Terence Tao 梳理 Green 对无 Siegel 零点证明的简化版本

For a given prime {p} , the Legendre symbol ${\left(\frac{D}{p}\right)}$ is defined as follows: if {D} is a quadratic residue modulo {p} , then ${\left(\frac{D}{p}\right) = 1}$ ; if {D} is a quadratic non-residue modulo {p} , then ${\left(\frac{D}{p}\right) = -1}$ ; and if {p} divides {D} , then ${\left(\frac{D}{p}\right) = 0}$ .

Let {D} be an integer that is not a perfect square. (One could require {D} 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 {+1} and {-1} . For instance, one can show that

$\displaystyle \sum_{p \leq x} \left(\frac{D}{p}\right) \frac{\log p}{p} = o\left(\log x\right)$

as ${x \rightarrow \infty}$ , where the sum is over primes {p} ; equivalently (by Mertens’ theorem), However, most of the known proofs of this asymptotic are ineffective with regards to the decay of the {o()} error. This is due to the infamous Siegel zero problem, arising from Dirichlet {L} -functions associated to {D} with a zero very close to {s=1} . A well known example comes when {D = -163} , which is a non-residue for all primes {p < 41} ; this is closely tied to Euler’s famous observation that ${n^2 + n + 41}$ is prime for all integers {n} with ${0 \leq n < 40}$ , as well as the fact that the ring of integers ${{\bf Z}[\sqrt{-163}]}$ 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 {L} -functions have real part strictly bounded away from {1} , has been established (with two separate papers giving a bound of {11/12} and {7/8} respectively) as part of a recent release by OpenAI. (In particular, the ${o(\log x)}$ error term in (1) can now be improved to a power saving ${O( |D|^{O(1)} x^{-1/8+o(1)})}$ .) 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

$\displaystyle \sum_{\varepsilon^{-2} \leq p \leq \varepsilon R : \left(\frac{D}{p}\right) = -1} \frac{\log p}{p} \leq \left(1 + O(\varepsilon)\right) \log N \ \ \ \ \ (2)$

whenever ${\varepsilon > 0}$ is a small quantity, ${N \geq |D|^{1/\varepsilon}}$ is a suitably large scale, and ${R := \lfloor N^{2 - \varepsilon}\rfloor}$ is a larger scale that is nearly quadratic in {N} . For ${\varepsilon}$ small enough, this improves by a factor of about ${\frac{1}{2}}$ upon the trivial bound

$\displaystyle \sum_{\varepsilon^{-2} \leq p \leq \varepsilon R : \left(\frac{D}{p}\right) = -1} \frac{\log p}{p} \leq \log R + O(1)$

coming from Mertens’ theorem; thus (2) is a non-trivial bound on the number of primes {p} at scales up to ${\varepsilon R}$ for which {D} is a quadratic non-residue. By optimizing in ${\varepsilon}$ , this gives the upper bound for (1); in fact with more care one concludes that

$\displaystyle \sum_{p \leq x : \left(\frac{D}{p}\right) = -1} \frac{\log p}{p} \leq \frac{1}{2} \log x + O(\log |D|) + O(\log\log x).$

(In fact, some extremely recent preprints of Jacob Pescador and Kevin Barretto have sharpened this bound further, with an upper bound of ${\frac{1}{2} \log x + \frac{1}{4} \log |D| + \log\log x + O(1)}$ .)

One can obtain the full strength of (1) 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 (1) to a classical shape ${\exp(- c \sqrt{\log x})}$ for some absolute constant {c>0} , assuming ${|D| \leq \exp(c \sqrt{\log x})}$ . 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 (2) 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. I also benefited from conversations with Ben Green, Alex Pascadi, Lior Silberman, Valentin Blomer, Gergely Harcos, Farrell Brumley, William Banks, and Edgar Assing.

— 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 {D} is negative; this makes ${{\bf Z}[\sqrt{D}]}$ 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 {D=-1} , so that ${|D| \geq 2}$ . I will also pretend at a later point that an arbitrarily chosen high-degree polynomial ${P(z_1,z_2)}$ 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

$\displaystyle |M| = \prod_p p^{v_p(M)}$

for any non-zero integer {M} , where ${v_p(M)}$ is the number of times {p} divides {M} . Taking logarithms, we obtain So, if one wanted to obtain a bound such as (2), one could try to construct some non-zero integer {M} for which one had some upper bound on the “Archimedean magnitude” of {M} , while also having lower bounds on the valuations ${v_p(M)}$ for primes {p} in the range of (2) for which {D} is a quadratic nonresidue. That is to say, one wants to find an {M} which is not too large, while also being divisible by many powers of such primes {p} , and also non-vanishing.

