <mo> ( </mo> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mi> x </mi> <msub> <mi> d </mi> <mi> j </mi> </msub> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <semantics> <mi> π </mi> <annotation encoding='Mathematica'> TagBox["\[Pi]", PrimePi] </annotation> </semantics> <mo> ( </mo> <msqrt> <mi> x </mi> </msqrt> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <msub> <mi> p </mi> <mi> k </mi> </msub> </mrow> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> p </mi> <mi> k </mi> </msub> <mo> ∈ </mo> <semantics> <mi> ℙ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalP]", Function[Primes]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> p </mi> <mi> k </mi> </msub> <mo> ≤ </mo> <msqrt> <mi> x </mi> </msqrt> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> π </ms> <ci> PrimePi </ci> </apply> <ms> ( </ms> <ms> x </ms> <ms> ) </ms> </list> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> ∑ </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> | </ms> <ms> n </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> μ </ms> <ci> MoebiusMu </ci> </apply> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ⌊ </ms> <apply> <ci> FractionBox </ci> <ms> x </ms> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> </apply> <ms> ⌋ </ms> </list> </apply> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> π </ms> <ci> PrimePi </ci> </apply> <ms> ( </ms> <apply> <ci> SqrtBox </ci> <ms> x </ms> </apply> <ms> ) </ms> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> ∏ </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> m </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> p </ms> <ms> k </ms> </apply> </list> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> p </ms> <ms> k </ms> </apply> <ms> ∈ </ms> <apply> <ci> TagBox </ci> <ms> ℙ </ms> <apply> <ci> Function </ci> <primes /> </apply> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> p </ms> <ms> k </ms> </apply> <ms> ≤ </ms> <apply> <ci> SqrtBox </ci> <ms> x </ms> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PrimePi", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["PrimePi", "[", SqrtBox["x"], "]"]], "+", RowBox[List["DivisorSum", "[", RowBox[List[RowBox[List[RowBox[List["MoebiusMu", "[", SubscriptBox["d", "j"], "]"]], " ", RowBox[List["Floor", "[", FractionBox["x", SubscriptBox["d", "j"]], "]"]]]], ",", RowBox[List["{", RowBox[List[SubscriptBox["d", "j"], ",", "n"]], "}"]]]], "]"]], "-", "1"]], "/;", RowBox[List[RowBox[List["n", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "m"], SubscriptBox["p", "k"]]]]], "&&", RowBox[List[SubscriptBox["p", "k"], "\[Element]", "Primes"]], "&&", RowBox[List[SubscriptBox["p", "k"], "\[LessEqual]", SqrtBox["x"]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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