there is actually something rather special about the questions mathematicians focus on but the jargon and technical detail get in the way. ultimately, the reason prime numbers are so fascinating is because they turn up in weird places, with strange properties, and connect wildly different branches of mathematics together.

the Riemann Hypothesis is an example of this. it starts from a pair of observations:
1. the sum of the logarithms of the prime numbers less than n approximates the graph of y = x to absurd precision. there's no obvious reason why this should be true but the result just kind of falls out if you attempt to estimate the sum, even in really crude ways.
2. the sum of the reciprocals of the squares was calculated by Euler centuries ago. the way he solved the problem was to connect two different ways of calculating sin(x) -- the Taylor series and a new idea he developed which you may have tried in a calculus class. namely, he wrote down sin(x) as a infinite product of its roots. one result that happens to fall out of this comparison between the infinite sum and the infinite product is that you can factor equations like sum(1/n^2) over all the integers into a product of the reciprocals of the primes.

this second fact leads very directly to a proof of the first through some clever algebra but it's far more general. in fact, we not only get exact values for sum(1/n^2), but exact values for all expressions of the form sum(1/n^s) where s is a positive even integer -- odd powers remain an unsolved problem to this very day. in fact, these sums are hiding in the Taylor expansion of sin(x). so somehow, we've connected a seemingly arbitrary question about how to take a certain kind of sum of the integers, connected it to a product of primes, and tied both to an analytical function -- sin(x). already bizarre.

but it gets even stranger. if you let s vary as a complex number and not just as a positive number > 1, this very same approach lets you calculate the sums for negative even powers as well. remember, we're calculating sum(1/n^s) -- how can we get any result but infinity for sum(1/n^-2) -- i.e. sum(n^2)? more perplexing is that the value you can calculate from algebra and calculus over the complex numbers, following Euler's method, and Riemann's extension of it, is 0 for every negative even power. this is also where the notion that the sum of the integers is -1/12 comes from. this is the Riemann Zeta function.

so now, we've connected a seemingly arbitrary sum of the integers to an analytic function over the complex numbers via prime numbers. and worse still, if we connect this back to the first point, we discover that this Zeta function tells us the order of the error term on the estimate for sum(log p, p < x). remember, this function looks like the graph of y = x. if the Zeta function has a zero at s = 1/2, we have an exact formula for that sum and the error term is precisely O(x^1/2). we can even calculate the constant factors. this fixes all the prime numbers. it in fact gives you an exact formula to calculate them, not as a recursive function, but directly. all because some asshole in the 1700s posed a problem no one could answer until Euler: what's sum(1/n^2)?

so this is why prime numbers fascinate mathematicians. nothing we understand about them is obvious. but being able to answer seemingly arbitrary questions about them unlocks whole fields of mathematics. and new techniques in completely unrelated fields of mathematics suddenly answer deep questions about the structure of the natural numbers.

if this topic interests you, I highly recommend this series of videos from zetamath: https://www.youtube.com/watch?v=oVaSA_b938U&list=PLbaA3qJlbE... -- he walks you through the basic algebra and calculus that lead to all of these conclusions. I can't recommend this channel enough.