Riemann zeta function: Zeros (formula 10.01.30.0009)

Zeta

$Mathematica Notation$

Zeta Functions and Polylogarithms

Zeta[s]

Zeros

Sums over zeros

http://functions.wolfram.com/10.01.30.0009.01

Input Form

Subscript[Î–,1]ï±(-(1/2))*n*(Log[4*Pi] + EulerGamma) - Sum[Î·[j - 1]*Binomial[n, j], {j, 1, n}] - Sum[(-1)^(j - 1)*Binomial[n, j]*(1 - 2^(-j))*Zeta[j], {j, 2, n}] + 1/; Subscript[Î–,1]ï±Limit[Sum[1/Subscript[Ï,k],Abs[Subscript[Ï,k]<T]], Tï”¢Infinity]&&Zeta[Subscript[Ï,k]]ï±0&& Re[Subscript[Ï,k]]â‰ 0&&nâˆˆIntegers&&n>0&& Log[s*Zeta[s + 1]] == -Sum[(Subscript[Î·, k]*s^(k + 1))/(k + 1),{k, 0, Infinity}]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[CapitalRho]", "n"], "\[Equal]", RowBox[List[RowBox[List["(", " ", RowBox[List["-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[SubscriptBox["\[Eta]", RowBox[List["j", "-", "1"]]], RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]]]]]]]], ")"]], "+", "\[IndentingNewLine]", "1", "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["4", "Pi"]], "]"]], "+", "EulerGamma"]], ")"]], FractionBox["n", "2"]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "2"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "-", "1"]]], RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["2", RowBox[List["-", "j"]]]]], ")"]], RowBox[List["Zeta", "[", "j", "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["\[CapitalRho]", "n"], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List[RowBox[List[UnderscriptBox["\[Sum]", RowBox[List[RowBox[List["\[LeftBracketingBar]", SubscriptBox["\[Rho]", "k"], "\[RightBracketingBar]"]], "<", "T"]]], "1"]], "-", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", FractionBox["1", SubscriptBox["\[Rho]", "k"]]]], ")"]], "n"]]], ",", RowBox[List["T", "\[Rule]", "\[Infinity]"]]]], "]"]]]], "\[And]", RowBox[List[RowBox[List["Zeta", "[", SubscriptBox["\[Rho]", "k"], "]"]], "\[Equal]", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", SubscriptBox["\[Rho]", "k"], "]"]], "\[NotEqual]", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List[RowBox[List["Log", "[", RowBox[List["s", " ", RowBox[List["Zeta", "[", RowBox[List["1", "+", "s"]], "]"]]]], "]"]], "\[Equal]", RowBox[List["-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SubscriptBox["\[Eta]", "k"], " ", SuperscriptBox["s", RowBox[List["k", "+", "1"]]]]], RowBox[List["k", "+", "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> <msub> <mi> Ρ </mi> <mi> n </mi> </msub> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[EulerGamma]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msub> <mi> η </mi> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mi> j </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <mo> - </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mi> j </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> j </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> j </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["j", Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> Ρ </mi> <mi> n </mi> </msub> <mo> ⩵ </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> T </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <mo> ( </mo> <mrow> <mrow> <munder> <mo> ∑ </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> ρ </mi> <mi> k </mi> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> = </mo> <mn> 1 </mn> </mrow> </munder> <mn> 1 </mn> </mrow> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msub> <mi> ρ </mi> <mi> k </mi> </msub> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∧ </mo> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> ρ </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[SubscriptBox["\[Rho]", "k"], Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mo> ⩵ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> ρ </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> <mo> ≠ </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["s", "+", "1"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mo> - </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <msub> <mi> η </mi> <mi> k </mi> </msub> <mo> ⁢ </mo> <msup> <mi> s </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> Ρ </ms> <ms> n </ms> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> n </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <ms> π </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> + </ms> <apply> <ci> TagBox </ci> <ms> ℽ </ms> <apply> <ci> Function </ci> <eulergamma /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> ∑ </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> n </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> η </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> TagBox </ci> <ms> n </ms> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> <list> <apply> <ci> TagBox </ci> <ms> j </ms> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> Binomial </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> ∑ </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 2 </ms> </list> </apply> <ms> n </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> TagBox </ci> <ms> n </ms> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> <list> <apply> <ci> TagBox </ci> <ms> j </ms> <ident /> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <apply> <ci> Binomial </ci> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Slot </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> SuperscriptBox </ci> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> j </ms> </list> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ζ </ms> <ms> ( </ms> <apply> <ci> TagBox </ci> <ms> j </ms> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <ci> BoxForm`e$ </ci> <apply> <ci> Zeta </ci> <ci> BoxForm`e$ </ci> </apply> </apply> </apply> </apply> </list> </apply> </list> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> Ρ </ms> <ms> n </ms> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <ms> Limit </ms> <ms> [ </ms> <apply> <ci> RowBox </ci> <list> <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> RowBox </ci> <list> <ms>  </ms> <apply> <ci> SubscriptBox </ci> <ms> ρ </ms> <ms> k </ms> </apply> <ms>  </ms> </list> </apply> <ms> = </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> 1 </ms> </list> </apply> <ms> - </ms> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> SubscriptBox </ci> <ms> ρ </ms> <ms> k </ms> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> n </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> T </ms> <ms> -> </ms> <ms> ∞ </ms> </list> </apply> <ms> ] </ms> </list> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ζ </ms> <ms> ( </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> ρ </ms> <ms> k </ms> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <ci> BoxForm`e$ </ci> <apply> <ci> Zeta </ci> <ci> BoxForm`e$ </ci> </apply> </apply> </apply> </apply> <ms> ⩵ </ms> <ms> 0 </ms> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Re </ms> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> ρ </ms> <ms> k </ms> </apply> <ms> ) </ms> </list> </apply> <ms> ≠ </ms> <ms> 0 </ms> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> ∈ </ms> <apply> <ci> SuperscriptBox </ci> <ms> ℕ </ms> <ms> + </ms> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> s </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ζ </ms> <ms> ( </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> s </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <ci> BoxForm`e$ </ci> <apply> <ci> Zeta </ci> <ci> BoxForm`e$ </ci> </apply> </apply> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> ∑ </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> <ms> ∞ </ms> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> η </ms> <ms> k </ms> </apply> <apply> <ci> SuperscriptBox </ci> <ms> s </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>

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

2003-08-21

Zeta[s,a]