Where to get peer-reviewed before submitting a paper to journal?

As a student where should he first approach for review of his paper, even more so a first one, before submitting to peer reviewed journal? Teachers tend to be busy with school and personal work, soliciting or emailing a person out of the blue may not be the ideal etiquette, while students in one's class may not have the required background. What to do in this situation?

Do any websites exist (other than arXiv) that allows for a process that critique papers?

Thank you.

There seems to be a disconnect here: If your result is anywhere near something that can be published in a journal, it will be infinitely more interesting for your professors to read than grading homework. It makes no sense to compare how busy they are with one, to how much time they'd have for the other. That's like thinking someone is so busy with washing dishes that they'd never have time to come eat a slice of your home-baked cake. Sure, the cake may or may not be horrible, but you're not going to judge that by how many dishes they wash.

So if you are enrolled as a student in a relevant program, the first and obvious people you should consult are your teachers. They will be excited and delighted if it turns out you have discovered something publishable, and if not, well, then it's part of their job to let you know in a gentle way.

As a practical matter, your first approach probably shouldn't be about the paper itself, but about the result. Instead of

Here is a paper I've written; would you please criticize it?

say something like

I seem to have discovered that derivations in intuitionistic propositional calculus have the same structure as certain objects in Church's 1936 characterization of computable functions, and this makes it much easier to discover formal proofs. Is that something that is already known?

Possible responses then include:

• "That sounds interesting. If you write it up, I would like to look at it." -- the best you can hope for at this stage.
• "Hmm, huh? Could you explain in more detail?" -- be prepared to attack the nearest whiteboard with a deeper explanation of the result.
• "It's not clear to me if that would be true, but if it is, what could we do with it?" -- as above, but with motivation that your result is interesting.
• "I don't actually know much about that area, but try talking to my colleague/grad student NN." -- now you have at least a referral.
• "Yes, that's known as the Such-And-Such theorem. It's a very nice observation, congratulations on rediscovering it. Here, let me recommend some advanced texts where you can read more about this if it interests you." -- bit of a bummer, but now you at least have a professor who will think highly of you, from which all manner of good stuff will follow in the fullness of time.
• "Yes; actually that's the main theorem in next semester's course. But since you're getting ahead of schedule, here is a riddle for you: what about classical propositional calculus?" -- okay, they can't all be winners.
• "Huh? No, that can't be true. In fact, consider the following counterexample ..." -- oops.