Thanks to all those who replied. I've asked around a bit (outside of these fora) and the consensus is that writing 3-4 papers before finishing a doctorate is more typical for a field like combinatorics, but not really in my field or most other fields of math.
Having one or two relatively significant papers is the advice I'm getting.
Question for Daniel: What is a VIGRE in relation to a person? I thought VIGRE is a NSF grant to math departments, and, therefore, would all American grad students in the department be VIGREs?
4 publications in pure mathematics before finishing the PhD is quite a lot, though more common now then it was decades ago. I know several recent VIGREs who did not have nearly this many when they applied (but the quality of their work in unpublished form was known and very good). - DvF