DLMF:25.10.E1 (Q7671)

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DLMF:25.10.E1
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    Statements

    Defining formula
    Z ⁢ ( t ) ≡ exp ⁡ ( i ⁢ ϑ ⁢ ( t ) ) ⁢ ζ ⁡ ( 1 2 + i ⁢ t ) , equal-by definition 𝑍 𝑡 𝑖 italic-ϑ 𝑡 Riemann-zeta 1 2 𝑖 𝑡 {\displaystyle{\displaystyle Z(t)\equiv\exp\left(i\vartheta(t)\right)\zeta% \left(\tfrac{1}{2}+it\right),}}
    0 references
    DLMF id
    DLMF:25.10.E1
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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.m2adec
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    Q10769
    Defining formula
    ≡ equal-by definition {\displaystyle{\displaystyle\equiv}}
    xml-id
    introduction.Sx4.p1.t1.r9.m2adec
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    exponential function
    Defining formula
    exp ⁡ z 𝑧 {\displaystyle{\displaystyle\exp\NVar{z}}}
    xml-id
    C4.S2.E19.m2adec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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    Q12158
    Defining formula
    Z ⁢ ( t ) 𝑍 𝑡 {\displaystyle{\displaystyle Z(t)}}
    xml-id
    C25.S10.XMD1.m1dec
    0 references
    Q12159
    Defining formula
    ϑ ⁢ ( t ) italic-ϑ 𝑡 {\displaystyle{\displaystyle\vartheta(t)}}
    xml-id
    C25.S10.XMD2.m1dec
    0 references
     
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