One of the most important open problems in mathematics is called the Riemann hypothesis. It states that the solutions of a certain equation `zeta(z)=0` are all of a particular type. Almost every living mathematician has tried to solve it at some point in their lives. The implications of the hypothesis are deep for the theory of numbers, for instance for the distribution of prime numbers.
In a recent paper some mathematicians claim they have put some stronger bounds on where those solutions can be. In this link Terrence Tao, one of the most acclaimed mathematicians alive speaks very highly about the paper.
IMHO, this is probably not of huge interest to not mathematicians just yet. It is an extremely technical result. And pending further review it might very well be wrong or incomplete.
There are lots of places you can read about the Riemann Hypothesis, its implications and its attempts to solve it.
As for why the Riemann Hypothesis itself is interesting, it is because this zeta function seems to have information about the primes inside it.
The eye opener for me is when a video about it connected the idea of the zeta function to how you might create a square wave by adding successive sine waves (of higher and higher frequency).
Imagine a function (call it `prime_count`), which gives you the number of prime numbers below it's argument.
If you tried to plot this function on a graph, it will look like a series of jaggard steps - a jump when you reach a new prime number. It turns out, if you rewrite the zeta function and decompose it (into it's infinitely many zeros - akin to doing a "taylor series" expansion for other functions), you will find that the more zeros you use, the closer you will get to a jaggard graph.
> The eye opener for me is when a video about it connected the idea of the zeta function to how you might create a square wave by adding successive sine waves (of higher and higher frequency).
The process of taking an arbitrary waveform and approximating it by adding together a series of sine and/or cosine waves with different amplitudes etc is called a Fourier transform. That is what Terrence Tao is talking about when he mentions trying this approach using Fourier analysis himself. [1] Fourier transforms are used all over the place, eg the discrete cosine transform (DCT) is part of JPEG, MP3 etc.
In a recent paper some mathematicians claim they have put some stronger bounds on where those solutions can be. In this link Terrence Tao, one of the most acclaimed mathematicians alive speaks very highly about the paper.
IMHO, this is probably not of huge interest to not mathematicians just yet. It is an extremely technical result. And pending further review it might very well be wrong or incomplete.
There are lots of places you can read about the Riemann Hypothesis, its implications and its attempts to solve it.