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.
I'm not a pure-maths type person but in my experience, if you get one answer by following simple, well understood maths (like "the sum of two positive integers is a positive integer") and another answer by "squinting and ignoring it", this doesn't mean the simple answer is wrong, it means you did something else wrong (like the hidden divide-by-zero present in your typical "proof that 1 = 2"). Paradoxes point to an error in the formulation of the question.
Why would you take the running average and how is that relevant to the sum?
1-1 = 0
0+1 = 1
1-1 = 0
0+1 = etc etc
...except that it doesn't; to me, it even looks more like it trends in the opposite direction, i.e. it is trying to stay away from 1/2.
Like two magnets repelling each other: if you were to hold them together and we call that 1/2 - but they are always trying to push away from each other!
Talking about it as summation might be misleading since that’s one of those concrete terms that mathematicians like to redefine without anyone’s approval. Picture we have a library that includes many tricks and approaches for taking an infinite series as input and outputs a number. We know it works as expected on every convergent series. But we forgot to put in any preconditions and we’ve let people input things that are not convergent series. But whoa in many cases we are still getting a number out of it, and it’s always the same answer no matter what we do. Maybe that’s something worth studying?
It's also probably worth noting that, if the series is genuinely summable, then Cesàro summation gives its sum.
In short:
Where the series obviously converge, use the summing formula.
Where the series is mis-behaving, use analytic continuation instead of resorting to weird infinite series re-ordering tricks (which I've always felt to be borderline offensive from a mathematical rigor pov).
My understanding of analytic continuation is that if a function f of the complex plane is sufficiently well behaved on a certain domain of the plane, it can be "extended" to the rest of the plane in a unique way that preserves the well-behavedness.
In the case of zeta, it can be shown that zeta obeys a functional equation that allows it to be extended everywhere.
A better explanation than mine is here:
https://math.stackexchange.com/questions/437883/what-is-the-...
Mathematicians aren't stupid, and more than any other profession, they value rigor. They know what they're doing.
It may not be wrong, but nothing is gained.
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.
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.