  We study multicorrelation sequences arising from systems with commuting
transformations. Our main result is a refinement of a decomposition result of
Frantzikinakis and it states that any multicorrelation sequences for commuting
transformations can be decomposed, for every $\epsilon>0$, as the sum of a
nilsequence $\phi(n)$ and a sequence $\omega(n)$ satisfying
$\lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N |\omega(n)|<\epsilon$ and
$\lim_{N\to\infty}\frac{1}{|\mathbb{P}\cap [N]|}\sum_{p\in \mathbb{P}\cap [N]}
|\omega(p)|<\epsilon$.

Dynamical Systems