I neglected to mention that I did try this. This switch is part of a bank of three switches in the same big box. All of the neutrals in the box are wired together. I tried connecting the switch's neutral to this bundle. When I turned to power on, the fan began to run at low speed, buzzing loudly, and the switch wouldn't turn the fan off. Should I separate the neutrals in the box so that the neutral of the switch is connected to only the neutral going to the fan?

By the way, what does that neutral do anyway?