[alexbeloi]

Not eli5, but a comparison to the Riemann Hypothesis (RH).

RH says the Riemann-zeta function has no zeros along the line (1/2) + iy in the complex plane.

The Lindelof hypothesis says that the number of zeros between (1/2) + iy and (1/2) + i(y+1) is much smaller (little-o) than log(y) as y grows.

So it can be thought of as a weaker version of RH, but still very very difficult. The fact that Lindelof has been an open problem for over a hundred years (and is an non-trivial weakening of RH) speaks to how difficult RH is as well.

Like RH, Lindelof implies things about primes, and also (like RH) has lots of implications about lots of interesting prime-like (irreducible) objects in different spaces.

[dbaupp]

I think you've flipped the condition: the RH says the Riemann zeta function _only_ has zeros along the line 1/2 + iy. (And, indeed, there are known zeros along this line: 1/2 + 14.135... i.)

The Lindelöf hypothesis is, apparently, equivalent to: the number of zeros with real part greater than 1/2+epsilon and imaginary part between y and y+1 is o(log(y)), for any epsilon > 0. That is, boxes of height 1 starting just off the critical line contain few zeros; the RH implies they contain zero.