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http://functions.wolfram.com/13.01.23.0003.01
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Sum[Subscript[p, 1][k]/Subscript[p, m][k], {k, 2, n}] \[Proportional]
PrimePi[n] (1 + o[n]) /;
n == Product[Subscript[p, j][k]^Subscript[n, j], {j, 1, m}] &&
Element[Subscript[p, j][k], Primes] && Element[Subscript[n, j],
Integers] && Subscript[n, j] > 0 && Subscript[p, j][k] <
Subscript[p, j + 1][k] && 1 <= j <= m - 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], "n"], FractionBox[RowBox[List[SubscriptBox["p", "1"], "[", "k", "]"]], RowBox[List[SubscriptBox["p", "m"], "[", "k", "]"]]]]], "\[Proportional]", RowBox[List[RowBox[List["PrimePi", "[", "n", "]"]], RowBox[List["(", RowBox[List["1", " ", "+", " ", RowBox[List["o", "[", "n", "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "m"], SuperscriptBox[RowBox[List[SubscriptBox["p", "j"], "[", "k", "]"]], SubscriptBox["n", "j"]]]]]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["p", "j"], "[", "k", "]"]], "\[Element]", "Primes"]], "\[And]", RowBox[List[SubscriptBox["n", "j"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["n", "j"], ">", "0"]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["p", "j"], "[", "k", "]"]], "<", RowBox[List[SubscriptBox["p", RowBox[List["j", "+", "1"]]], "[", "k", "]"]]]], "\[And]", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", RowBox[List["m", "-", "1"]]]]]]]]]]
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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> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mi> n </mi> </munderover> <mfrac> <mrow> <msub> <mi> p </mi> <mn> 1 </mn> </msub> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mrow> <msub> <mi> p </mi> <mi> m </mi> </msub> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> ∝ </mo> <mrow> <mrow> <semantics> <mi> π </mi> <annotation encoding='Mathematica'> TagBox["\[Pi]", PrimePi] </annotation> </semantics> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> o </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <msup> <mrow> <msub> <mi> p </mi> <mi> j </mi> </msub> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <msub> <mi> n </mi> <mi> j </mi> </msub> </msup> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mrow> <msub> <mi> p </mi> <mi> j </mi> </msub> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℙ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalP]", Function[List[], Primes]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> n </mi> <mi> j </mi> </msub> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> n </mi> <mi> j </mi> </msub> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <msub> <mi> p </mi> <mi> j </mi> </msub> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mrow> <msub> <mi> p </mi> <mrow> <mi> j </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mn> 1 </mn> <mo> ≤ </mo> <mi> j </mi> <mo> ≤ </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </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'> 2 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> p </ci> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> m </ci> </apply> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <ci> PrimePi </ci> <ci> n </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> o </ci> <ci> n </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <ci> n </ci> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <power /> <apply> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> <ci> k </ci> </apply> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> j </ci> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> <ci> k </ci> </apply> <primes /> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> j </ci> </apply> <integers /> </apply> <apply> <gt /> <apply> <ci> Subscript </ci> <ci> n </ci> <ci> j </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <lt /> <apply> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> j </ci> </apply> <ci> k </ci> </apply> <apply> <apply> <ci> Subscript </ci> <ci> p </ci> <apply> <plus /> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <ci> k </ci> </apply> </apply> <apply> <leq /> <cn type='integer'> 1 </cn> <ci> j </ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], "n_"], FractionBox[RowBox[List[SubscriptBox["p", "1"], "[", "k", "]"]], RowBox[List[SubscriptBox["p", "m"], "[", "k", "]"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["PrimePi", "[", "n", "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["o", "[", "n", "]"]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Equal]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "m"], SuperscriptBox[RowBox[List[SubscriptBox["p", "j"], "[", "k", "]"]], SubscriptBox["n", "j"]]]]]], "&&", RowBox[List[RowBox[List[SubscriptBox["p", "j"], "[", "k", "]"]], "\[Element]", "Primes"]], "&&", RowBox[List[SubscriptBox["n", "j"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["n", "j"], ">", "0"]], "&&", RowBox[List[RowBox[List[SubscriptBox["p", "j"], "[", "k", "]"]], "<", RowBox[List[SubscriptBox["p", RowBox[List["j", "+", "1"]]], "[", "k", "]"]]]], "&&", RowBox[List["1", "\[LessEqual]", "j", "\[LessEqual]", RowBox[List["m", "-", "1"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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