I believe I've seen reports that discussed a spike in lottery ticket sales when it's positive EV; I think there are some people that do watch for that and behave accordingly. Putting your "actually risk neutral" investment money into it isn't a terrible idea, but whether or not you have that investment money depends on your situation.
[edit] That is, you should split your financial resources into "expected to be needed in the next 3 months," "expected to be needed in the next year," "expected to be needed in the next 5 years," "expected to be needed in the next 20 years," and "risk neutral." How much of your current income flow to put into each of those buckets depends on your projected income, projected consumption, and so on, and is a fairly complicated calculation if you want to do it carefully. Even so, eyeballing it is better than not running those numbers.
So the jackpot in the Ohio lottery is around 25 million, and the chance of winning it is one in roughly 14 million, with tickets at 1 dollar a piece. It appears to me that roughly a quarter million tickets are sold each drawing; so, supposing you win, the probability of someone else also winning is 1 - (1 - 1/14e6)^{250000}=2%, which does not significantly reduce the expectation value of a ticket. So, unless I'm making a silly mistake somewhere, buying lottery tickets has positive expected value. (I find this counterintuitive; where are all the economists who should be picking up this free money? But I digress.)
I pointed this out to my wife, and said that it might be worth putting a dollar into it; and she very cogently asked, "Then why not make it 100 dollars?" Why not, indeed! Is there any sensible way of deciding how much to put into an option that has a positive expected value, but very low chance of payoff?