Currently, when my window is bigger than the buffer being displayed, blank lines are shown with a ~ in the line number column. By blank lines I obviously don't mean blank lines in the buffer itself, but lines that extend beyond the end of the buffer.
I'd prefer for the line number to be blank. Is this possible?
Changing the character (~) that Vim uses to show the lines beyond the end of file, is not possible without patching Vim source code. So, it's possible only to hide those tildes: Vim uses NonText highlight group for it. Unfortunately, this group used also for list chars (see list and listchars options). If you don't use list chars much you can make foreground color to be equal to background one for this group:
highlight NonText ctermfg=bg guifg=bg