How do you "maintain" connections?

Update when you're starting something new, just a quick email or message letting them now, update when you're starting to interview at their firm, if none of those apply then update once every few months for a simple coffee chat

Every 6-8 weeks with a real reason to reach out. Never "just checking in" with nothing behind it.

Best triggers: you got an interview, a superday, an offer, or you have a specific question they can answer.

When something good happens to them — promotion, deal news — a quick congrats goes a long way. 
Makes it feel like a real relationship not a one-way transaction.

Quality over frequency every time.

Yeah echo this, also if you build out a tracker you can have it alert you with formatting when it’s time to reach out. I would also put notes in that same tracker. So, if for instance, they told you something to do to prepare, or if they told you about a deal you can follow up with more targeted info

Do it when you actually have something to say. Sending random “just checking in” emails every few weeks gets annoying fast unless the person really likes you. Best cadence is usually every couple months, or when there is a real update like internship news, recruiting progress, club leadership, or a reason their earlier advice helped.

Basically keep the relationship warm, not forced. Better to send 3 good updates over a year than 10 nothing emails.