The n-Category Café

Since 1897, we’ve known all the nontrivial zeros of the Riemann zeta function lie in this strip in the complex plane:

0 < Re(z) < 1

The quasi-Riemann hypothesis says they lie in some narrower strip

a ≤ Re(z) ≤ 1 - a

The Riemann hypothesis claims all these zeros lie exactly on the line

Re(z) = 1/2

The closer to this line we can show these zeros are, the more regular the distribution of primes must be!

First, remember the number of primes less than n is asymptotically n / (ln n). But a more precise estimate of the number of primes less than n is a function called li(n), which is the integral of 1/ln(x) from 0 to n.

The difference between li(n) and n / (ln n) is about n / (ln n)². This is a lot bigger than the errors I’m going to talk about.

Before the proof of the quasi-Riemann hypothesis, we knew things like this: if π(n) is the number of primes less than n, then

|π(n) - li(n)| ≤ n / (ln n)¹⁰⁰⁰

if n is large enough. Indeed, we could replace the number 1000 by any other large number.

The Riemann hypothesis is equivalent to this: if π(n) is the number of primes less than n, then

|π(n) - li(n)| ≤ C n^p

for any p > 1/2, and some C that depends on p. We know we can’t get p all the way down to 1/2.

So crudely, the Riemann hypothesis brings the error bound down from “a lot less than n, but still bigger than any smaller power of n” to “not much more than √n”.

The quasi-Riemann hypothesis does the same thing but replacing √n with some bigger power of n, namely n^a, if all the nontrivial zeros of the Riemann zeta function lie in the strip

a ≤ Re(z) ≤ 1 - a

In the recent massive drop of AI-generated results there’s a long, complicated proof that the quasi-Riemann hypothesis holds for a = 1/8:

• The quasi-Riemann hypothesis, 199 pages.

If this proof turns out to be correct, then we’ll know

|π(n) - li(n)| ≤ C n^p

for any p > 7/8.

But there’s also a much simpler proof that a = 1/12 works:

• The quasi-Riemann hypothesis, alternate proof, 49 pages.

A bunch of mathematicians have been writing about this, developing similar results—so we can feel pretty sure this is true. Thus, at this point, number theorists are confident that

|π(n) - li(n)| ≤ C n^p

for any p > 11/12.

But the quasi-Riemann hypothesis has other consequences, too—especially combined with the recent proof that a bunch of L-functions have no ‘Siegel zeros’. One of the cuter, less technical results is called ‘Goldbach conjecture on average’.

You can count, for each even number up to n, the number of ways to write it as a sum of two primes, and add these counts up. The total is close to a simple formula, and the question is how close. The quasi-Riemann hypothesis puts a bound on the error, thus proving Goldbach’s conjecture on average. I’ll spare you the details. Unfortunately, this bound is not good enough to prove Goldbach’s actual conjecture.

There are also plenty of more technical yet more profound implications of the new work on the quasi-Riemann hypothesis. These rely on not just the statement, but the proof method. There are various other important functions in number theory—zeta functions and L-functions—that we can now prove have zeros in a half-plane Re(z) > 1 - a for some fixed a > 0. Write QRH(a) for this statement, so OpenAI’s 7/8 result is QRH(1/8) for the Riemann zeta function.

A recent paper by Frei, Sofos and Widmer summarizes some of what’s been done so far:

• Christopher Frei, Efthymios Sofos and Martin Widmer, Some applications of the quasi-Riemann hypothesis to class groups and small generators of number fields.

For example, OpenAI claims QRH(1/8) for all Dirichlet L-functions and all finite-order Hecke L-functions of ℚ√(−3), with a second preprint giving QRH(1/12). OpenAI also claims QRH(10^-6) for all finite-order Hecke L-functions of cyclotomic fields containing the 12th roots of unity. Most exciting to me, a preprint by Cureton and Thorner claims QRH(1/32) for the Dedekind zeta functions of all number fields:

• Dean Cureton and Jesse Thorner, A quasi-Riemann hypothesis for Dedekind zeta functions.

You may not consider any of these implications “staggering”. But that probably means you’re not an analytic number theorist. According to Joshua Zelinsky, there are hundreds of papers deriving results that can be improved using the quasi-Riemann hypothesis. Mathematicians have been working for over a century to make these things happen.