Not by law, of course. But I'd still do it. Your invitation will have a higher rate of success if it is a real invitation and you make her your guest. It's much more charming that way because it shows you are willing to take a risk or make an investment to have dinner with her. It will take some pressure off. After all, she could think "Wait a minute. I don't even know this man. He asks me to have dinner with him at a place of his choice and then I should pay half or at least my part. I don't think so."

If she finds the proposition too "close" because this would essentially be a spontaneous date and she refuses because of that, you can still offer to split the bill or go each on his/her own account.

The initial hit is a pretty strong one. I am not proposing to chat her up for a long time before you ask the dinner question. Maybe introduce yourself, tell her what you are doing in town, ask a question or two of her (you could ask if she knows a nice place to eat) and then offer to join you for dinner. No more than 2-3 minutes, after all your priority is food intake not dating. So because the initial hit is a daring proposition, you need to soften the blow and be as much of a gentleman as you can during the entire process.

It should show from your attitude that you are not in it to seduce her but to have a nice dinner with a charming companion (her!). She should feel safe. It'll be exciting enough anyway. If anything does come from it you got lucky. But do not aim for that. What gives you a higher chance is precisely that you make it obvious that you are not hitting her up to sleep with her.

Note that I am not giving you abstract dating advice. I've done this myself and have had very nice evenings due to it.

To come back to the initial payment question. I was once walking in the streets of Lugano, Switzerland with my mother. I was 18 at the time. There were three very pretty girls coming towards us. I must have gotten very big eyes because all three gave me really nice smiles. When I sat down to dinner with my mother in a restaurant, I told her that I was grateful for her company but I could just as well imagine to have dinner here with those three beauties. (I am such an ungrateful son!) She laughed and said: Yes, my son. But then you would also have to pick up the tab for all of them. That calmed my enthusiasm.

Till