A note on infinite divisibility of zeta distributions

The Riemann zeta distribution, defined as the one whose characteristic function is the normalised Riemann zeta function, is an interesting example of an infinitely divisible distribution. The infinite divisibility of the distribution has been proved with recourse to the Euler product of the Riemann zeta function. In this paper, we look at multiple zeta-star function, which is a multi-dimensional generalisation of the Riemann zeta function and is believed to have no Euler product, and show that the corresponding distribution is not infinitely divisible.