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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<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> <mrow> <munder> <mo> &#8721; </mo> <mrow> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> | </mo> <mi> n </mi> </mrow> </munder> <mrow> <mrow> <semantics> <mi> &#956; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Mu]&quot;, MoebiusMu] </annotation> </semantics> <mo> ( </mo> <msub> <mi> d </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mi> x </mi> <msub> <mi> d </mi> <mi> j </mi> </msub> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#960; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Pi]&quot;, 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> &#10869; </mo> <mrow> <munderover> <mo> &#8719; </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> &#8743; </mo> <mrow> <msub> <mi> p </mi> <mi> k </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> p </mi> <mi> k </mi> </msub> <mo> &#8804; </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> &#960; </ms> <ci> PrimePi </ci> </apply> <ms> ( </ms> <ms> x </ms> <ms> ) </ms> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderscriptBox </ci> <apply> <ci> ErrorBox </ci> <ms> &#8721; </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> &#956; </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> &#8970; </ms> <apply> <ci> FractionBox </ci> <ms> x </ms> <apply> <ci> SubscriptBox </ci> <ms> d </ms> <ms> j </ms> </apply> </apply> <ms> &#8971; </ms> </list> </apply> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#960; </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> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8719; </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> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> p </ms> <ms> k </ms> </apply> <ms> &#8712; </ms> <apply> <ci> TagBox </ci> <ms> &#8473; </ms> <apply> <ci> Function </ci> <primes /> </apply> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> p </ms> <ms> k </ms> </apply> <ms> &#8804; </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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2001-10-29