Is there ever an odds to payout ratio that makes purchasing a lottery ticket a mathematically rational choice?

I realize that you have to factor in multiple winners splitting the prize, the odds that will happen etc. Let’s have the most basic case, where you buy a ticket for $1 with 1 number between 1 and 10 and the winner gets $100 trillion. In such cases, the mathematician would buy as many of each number as he could possibly afford. Of course everyone else would be doing the same. Even still, I suspect that such a lottery would still net a nice return even split several billion ways.

So how would a rational mathematician calculate when the odds are actually in her favor to win, given that there could be many other winners?