Which sentence is grammatically correct?
Using Google's search results:
So, I inferred that both variants are popular. Is this method (comparing the number of results by Googling) sound?
Speaking as a longtime computer geek, files are said to reside on a hard drive or on a certain machine. Since I know that a website is made of computer files, I think of a website as being on a machine and so any part of the website is also on that machine.
A website's "address" is a "Uniform Resource Locator" or URL. These terms connote the idea of website as "place". In this case the file would be at the website location.
So, if you think of a website as it is actually constructed (files stored on hard drives), you will probably say on.
If you think in the location metaphor, you will probably say at.
Personally I find myself saying "on" to other geeks and falling into "at" when speaking with normal people.
For some purposes, the method is sound.
What's in use is generally what's correct, or what will become correct if it's used long enough. Even if it's not strictly correct, if it's very commonly used it's still usable, people will understand it, and few will react to any inaccuracies. If it's very popular, it's simply good enough for most purposes.
If you are writing something where you really need to be correct, looking at what's popular is not a sure method to find that out. Something that is incorrect can still become more popular than the correct form.
In the case of "website", it's not so clear what's correct, as a website is a rather abstract phenomenon. If you think of it as a site on the web, "at" would be correct, but if you think of it as a page on the web, "on" would be correct.
Something is sound if, and only if, it the premises are true and the conclusion necessarily follows from the premises.
If you're using the number of results returned to determine proper usage, there might be a high correlation between number of results and usage, but it's not a sound method. For a counterexample, consider the British and American spellings of words: if color had more results than colour, inferring that color is the proper word and colour is wrong would be incorrect.
If you're using the number of results to determine popularity, it's not a sound method, either. Firstly, the number is without context: to provide meaning, you'd need something else to compare it to. When you start doing that, you run into the problem mentioned above. If you're inferring that because it has over n results, it must be popular, realize that Google only provides a basic estimate that can change due to a number of different factors, including doing something as simple as browsing page 2 of the results.
Additionally, certain keywords are given undue weight due to their SEO value; a historical example is the keyword "mesolthelioma": a relatively obscure term for malignant tumor that caused by exposure to asbestos. It's particularly popular in search results because of the high settlement rate for asbestos exposure compensation cases.
Keywords like that highlight the reality of Google results: they're not the sum total of human knowledge and are subject to gaming. The only sound conclusion one can reach from the number of Google results is that Google has indexed a lot of pages that contain or relate to that keyword, not that the term is particularly popular or that it's correct.
I think a key point missed by the answers so far is that in the examples you listed:
The subject is "The papers" is the subject, and "website" is the direct object (and now I'm second guessing whether direct object is the correct term).
So it seems to me that the papers exist on a website. If an address had been given instead then at would be appropriate.
