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PrimePi




Mathematica Notation

Traditional Notation









Number Theory Functions > PrimePi[x] > Representations through equivalent functions > With related functions





http://functions.wolfram.com/13.04.27.0001.01









  


  








Input Form





PrimePi[x] == Sum[1, {i, 1, n}] /; Inequality[Prime[n], LessEqual, x, Less, Prime[n + 1]]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PrimePi", "[", "x", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "n"], "1"]]]], "/;", RowBox[List[RowBox[List["Prime", "[", "n", "]"]], "\[LessEqual]", "x", "<", RowBox[List["Prime", "[", RowBox[List["n", "+", "1"]], "]"]]]]]]]]








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> &#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> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mn> 1 </mn> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> prime </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#8804; </mo> <mi> x </mi> <mo> &lt; </mo> <mrow> <mi> prime </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </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> i </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Inequality </ci> <apply> <ci> prime </ci> <ci> n </ci> </apply> <leq /> <ci> x </ci> <lt /> <apply> <ci> prime </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </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["i", "=", "1"]], "n"], "1"]], "/;", RowBox[List[RowBox[List["Prime", "[", "n", "]"]], "\[LessEqual]", "x", "<", RowBox[List["Prime", "[", RowBox[List["n", "+", "1"]], "]"]]]]]]]]]]


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





2001-10-29