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I'm writing a paper with a professor. I want to put a link to my website in the paper. I am first author, so I'm wondering if the following is okay:

Bob Smith, Slifer University, https://BobSmith.io
Peter Bob, Georgia Tech of Mechanical Engineering

Suppose I'm Bob Smith and the professor is Peter Bob.

This way people can see my other work and potentially hire me. I'd also mention, this is a quite well known professor, so don't say "No one will look at your website".

Some papers I've written, I've included, in the header with my name and email address, a link to my website, though not to the paper. Just a link to the front page. On joint papers we have all done this.

But only consider doing something like this if everyone approves, including other authors and the journal/conference authority.

But, if you don't own the site (say the university has provided you a web space), I would recommend against it as your affiliation will change eventually and you will lose control of your university owned web site. If your link is to a personal site (as you indicate) that you own and intend to maintain over the long run, say 50 years, then it might be fine.

However, Google and other search engines can find you and your work easily enough. The link isn't really necessary. If the paper has a reference on the site then a search with your name and some part of the title will probably turn it up easily enough. So, I think the advantage would be quite small.

Since the OP has commented that they've never seen an example of a published paper including the URL of an author's website, here is an example of such a paper:

Local Selectivity of Orders in Central Simple Algebras, Benjamin Linowitz and Thomas R. Shemanske, Int. J. Number Theory 13 (2017), no. 4, 853-884.

Our website URLs appear with our institutional information on the last page of the paper.

By the way, here's another lesson to draw from this example: try to use email addresses and website URLs that are permanent. By the time this paper was published I had already moved to a different institution and the University of Michigan had deleted the email and web accounts I listed in the paper.