A sketch of the proof of non-existence of Siegel zeroes

For a given prime , the Legendre symbol is defined as follows: if is a quadratic residue modulo , then ; if is a quadratic non-residue modulo , then ; and if divides , then .

Let be an integer that is not a perfect square. (One could require to be a fundamental discriminant if desired, although it turns out to be immaterial for the arguments here.) By combining quadratic reciprocity with the prime number theorem in arithmetic progressions, one can show that the Legendre symbol is asymptotically balanced between and . For instance, one can show that

as , where the sum is over primes ; equivalently (by Mertens’ theorem),

However, most of the known proofs of this asymptotic are ineffective with regards to the decay of the error. This is due to the infamous Siegel zero problem, arising from Dirichlet -functions associated to with a zero very close to . A well known example comes when , which is a non-residue for all primes ; this is closely tied to Euler’s famous observation that is prime for all integers with , as well as the fact that the ring of integers has unique factorization. For further discussion of these topics, see this previous set of lecture notes.

Recently, the Siegel zero problem has been solved in a number of ways. In fact, a stronger “quasi-Riemann hypothesis”, showing that the zeroes of all Dirichlet -functions have real part strictly bounded away from , has been established (with two separate papers giving a bound of and respectively) as part of a recent release by OpenAI. (In particular, the error term in can now be improved to a power saving .) These arguments are analytic in nature, using identities relating the Möbius function to the Fourier coefficients of certain automorphic forms; they are still in the process of being fully digested, and I will not be discussing them further here. However, there was a third paper in the same release that did not establish the quasi-Riemann hypothesis, but instead excluded Siegel zeroes directly. This paper was significantly shorter (nine pages) and proceeded almost entirely via algebraic number theory and the geometry of numbers.

Recently, various digestions and simplifications of the argument have been produced by Banks, by Kerr, and by Green. I would like here to present a sketch of Green’s formulation (with some small notational changes), which I find particularly elegant. The key bound he establishes is whenever is a small quantity, is a suitably large scale, and is a larger scale that is nearly quadratic in . For small enough, this improves by a factor of about upon the trivial bound

coming from Mertens’ theorem; thus is a non-trivial bound on the number of primes at scales up to for which is a quadratic non-residue. By optimizing in , this gives the upper bound for ; in fact with more care one concludes that

(In fact, some extremely recent preprints of Jacob Pescador and Kevin Barretto have sharpened this bound further, with an upper bound of .)

One can obtain the full strength of using standard analytic number theory methods (omitted here, but see for instance these previous notes) to rule out Siegel zeroes, which can then in turn be used to upgrade the error term in to a classical shape for some absolute constant , assuming . Besides this, though, none of the classical analytic number theory methods are utilized in this proof. In retrospect this makes sense, as the infamous “parity problem” blocks such methods from getting results such as without some parity-breaking input: in this case, said input comes from the polynomial method, whereas in the QRH papers, it appears to be some (still poorly understood) automorphic miracle that is breaking the parity barrier.

AI disclosure: a chat session with the new SAIR open math model was used to help understand the preprints.

— 1. High-level description of proof —

I will make some minor assumptions and “cheats” to simplify the exposition. Firstly I will restrict to the case when is negative; this makes have one complex embedding rather than two real embeddings, which allows for a very small simplification to the notation. For extremely minor notational reasons I will also exclude , so that . I will also pretend at a later point that an arbitrarily chosen high-degree polynomial is irreducible, which is of course not always the case, but allows one to simplify a certain algebraic calculation, the general case of which is handled in Green’s writeup.

The overall strategy is that of the “adelic polynomial method”, which appears in various guises (though usually not with this name) throughout the literature, for instance in the Bombieri–Vaaler paper on Siegel’s lemma (I would be interested in further historical information on the development of the method). We will not actually use any of the machinery of the adeles in this post, though experts will readily see them lying just beneath the surface.

The first observation is just the fundamental theorem of arithmetic, which gives the factorization

for any non-zero integer , where is the number of times divides . Taking logarithms, we obtain

So, if one wanted to obtain a bound such as , one could try to construct some non-zero integer for which one had some upper bound on the “Archimedean magnitude” of , while also having lower bounds on the valuations for primes in the range of for which is a quadratic residue. That is to say, one wants to find an which is not too large, while also being divisible by many powers of such primes , and also non-vanishing.

This quantity will be constructed geometrically, being essentially the covolume of a certain lattice (actually the square of that covolume, to make it an integer). Let be a small multiple of . First let be the set of polynomials with integer coefficients that are of degree less than in the two variables , by which we mean that they are of degree less than in the variable, and less than in the variable. The behavior in will be more important for us than that in , and this is reflected in the disparity in sizes between the two degrees .

Clearly is a lattice of rank . We will evaluate this lattice on the grid

The idea of evaluating a lattice or vector space of polynomials on various carefully selected sets such as lies at the heart of the polynomial method, with a focus often being made on the extent to which such polynomials can vanish on . In many variants of the polynomial method, one also analyzes the extent to which such polynomials can vanish with multiplicity on ; Stepanov’s method (discussed for instance in this previous post) is one well-known example. One of the innovations in Green’s treatment is to also take advantage of multiplicity to achieve the non-trivial bound ; previous treatments did not do this, and had to work with a (somewhat artificial) extension of , such as , in order to get comparably non-trivial bounds.

Specifically, for each polynomial in , we consider the evaluations

of the first horizontal derivatives of at each point of , where the optimal choice of turns out to be , though as these two quantities play conceptually different roles we will continue to denote them by different symbols.

