<p>Grades and coursework are much more important in math grad school admissions than in lab science, since math homework is a lot closer to math research than, say chemistry coursework is to chemistry research.
If you’re an undegrad in a school with a math department that sends lot of people to grad school, take lots of graduate classes, get top grades in them, and get great letters of recommendation from well known professors, it is possible to get into “elite” programs without any research experience. Since recommendation letters are the most important factor in grad school admissions, and succesfully working on a research project with a prof is a great way to convince them that you are really smart, research experience is useful. Whether or not you have publications in itself doesn’t really matter much, since it depends on so many factors other than the quality of your research.
Also, to the OP, why are you so set on Harvard specifically?</p>