I found the following problem in a captcha: (and I was really surprised, I expected just regular blurred or distorted text)
What does that mean, and what would the solution be?
EDIT: It looks, from comments and answers, that this is a consequence of Riemann Hypothesis. I would appreciate if someone could outline the proof of that fact. Cheers!
On
The Riemann hypothesis is one of the seven Clay Mathematics Institute Millennium Prize Problems. It hasn't been solved as of today and the resolution of this is way beyond your average Internet user knowledge of mathematics. You can tell how difficult this problem is with the enormous hype that surrounded the resolution of Poincaré's Conjecture, the only Millenium Problem that has been solved.
So my guess is that it is joke, either on your friend's side (a montage), or on the webpage's side (like a way to tell you that as a non-premium user, you can't access the file).
The Riemann Hypothesis is arguably the most famous problem in mathematics. Its usual statement involves the zeroes of a function $\zeta:\mathbb C\to\mathbb C$ defined by $$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}.$$ Although it looks like it's only in the world of complex numbers, it turns out to have much deeper implications in number theory and exaplaining the behavior of the prime numbers. It's an open problem, however - nobody has solved it yet. We have however proved that many things assuming it, and the statement here is one of these.
The symbol $\pi(x)$ is the prime counting function, it counts the number of prime numbers less than or equal to $x$. So for instance $\pi(6) = 3$, because of the $3$ primes $2,3,5$. I assume you understand the integral. The $O(x^{1/2+\varepsilon})$ is just saying that the expression roughly acts like $x^{1/2+\varepsilon}$ as $x$ grows.
The problem is very hard - it's even worth a million dollars. I would go ahead and assume it's a scam or a joke.