This quantity {M} will be constructed geometrically, being essentially the covolume of a certain lattice (actually the square of that covolume, to make it an integer). Let ${S := \lfloor \varepsilon R \rfloor}$ be a small multiple of {R} . First let ${\Lambda}$ be the set of polynomials ${P(z_1,z_2)}$ with integer coefficients that are of degree less than {(R,S)} in the two variables ${z_1,z_2}$ , by which we mean that they are of degree less than {R} in the ${z_1}$ variable, and less than {S} in the ${z_2}$ variable. The behavior in ${z_1}$ will be more important for us than that in ${z_2}$ , and this is reflected in the disparity in sizes between the two degrees {R,S} .

Clearly ${\Lambda}$ is a lattice of rank {RS} . We will evaluate this lattice on the ${N \times N}$ grid

$\displaystyle E := \{ (m + n\sqrt{D}, m - n\sqrt{D}): m,n \in {\bf Z}, 0 \leq m, n < N \}.$

The idea of evaluating a lattice or vector space of polynomials on various carefully selected sets such as {E} lies at the heart of the polynomial method, with a focus often being made on the extent to which such polynomials can vanish on {E} . In many variants of the polynomial method, one also analyzes the extent to which such polynomials can vanish with multiplicity on {E} ; 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 (2); previous treatments did not do this, and had to work with a (somewhat artificial) extension of ${{\bf Z}[\sqrt{D}]}$ , such as ${{\bf Z}[\sqrt{D}, \sqrt{2}]}$ , in order to get comparably non-trivial bounds.

Specifically, for each polynomial {P} in ${\Lambda}$ , we consider the evaluations

$\displaystyle TP := \left( \partial_{z_1}^j P(z_1, z_2) \right)_{0 \leq j < J; (z_1,z_2) \in E}$

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

The tuple {TP} is an element of ${{\bf Z}[\sqrt{D}]^{J \times N^2}}$ , and so ${T\Lambda}$ is some additive subgroup of ${{\bf Z}[\sqrt{D}]^{J \times N^2}}$ . A key fact is:

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

We will prove this later using the theory of resultants.

Thus ${T\Lambda}$ is a lattice of rank {RS} . To this lattice, we can prescribe a (squared) covolume {M} as follows. Let ${v_1,\dots,v_{RS}}$ be some basis for this lattice, then form the ${RS \times RS}$ Gram matrix

$\displaystyle \left( \langle v_i, v_j \rangle \right)_{1 \leq i,j \leq RS}$

where we embed ${{\bf Z}[\sqrt{D}]^{N \times N^2}}$ into ${{\bf C}^{N \times N^2} \equiv {\bf R}^{2 \times N \times N^2}}$ 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 {M} to be this determinant. Geometrically, one can think of {M} as the square of the volume of a fundamental domain of ${T\Lambda}$ .

Some basic facts about {M} . Firstly, the injectivity in Lemma 1 shows that {M} is non-zero. As all the coefficients of the basis vectors ${v_i}$ lie in the ring ${{\bf Z}[\sqrt{D}]}$ , the inner products ${\langle v_i, v_j \rangle}$ are all integer, and so {M} is also. Hence {M} is a non-zero integer. (It is also positive, although this is irrelevant for us.) In particular, the fundamental theorem of arithmetic gives (3).

It is easy to bound the Archimedean size of {M} :

Lemma 2 (Archimedean bound) We have
$\displaystyle \log |M| \leq (1 + O(\varepsilon) ) R^2 S \log N.$

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 {p} is an (odd) prime for which ${\left(\frac{D}{p}\right) = -1}$ . From Fermat’s little theorem we have

$\displaystyle D^{\frac{p-1}{2}} = -1 \hbox{ mod } p$

and therefore

$\displaystyle \sqrt{D}^p = - \sqrt{D} \hbox{ mod } p {\bf Z}[\sqrt{D}].$

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

$\displaystyle (m + n \sqrt{D})^p = m - n \sqrt{D} \hbox{ mod } p {\bf Z}[\sqrt{D}]$

for any integers {m,n} . In particular, from a {p} -adic point of view, the grid {E} lies close to the curve

