Is this analytic function linked to Barnes multiple gamma function ? Is it entire? I came across this series of following analytic functions 
$$
f_n\ :\ z\mapsto\frac{1}{\Big(\Pi_{k=0}^{n-1}\Gamma(1+
e^{2i\frac{k}{n}\pi}z)\Big)^{\frac{1}{n}}}
$$
One can get easily a local (around zero) expansion (as an exponential, hence my question) with convergence radius $R=1$ and I wonder whether it can be continued as an entire function. 
Is it linked to some multiple gamma function defined by Barnes ? 
 A: This is the problem of n-th root of an entire function within the space of entire functions. For $n\geq 2$, there is no solution as soon as the original function has (at least) a simple zero. Considering 
$$
g_n\ :\ z\mapsto\frac{1}{\Big(\Pi_{k=0}^{n-1}\Gamma(1+
e^{2i\frac{k}{n}\pi}z)\Big)}
$$
we have $g_n=f_n^n$ for $|z|<1$. The function $g_n$ is entire and has zeroes at the points such that 
$$
1+
e^{2i\frac{k}{n}\pi}z\in \mathbb{Z}_{\leq 0}
$$
for some $0\leq k<n$, this is the following disjoint union (each term of the union being the set of 
- simple - zeroes of the corresponding factor with the same $k$) 
$$
Z_n=\cup_{0\leq k\leq n-1} 
\{ne^{2i\pi\frac{(n-k)}{n}}\}_{n\leq -1}
$$
hence the zeroes are simple whence the impossibility that $f_n$ be entire.
A: Using a product representation of the gamma function your term is holomorphic in $\mathbb{C}$ without $-z\in\mathbb{N}$.
For $n\ge 2$ and on the right side $\,|z|<1\,$ (and $z=1)$ your term is equivalent to: 
$$\left(\prod\limits_{k=1}^\infty\left(1+(-1)^{n-1}\left(\frac{z}{k}\right)^n\right)\right)^{\frac{1}{n}}=\exp\left(-\sum\limits_{k=1}^\infty\frac{(-z)^{nk}\zeta(nk)}{nk}\right)$$  
