I'm confused about the existence of the prime counting function. When I search for information about pi(x), I turn up a lot of information on approximations and algorithms for finding pi(x) but there doesn't seem to be any clear cut formula, yet it seems to exist? If there exists formula for the prime counting function, is it that they just aren't very friendly to work with and therefore we resort back to using the approximations? If so, what makes them hard to work with exactly? Thank you.