  In a previous paper of the second author with K. Ono, surprising
multiplicative properties of the partition function were presented. Here, we
deal with $k$-regular partitions. Extending the generating function for
$k$-regular partitions multiplicatively to a function on $k$-regular
partitions, we show that it takes its maximum at an explicitly described small
set of partitions, and can thus easily be computed. The basis for this is an
extension of a classical result of Lehmer, from which an inequality for the
generating function for $k$-regular partitions is deduced which seems not to
have been noticed before.
