# Riemann zeta function

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##### 1: 25.1 Special Notation
 $k,m,n$ nonnegative integers. …
The main function treated in this chapter is the Riemann zeta function $\zeta\left(s\right)$. … The main related functions are the Hurwitz zeta function $\zeta\left(s,a\right)$, the dilogarithm $\mathrm{Li}_{2}\left(z\right)$, the polylogarithm $\mathrm{Li}_{s}\left(z\right)$ (also known as Jonquière’s function $\phi\left(z,s\right)$), Lerch’s transcendent $\Phi\left(z,s,a\right)$, and the Dirichlet $L$-functions $L\left(s,\chi\right)$.
##### 2: 8.22 Mathematical Applications
###### §8.22(ii) RiemannZetaFunction and Incomplete RiemannZetaFunction
If $\zeta_{x}(s)$ denotes the incomplete Riemann zeta function defined by …so that $\lim_{x\to\infty}\zeta_{x}(s)=\zeta\left(s\right)$, then
8.22.3 $\zeta_{x}(s)=\sum_{k=1}^{\infty}k^{-s}P\left(s,kx\right),$ $\Re s>1$.
For further information on $\zeta_{x}(s)$, including zeros and uniform asymptotic approximations, see Kölbig (1970, 1972a) and Dunster (2006). …
##### 3: 25.7 Integrals
###### §25.7 Integrals
For definite integrals of the Riemann zeta function see Prudnikov et al. (1986b, §2.4), Prudnikov et al. (1992a, §3.2), and Prudnikov et al. (1992b, §3.2).
##### 4: 25.17 Physical Applications
###### §25.17 Physical Applications
See Armitage (1989), Berry and Keating (1998, 1999), Keating (1993, 1999), and Sarnak (1999). …
##### 5: 25.20 Approximations
• Cody et al. (1971) gives rational approximations for $\zeta\left(s\right)$ in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are $0.5\leq s\leq 5$, $5\leq s\leq 11$, $11\leq s\leq 25$, $25\leq s\leq 55$. Precision is varied, with a maximum of 20S.

• Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of $s\zeta\left(s+1\right)$ and $\zeta\left(s+k\right)$, $k=2,3,4,5,8$, for $0\leq s\leq 1$ (23D).

• Antia (1993) gives minimax rational approximations for $\Gamma\left(s+1\right)F_{s}(x)$, where $F_{s}(x)$ is the Fermi–Dirac integral (25.12.14), for the intervals $-\infty and $2\leq x<\infty$, with $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$. For each $s$ there are three sets of approximations, with relative maximum errors $10^{-4},10^{-8},10^{-12}$.

• ##### 6: 25.3 Graphics
###### §25.3 Graphics Figure 25.3.1: Riemann zeta function ζ ⁡ ( x ) and its derivative ζ ′ ⁡ ( x ) , - 20 ≤ x ≤ 10 . Magnify Figure 25.3.2: Riemann zeta function ζ ⁡ ( x ) and its derivative ζ ′ ⁡ ( x ) , - 12 ≤ x ≤ - 2 . Magnify Figure 25.3.3: Modulus of the Riemann zeta function | ζ ⁡ ( x + i ⁢ y ) | , - 4 ≤ x ≤ 4 , - 10 ≤ y ≤ 40 . Magnify 3D Help Figure 25.3.6: Z ⁡ ( t ) , 10000 ≤ t ≤ 10050 . Magnify
##### 7: 25.18 Methods of Computation
###### §25.18(i) Function Values and Derivatives
Calculations relating to derivatives of $\zeta\left(s\right)$ and/or $\zeta\left(s,a\right)$ can be found in Apostol (1985a), Choudhury (1995), Miller and Adamchik (1998), and Yeremin et al. (1988). …
###### §25.18(ii) Zeros
Most numerical calculations of the Riemann zeta function are concerned with locating zeros of $\zeta\left(\frac{1}{2}+it\right)$ in an effort to prove or disprove the Riemann hypothesis, which states that all nontrivial zeros of $\zeta\left(s\right)$ lie on the critical line $\Re s=\frac{1}{2}$. …
##### 8: 27.4 Euler Products and Dirichlet Series
The completely multiplicative function $f(n)=n^{-s}$ gives the Euler product representation of the Riemann zeta function $\zeta\left(s\right)$25.2(i)): … The Riemann zeta function is the prototype of series of the form …
27.4.5 $\sum_{n=1}^{\infty}\mu\left(n\right)n^{-s}=\frac{1}{\zeta\left(s\right)},$ $\Re s>1$,
27.4.7 $\sum_{n=1}^{\infty}\lambda\left(n\right)n^{-s}=\frac{\zeta\left(2s\right)}{% \zeta\left(s\right)},$ $\Re s>1$,
27.4.12 $\sum_{n=1}^{\infty}\Lambda\left(n\right)n^{-s}=-\frac{\zeta'\left(s\right)}{% \zeta\left(s\right)},$ $\Re s>1$,
##### 9: 25.4 Reflection Formulas
###### §25.4 Reflection Formulas
25.4.4 $\xi\left(s\right)=\tfrac{1}{2}s(s-1)\Gamma\left(\tfrac{1}{2}s\right)\pi^{-s/2}% \zeta\left(s\right).$
25.4.5 $(-1)^{k}{\zeta}^{(k)}\left(1-s\right)=\frac{2}{(2\pi)^{s}}\sum_{m=0}^{k}\sum_{% r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}\left(\Re% \left(c^{k-m}\right)\cos\left(\tfrac{1}{2}\pi s\right)+\Im\left(c^{k-m}\right)% \sin\left(\tfrac{1}{2}\pi s\right)\right){\Gamma}^{(r)}\left(s\right){\zeta}^{% (m-r)}\left(s\right),$
##### 10: 25.8 Sums
###### §25.8 Sums
25.8.1 $\sum_{k=2}^{\infty}\left(\zeta\left(k\right)-1\right)=1.$
25.8.2 $\sum_{k=0}^{\infty}\frac{\Gamma\left(s+k\right)}{(k+1)!}\left(\zeta\left(s+k% \right)-1\right)=\Gamma\left(s-1\right),$ $s\neq 1,0,-1,-2,\dots$.
25.8.4 $\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}(\zeta\left(nk\right)-1)=\ln\left(\prod_{% j=0}^{n-1}\Gamma\left(2-e^{(2j+1)\pi i/n}\right)\right),$ $n=2,3,4,\dots$.
25.8.9 $\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{(2k+1)2^{2k}}=\frac{1}{2}-\frac% {1}{2}\ln 2.$