# series representation

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##### 1: 1.17 Integral and Series Representations of the Dirac Delta
###### §1.17(iii) SeriesRepresentations
By analogy with §1.17(ii) we have the formal series representation
##### 2: 25.16 Mathematical Applications
25.16.1 $\psi\left(x\right)=\sum_{m=1}^{\infty}\sum_{p^{m}\leq x}\ln p,$
25.16.6 $H\left(s\right)=-\zeta'\left(s\right)+\gamma\zeta\left(s\right)+\frac{1}{2}% \zeta\left(s+1\right)+\sum_{r=1}^{k}\zeta\left(1-2r\right)\zeta\left(s+2r% \right)+\sum_{n=1}^{\infty}\frac{1}{n^{s}}\int_{n}^{\infty}\frac{\widetilde{B}% _{2k+1}\left(x\right)}{x^{2k+2}}\mathrm{d}x,$
25.16.7 $H\left(s\right)=\frac{1}{2}\zeta\left(s+1\right)+\frac{\zeta\left(s\right)}{s-% 1}-\sum_{r=1}^{k}\genfrac{(}{)}{0.0pt}{}{s+2r-2}{2r-1}\zeta\left(1-2r\right)% \zeta\left(s+2r\right)-\genfrac{(}{)}{0.0pt}{}{s+2k}{2k+1}\sum_{n=1}^{\infty}% \frac{1}{n}\int_{n}^{\infty}\frac{\widetilde{B}_{2k+1}\left(x\right)}{x^{s+2k+% 1}}\mathrm{d}x.$
##### 4: 25.11 Hurwitz Zeta Function
###### §25.11(iv) SeriesRepresentations
25.11.8 $\zeta\left(s,\tfrac{1}{2}a\right)=\zeta\left(s,\tfrac{1}{2}a+\tfrac{1}{2}% \right)+2^{s}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}},$ $\Re s>0$, $s\neq 1$, $0.
25.11.9 $\zeta\left(1-s,a\right)=\frac{2\Gamma\left(s\right)}{(2\pi)^{s}}\*\sum_{n=1}^{% \infty}\frac{1}{n^{s}}\cos\left(\tfrac{1}{2}\pi s-2n\pi a\right),$ $\Re s>1$, $0.
###### §25.11(x) Further SeriesRepresentations
25.11.36 $L\left(s,\chi\right)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}=k^{-s}\sum_{r=1}% ^{k-1}\chi(r)\zeta\left(s,\frac{r}{k}\right),$ $\Re s>1$,
##### 5: 25.12 Polylogarithms
25.12.10 $\mathrm{Li}_{s}\left(z\right)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{s}}.$
##### 6: 12.18 Methods of Computation
These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …
##### 7: 25.15 Dirichlet $L$-functions
25.15.1 $L\left(s,\chi\right)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}},$ $\Re s>1$,
##### 8: 25.14 Lerch’s Transcendent
25.14.1 ${\Phi\left(z,s,a\right)\equiv\sum_{n=0}^{\infty}\frac{z^{n}}{(a+n)^{s}}},$ $|z|<1$; $\Re s>1,|z|=1$.
##### 9: Bibliography E
• J. A. Ewell (1990) A new series representation for $\zeta(3)$ . Amer. Math. Monthly 97 (3), pp. 219–220.
• ##### 10: 13.12 Products
For integral representations, integrals, and series containing products of $M\left(a,b,z\right)$ and $U\left(a,b,z\right)$ see Erdélyi et al. (1953a, §6.15.3).