The Riemann zeta function is the object for one of the most famous unsolved problems in math, the Riemann Hypothesis. It's a big enough deal, in fact, to have a $1,000,000 prize offered for its resolution from the Clay Math Institute, although the money probably pales in importance compared to the mark you would make in mathematical history by resolving it. 3blue1brown viewers may be familiar with walking through how the function itself is defined. The million dollar question is whether all "non-trivial zeros" of this function lie on the line 0.5 + it, where t ranges over all real numbers. This is the so-called "critical line".
Illustrated on this shirt is the chaotic behavior of the zeta function on this critical line (for t ranging from 0 to 50, if you're curious), as animated . Why do mathematicians care? Each point where this spiraling line crosses zero corresponds, in some sense, to a prominent frequency in the pattern of prime numbers, making the collection of these zeros something like a musical chord which encodes patterns in the primes, and no one fully understands it.
And tell you what, if you do solve the Riemann hypothesis, I'll send you a free shirt.