relation to Riemann zeta function

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1: 25.17 Physical Applications

§25.17 Physical Applications

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2: 8.22 Mathematical Applications

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§8.22(ii) Riemann Zeta Function and Incomplete Riemann Zeta Function

… ►The Debye functions ${\int }_0^x{t}^n{\left({\mathrm{e}}^t-1\right)}^{-1}dt$ and ${\int }_x^{\mathrm{\infty }}{t}^n{\left({\mathrm{e}}^t-1\right)}^{-1}dt$ are closely related to the incomplete Riemann zeta function and the Riemann zeta function. …

3: 20.9 Relations to Other Functions

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§20.9(iii) Riemann Zeta Function

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4: 24.17 Mathematical Applications

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§24.17(iii) Number Theory

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5: 24.4 Basic Properties

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§24.4(ix) Relations to Other Functions

►For the relation of Bernoulli numbers to the Riemann zeta function see §25.6, and to the Eulerian numbers see (26.14.11).

6: 25.12 Polylogarithms

… ►The special case $z=1$ is the Riemann zeta function: $\zeta \left(s\right)={\mathrm{Li}}_s\left(1\right)$. … ►Further properties include …

7: 25.6 Integer Arguments

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§25.6(i) Function Values

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8: 25.11 Hurwitz Zeta Function

… ►The Riemann zeta function is a special case: …

9: 25.10 Zeros

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§25.10(i) Distribution

… ►The functional equation (25.4.1) implies $\zeta \left(-2n\right)=0$ for $n=1,2,3,\mathrm{\dots }$. … ►Calculations relating to the zeros on the critical line make use of the real-valued function … ►

§25.10(ii) Riemann–Siegel Formula

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10: 25.16 Mathematical Applications

… ►which is related to the Riemann zeta function by …