DLMF:25.2.E8 (Q7603): Difference between revisions

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‎Changed an Item: Add constraint
‎Changed an Item: Add constraint
Property / Symbols used
 
Property / Symbols used: Q10811 / rank
 
Normal rank
Property / Symbols used: Q10811 / qualifier
 
Defining formula:

ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}

\realpart@@
Property / Symbols used: Q10811 / qualifier
 
xml-id: C1.S9.E2.m1aedec

Revision as of 13:21, 2 January 2020

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DLMF:25.2.E8
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    Statements

    Defining formula
    ζ ⁡ ( s ) = ∑ k = 1 N 1 k s + N 1 - s s - 1 - s ⁢ ∫ N ∞ x - ⌊ x ⌋ x s + 1 ⁢ d x , Riemann-zeta 𝑠 superscript subscript 𝑘 1 𝑁 1 superscript 𝑘 𝑠 superscript 𝑁 1 𝑠 𝑠 1 𝑠 superscript subscript 𝑁 𝑥 𝑥 superscript 𝑥 𝑠 1 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+% \frac{N^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\left\lfloor x\right\rfloor}{x^{% s+1}}\mathrm{d}x,}}
    0 references
    DLMF id
    DLMF:25.2.E8
    0 references
    constraint
    ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re s>0}}
    0 references
    ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re s>0}}
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    N = 1 , 2 , 3 , … 𝑁 1 2 3 … {\displaystyle{\displaystyle N=1,2,3,\dots}}
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    Symbols used
    Riemann zeta function
    Defining formula
    ζ ⁡ ( s ) Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}
    xml-id
    C25.S2.E1.m2afdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1adec
    0 references
    Q11169
    Defining formula
    ⌊ x ⌋ 𝑥 {\displaystyle{\displaystyle\left\lfloor\NVar{x}\right\rfloor}}
    xml-id
    introduction.Sx4.p1.t1.r17.m4adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3adec
    0 references
    Q10811
    Defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aedec
    0 references
     
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