Can someone eli5 the lindelof hypothesis?
zeta(s) = 1 + 1/2^s + 1/3^s + 1/4^s + 1/5^s + 1/6^s + ....
For example,
zeta(2) = 1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + ... = pi^2/6.
As a partially tongue-in-cheek example,
zeta(-1) = 1 + 2 + 3 + 4 + 5 + 6 + ... = -1/12.
Obviously it doesn't make sense to add all the positive integers (the series doesn't converge), but if you squint and ignore this, and just do the arithmetic a certain way, you get -1/12.
The original definition I gave is valid when s is a complex number with real part greater than 1. But the Riemann zeta function can be proved to have analytic continuation: zeta(s) makes sense for any complex number s, other than 1. For example, zeta(-1) really equals -1/12.
The zeta function is easy to understand when the real part is greater than 1: the formula I described is enough. Because of the so-called functional equation, it is also easy to understand when the real part is less than 0. But it is in the middle that all of its secrets lie. For example, the notoriously unsolved Riemann Hypothesis stipulates that the "nontrivial" zeroes all have real part 1/2.
The Lindelof Hypothesis stipulates that the zeta function grows very slowly along this line (real part = 1/2). It is very closely related to the Riemann Hypothesis. More technical, and of less direct interest to nonspecialists, but in the same family of problems.
As an example of how much mathematicians care about this, here are the Google search results for "subconvexity bound":
https://www.google.com/search?q=subconvexity+bound
A "subconvexity bound" is any result which approaches the Lindelof Hypothesis, for either the Riemann zeta function or a more general "L-function". A lot of ink has been spilled on proving results weaker than what Fokas is claiming.