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Is linkedin useful for academic mathematicians?

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No.

Topic specific mailing lists is the place to be.
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Not useful for that imho. It's for "business people" to network. There are special internet groups including math, but they are more for business humans who like/use math. So you will learn about the ramanujan taxi number story or view a humorous gif about the relationship between e pi and i. Or a link to an article in quanta.

If, like me, you end up in business you'll want to have an account with your resume. One of the most useful things is to "friend" a new colleague so that the colleague can be sure to know which fancy degree you got from which fancy  school, and that you were head of Blah and Bleh Co. You will also see if a former coworker is now in a cool new position, in which case you may want to "reach out".

P. S. The other notable thing is that  Often the career change notes  are of the form "congratulate Joe Schmidlap on being named Managing director and ceo of Schmidco Consulting". This usually means that Joe got fired and just set up an s-corp.
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As a pure mathematician, one of your roles will be mentoring students. Some of those students will go into applied math or industry jobs, and they will have a use for LinkedIn. On LinkedIn, it looks good for students to be connected to their professors, as it indicates some deeper-than-just-attended connection.

So get on LinkedIn, and encourage your students to "Link" to you. Do it for the children.
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I have a linkedin account from a brief period when I worked "in industry". When I subsequently did two (pure/theory) postdocs, I updated my details to reflect that fact. I haven't actively used it to look for jobs or connections much, but it seems unlikely to be useful to me. Linkedin is too clogged up with other stuff, there are more direct ways to eg advertise jobs in academia. On the other hand Vaclav Chvatal sent me a (probably automated) message on there about his new book, which is about Erdos, nothing to do with industry or anything. So at least someone out there (maybe Chvatal, maybe some literary agent or something) thinks it might be useful...
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LinkedIn I don't see as being particularly useful, unless you're looking to do some kind of industrial links/collaborations. Researchgate is pretty useful, but more at the level of keeping on top of recent publications rather than networking
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Nah.
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Tbh not sure about academia, but in ML, I feel like twitter is the social network.
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If you haven’t already, then You might want to look into ResearchGate.
by
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I do see more academic mathematicians joining LinkedIn, but I haven’t seen much activity. I think this could change. It doesn’t hurt to join and connect with other mathematicians you see on it. And, if you have a reasonable excuse, message them. They might not reply simply because they never check LinkedIn for messages, but you never know. Just don’t rely on it.
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If you happen to already know a topic you find interesting, see if you can find an advisor who is still actively publishing, read through some of his work (probably his thesis is the best place to start) and then send him an email introducing yourself and showing your interest. See if he is taking students on now and also also for some recommendations on what papers of his you should focus on to prepare.

Any professor would be very happy to help a student who shows enthusiasm for their work. Good luck!!

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