The tuple is an element of , and so is some additive subgroup of . A key fact is:

Lemma 1 (Injectivity) is injective on . In other words, if is a non-zero element of , then cannot vanish to (horizontal) order at every point of .

We will prove this later using the theory of resultants.

Thus is a lattice of rank . To this lattice, we can prescribe a (squared) covolume as follows. Let be some basis for this lattice, then form the Gram matrix

where we embed into in the usual fashion with the Euclidean inner product for the purpose of defining the (real) inner product. This Gram matrix depends on the choice of basis, but the determinant does not; we then take to be this determinant. Geometrically, one can think of as the square of the volume of a fundamental domain of .

Some basic facts about . Firstly, the injectivity in Lemma shows that is non-zero. As all the coefficients of the basis vectors lie in the ring , the inner products are all integer, and so is also. Hence is a non-zero integer. (It is also positive, although this is irrelevant for us.) In particular, the fundamental theorem of arithmetic gives .

It is easy to bound the Archimedean size of :

Lemma 2 (Archimedean bound) We have

This will be a straightforward application of Hadamard’s inequality, using the obvious monomial basis, we prove this later in the post.

Now for the arithmetic input. Suppose that is an (odd) prime for which . From Fermat’s little theorem we have

and therefore

From further application of Fermat’s little theorem (and the Frobenius automorphism), this implies that

for any integers . In particular, from a -adic point of view, the grid lies close to the curve

If the grid happened to lie exactly on this curve, then the lattice would degenerate (contradicting Lemma ) and would now vanish. This of course does not happen; but the fact that is -adically close to the curve will force to be -adically close to zero, or equivalently give a useful lower bound on :

Lemma 3 (Non-Archimedean bound) If and , then

We prove this bound later in the post. Inserting Lemma and Lemma into and cancelling the common factor of , we obtain as required.

— 2. Injectivity —

Now we establish Lemma . From my understanding, these sorts of non-vanishing theorems go back to the work of Masser (and the Kerr preprint explicitly uses Masser’s bounds to obtain an analogue of this lemma), which initially used a fair amount of ring theory, but a more elementary alternate approach based on the interpolating polynomial method was introduced by Laurent, an adaptation of which is used in the OpenAI release. Green’s approach is simpler still, based largely on the theory of resultants to project the (high-order) vanishing of a two-variable polynomial to a (high-order) vanishing of a one-variable polynomial.

As is an grid, a direct application of the Schwartz–Zippel lemma tells us that a non-zero degree polynomial can only vanish on at most of the points of . This is not directly useful for us because the degree of can be as large as , and so this does not prevent from vanishing on all of . And indeed, if we replaced by the standard grid

then the degree polynomial already vanishes on , as does all of the horizontal derivatives of this polynomial. So to proceed here we have to somehow use the fact that the grid is “skew”; in particular we will take advantage of the fact that the second coordinates of this grid are all distinct, in contrast to the situation with .

Let be a non-zero polynomial of degree less than . To avoid technicalities we will assume that is irreducible, and not solely a function of , so that is not identically zero; the general case is treated in Ben Green’s note.

Suppose that vanishes horizontally to order at least at a point , then by performing a Taylor expansion around that point we can factor as

for some polynomials of degree less than and respectively. If we take one derivative in the direction, then vanishes to order at least at the same point, thus

for some polynomials of degree less than and respectively.

Since and both vanish to high order at , it is natural to look at the horizontal resultant of these two polynomials (which we now view a -valued polynomials of ), defined as the determinant of the polynomials

each of which we view as vectors of coefficients in . Note that all the polynomials in this list take the form

for some polynomials of degree less than and respectively (here we drop some irrelevant lower order terms). By multilinearity of the determinant, the resultant now splits into a linear combination of various determinants involving expressions of the form and expressions of the form . As is of degree less than in the variable, any contribtion with must vanish, and so we need to have at least factors of the form . We conclude that the resultant is divisible by .

On the other hand, a straightforward degree count shows that this resultant is a polynomial of of degree . The hypothesis that is irreducible and implies that the resultant is non-zero. So if vanished to order at least at each of the points of , then the resultant would have to be divisible by different factors of (here we use the fact that the are all distinct), leading to the inequality

which is false for small enough by the choice of , , , and . This proves Lemma .

— 3. Archimedean bound —

The Archimedean bound is straightforward. For any basis of , the quantity is simply the square of the -dimensional parallelepiped with edges , viewed as elements of . By the Hadamard’s inequality, we thus have

(using the usual Euclidean norm), or upon taking logarithms

Applying the sampling operator to the monomial basis , we thus have

Each point in the grid is of size by choice of . Every component of then has size , and so

or on taking logs and using the construction of

Summing the arithmetic series in we obtain

By construction we have and , so Theorem follows.

— 4. Non-Archimedean bound —

Let be a prime with and . As discussed in the high-level overview, the polynomial takes values in on the entire grid . This has the following consequence: any polynomial which is divisible by for some will take values in . Furthermore, the same is true for any iterated -derivative of : note from the chain rule that applying to will lower the exponent by , thus removing one factor of , but also produce an additional factor of which restores that missing factor. We conclude that every coefficient of then lies in .

To use this, we replace the standard monomial basis , , of with the modified basis

for , , where

Note that such a basis element still lies in , and is of the form plus a linear combination of monomials that are lower than this in the lexicographical ordering. So this modified basis remains a basis of . Applying , we see from the previous discussion that each of the basis elements is divisible by . Computing the Gram matrix in this basis, we conclude that

For , one has by the hypotheses on and the choice of . We thus have

and Theorem then follows by summing the arithmetic series.