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In mathematics, the Arakawa–Kaneko zeta function is a generalisation of the Riemann zeta function which generates special values of the polylogarithm function.

Definition

The zeta function \xi_k(s) is defined by

$$\xi_k(s) = \frac{1}{\Gamma(s)} \int_0^{+\infty} \frac{t^{s-1}}{e^t-1}\mathrm{Li}_k(1-e^{-t}) \, dt \$$

where Lik is the k-th polylogarithm

$$\mathrm{Li}_k(z) = \sum_{n=1}^{\infty} \frac{z^n}{n^k} \ .$$

Properties

The integral converges for $$\Re(s) > 0$$and $$\xi_k(s)$$ has analytic continuation to the whole complex plane as an entire function.

The special case k = 1 gives $$\xi_1(s) = s \zeta(s+1)$$ where $$\zeta$$ is the Riemann zeta-function.

The special case s = 1 remarkably also gives $$\xi_k(1) = \zeta(k+1)$$ where $$\zeta$$ is the Riemann zeta-function.

The values at integers are related to multiple zeta function values by

$$\xi_k(m) = \zeta_m^*(k,1,\ldots,1)$$

where

$$\zeta_n^*(k_1,\dots,k_{n-1},k_n)=\sum_{0<m_1<m_2<\cdots<m_n}\frac{1}{m_1^{k_1}\cdots m_{n-1}^{k_{n-1}}m_n^{k_n}} \ .$$

References

Kaneko, Masanobou (1997). "Poly-Bernoulli numbers". J. Théor. Nombres Bordx. 9: 221–228. Zbl 0887.11011.
Arakawa, Tsuneo; Kaneko, Masanobu (1999). "Multiple zeta values, poly-Bernoulli numbers, and related zeta functions". Nagoya Math. J. 153: 189–209. MR 1684557. Zbl 0932.11055.
Coppo, Marc-Antoine; Candelpergher, Bernard (2010). "The Arakawa–Kaneko zeta function". Ramanujan J. 22: 153–162. Zbl 1230.11106.

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