system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: What is the riemann zeta function and is ζ(3) so important?
gpt
Without going into specifics (it is ELI5 after all), the zeta function is a differentiable (read: nice) function living on the complex plane (read: put in a complex number, get out a complex number). The definition for some of the points is nothing spectacular, however, you can show that the zeta function has very odd and unexpected connections to other areas of Mathematics, even connecting areas of Mathematics that no one would have expected to be able to connect. Prime numbers are one example of this. In fact, the connection that the zeta function brings (or would bring if we knew Riemann's conjecture to be true) is so strong you could compare the influence of the discovery of that proof to mathematicians to the discovery of the Higgs boson for physicists.

Due to the nature of the function it is exceedingly hard to know more than "it exists and it relates to that" (even for the explicitly defined values you can tell little more), but since for real values greater than one the function is defined in a really simple way you can unsurprisingly find connections to some areas (especially for real or even integers values). Since the zeta function relates to prime numbers (zeta(s) is also denoted as the product of (1/1-p\^(-s)) of all prime numbers p) and those make up all of our security systems (RSA - if you want to search) and since knowledge about the zeta function equals knowledge about prime numbers (specifically, you can show that 1/zeta(3) is the probability that iirc. 3 randomly chosen integers are at least co-prime) any such knowledge is very important to everyday life (if RSA were suddenly solvable or more easily computable, most of the internet communication would need a complete restructure to function). 

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Apart from the connection to prime numbers the zeta function (especially t -> z(1+it)) may have connections to highly chaotic systems, but i don't know enough about that to make a qualified statement on that.

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TL;DR: The definition is mostly simple, it relates to prime numbers, we know nothing about prime numbers, if we know more about zeta we konw more about prime numbers.