NP-hardness of a special case of Number Partitioning
(This is a little more detailed version of my comments on the question.)

As turkistany suggested in a comment on the question, this problem is NP-hard for every constant k≥3 by a reduction from the k-partition problem.  The reduction does not change instances at all: just note that the maximum of the product of the sums is at least (∑ai/k)m if and only if the numbers can be partitioned into m sets each with size k whose sums are all equal.

Note that the input to the k-partition problem is usually defined to be km numbers which may not be all distinct, and this is essential in the standard proof of its NP-completeness (such as the one in Garey and Johnson).  Therefore, the reduction above only proves the NP-hardness of a slight generalization of the current problem where the input is allowed to be a multiset instead of a set.  However, this gap can be filled because the k-partition problem remains NP-complete even if the numbers in the input are required to be all distinct; see [HWW08] for the case of k=3 (see also Serge Gaspers’s answer to another question), which can be modified easily for larger values of k.

In addition, everything stated here remains NP-complete/NP-hard even when the numbers in the input are given in unary.

[HWW08] Heather Hulett, Todd G. Will, Gerhard J. Woeginger.  Multigraph realizations of degree sequences: Maximization is easy, minimization is hard.  Operations Research Letters, 36(5):594–596, Sept. 2008.  http://dx.doi.org/10.1016/j.orl.2008.05.004