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PrimePi






Mathematica Notation

Traditional Notation









Number Theory Functions > PrimePi[x] > Series representations > Other series representations





http://functions.wolfram.com/13.04.06.0012.01









  


  










Input Form





PrimePi[x] == R[x] + Sum[R[Subscript[\[Rho], k]], {Subscript[\[Rho], k], -Infinity, Infinity}] /; R[x] == Sum[(MoebiusMu[k] LogIntegral[x^(1/k)])/k, {k, 1, Infinity}] == 1 + Sum[Log[x]^k/(k Zeta[k + 1] k!), {k, 1, Infinity}] && Zeta[Subscript[\[Rho], k]] == 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PrimePi", "[", "x", "]"]], "\[Equal]", RowBox[List[RowBox[List["R", "[", "x", "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["\[Rho]", "k"], "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List["R", "[", SubscriptBox["\[Rho]", "k"], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["R", "[", "x", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["MoebiusMu", "[", "k", "]"]], " ", RowBox[List["LogIntegral", "[", SuperscriptBox["x", RowBox[List["1", "/", "k"]]], "]"]]]], "k"]]], "\[Equal]", RowBox[List["1", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[SuperscriptBox[RowBox[List["Log", "[", "x", "]"]], "k"], RowBox[List["k", " ", RowBox[List["Zeta", "[", RowBox[List["k", "+", "1"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]]]], "\[And]", RowBox[List[RowBox[List["Zeta", "[", SubscriptBox["\[Rho]", "k"], "]"]], "\[Equal]", "0"]]]]]]]]










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> <mrow> <mi> R </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> &#961; </mi> <mi> k </mi> </msub> <mo> = </mo> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mi> R </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> &#961; </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mi> R </mi> <mo> &#8289; </mo> <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> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <mrow> <semantics> <mi> &#956; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Mu]&quot;, MoebiusMu] </annotation> </semantics> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> li </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> x </mi> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> k </mi> </mrow> </msup> <mo> ) </mo> </mrow> </mrow> <mi> k </mi> </mfrac> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <msup> <mi> log </mi> <mi> k </mi> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;1&quot;]], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> &#961; </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[SubscriptBox[&quot;\[Rho]&quot;, &quot;k&quot;], Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> PrimePi </ci> <ci> x </ci> </apply> <apply> <plus /> <apply> <ci> R </ci> <ci> x </ci> </apply> <apply> <sum /> <bvar> <apply> <ci> ZetaZero </ci> <ci> k </ci> </apply> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <ci> R </ci> <apply> <ci> ZetaZero </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> R </ci> <ci> x </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> MoebiusMu </ci> <ci> k </ci> </apply> <apply> <ci> LogIntegral </ci> <apply> <power /> <ci> x </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <apply> <ln /> <ci> x </ci> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <ci> k </ci> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <factorial /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <eq /> <apply> <ci> Zeta </ci> <apply> <ci> ZetaZero </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> 0 </cn> </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[RowBox[List["R", "[", "x", "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["\[Rho]", "k"], "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List["R", "[", SubscriptBox["\[Rho]", "k"], "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["R", "[", "x", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["MoebiusMu", "[", "k", "]"]], " ", RowBox[List["LogIntegral", "[", SuperscriptBox["x", RowBox[List["1", "/", "k"]]], "]"]]]], "k"]]], "\[Equal]", RowBox[List["1", "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[SuperscriptBox[RowBox[List["Log", "[", "x", "]"]], "k"], RowBox[List["k", " ", RowBox[List["Zeta", "[", RowBox[List["k", "+", "1"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]]]], "&&", RowBox[List[RowBox[List["Zeta", "[", SubscriptBox["\[Rho]", "k"], "]"]], "\[Equal]", "0"]]]]]]]]]]










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





2001-10-29





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