In mathematics, the Arakawa–Kaneko zeta function is a generalisation of the Riemann zeta function which generates special values of the polylogarithm function.
The zeta function
\xik(s)
\xik(s)=
| 1 | |
| \Gamma(s) |
| +infty | |
| \int | |
| 0 |
| ts-1 | |
| et-1 |
| -t | |
| Li | |
| k(1-e |
)dt
where Lik is the k-th polylogarithm
Lik(z)=
| infty | |
| \sum | |
| n=1 |
| zn | |
| nk |
.
The integral converges for
\Re(s)>0
\xik(s)
The special case k = 1 gives
\xi1(s)=s\zeta(s+1)
\zeta
The special case s = 1 remarkably also gives
\xik(1)=\zeta(k+1)
\zeta
The values at integers are related to multiple zeta function values by
\xik(m)=
| *(k,1,\ldots,1) | |
| \zeta | |
| m |
where
| *(k | |
| \zeta | |
| 1,...,k |
n-1,kn)=\sum
| 0<m1<m2< … <mn |
| 1 | |||||||||||||||||||||
|
.