$\displaystyle \{ (z_1,z_2): z_1^p - z_2 = 0 \}.$

If the grid {E} happened to lie exactly on this curve, then the lattice ${T\Lambda}$ would degenerate (contradicting Lemma 1) and {M} would now vanish. This of course does not happen; but the fact that {E} is {p} -adically close to the curve will force {M} to be {p} -adically close to zero, or equivalently give a useful lower bound on ${v_p(M)}$ :

Lemma 3 (Non-Archimedean bound) If ${\varepsilon^{-2} \leq p \leq \varepsilon R}$ and ${\left(\frac{D}{p}\right) = -1}$ , then
$\displaystyle v_p(M) \geq (1-O(\varepsilon)) \frac{R^2 S }{p}.$

We prove this bound later in the post. Inserting Lemma 2 and Lemma 3 into (3) and cancelling the common factor of ${R^2 S}$ , we obtain (2) as required.

— 2. Injectivity —

Now we establish Lemma 1. 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 {E} is an ${N \times N}$ grid, a direct application of the Schwartz–Zippel lemma tells us that a non-zero degree {d} polynomial can only vanish on at most {dN} of the points of {E} . This is not directly useful for us because the degree of {P} can be as large as ${R+S \asymp N^{2 - O(\varepsilon)}}$ , and so this does not prevent {P} from vanishing on all of {E} . And indeed, if we replaced {E} by the standard ${N \times N}$ grid

$\displaystyle E' := \{ (m, n): 0 \leq m,n < N \}$

then the degree {N} polynomial ${\prod_{0 \leq n < N} (z_2-n)}$ already vanishes on {E'} , as does all of the horizontal derivatives of this polynomial. So to proceed here we have to somehow use the fact that the grid {E} is “skew”; in particular we will take advantage of the fact that the ${N^2}$ second coordinates ${m - n \sqrt{D}}$ of this grid are all distinct, in contrast to the situation with {E'} .

Let ${P(z_1,z_2)}$ be a non-zero polynomial of degree less than {(R,S)} . To avoid technicalities we will assume that {P} is irreducible, and not solely a function of ${z_2}$ , so that ${\partial_{z_1} P}$ is not identically zero; the general case is treated in Ben Green’s note.

Suppose that {P} vanishes horizontally to order at least {J} at a point ${(z_1^0, z_2^0)}$ , then by performing a Taylor expansion around that point we can factor {P} as

$\displaystyle P(z_1,z_2) = (z_1 - z_1^0)^J Q(z_1,z_2) + (z_2 - z_2^0) U(z_1,z_2)$

for some polynomials {Q,U} of degree less than {(R-J,S)} and {(R,S-1)} respectively. If we take one derivative in the ${z_1}$ direction, then ${\partial_{z_1} P(z_1,z_2)}$ vanishes to order at least {J-1} at the same point, thus

$\displaystyle \partial_{z_1} P(z_1,z_2) = (z_1 - z_1^0)^{J-1} \tilde Q(z_1,z_2) + (z_2 - z_2^0) \tilde U(z_1,z_2)$

for some polynomials ${\tilde Q,\tilde U}$ of degree less than {(R-J-1,S)} and {(R-1,S-1)} respectively.

Since {P} and ${\partial_{z_1} P}$ both vanish to high order at ${(z_1^0, z_2^0)}$ , it is natural to look at the horizontal resultant ${\mathrm{Res}_{z_1}(P, \partial_{z_1} P) \in {\bf Z}[z_2]}$ of these two polynomials (which we now view as ${{\bf Z}[z_2]}$ -valued polynomials of ${z_1}$ ), defined as the determinant of the {(R-1) + (R-2)} polynomials

$\displaystyle P, z_1 P, \dots, z_1^{R-2} P, \partial_{z_1} P, z_1 \partial_{z_1} P, \dots, z_1^{R-3} \partial_{z_1} P$

each of which we view as vectors of {(R-1) + (R-2)} coefficients in ${{\bf Z}[z_2]}$ . Note that all the polynomials in this list take the form

$\displaystyle (z_1 - z_1^0)^{J-1} Q'(z_1,z_2) + (z_2 - z_2^0) U'(z_1,z_2)$

for some polynomials {Q', U'} of degree less than {(2R-J, S)} and {(2R, S)} 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 {k} expressions of the form ${(z_1 - z_1^0)^{J-1} Q'(z_1,z_2)}$ and {(R-1)+(R-2)-k} expressions of the form ${(z_2 - z_2^0) U'(z_1,z_2)}$ . As {Q'} is of degree less than {2R-J} in the ${z_1}$ variable, any contribution with {k > 2R-J} must vanish, and so we need to have at least {J-O(1)} factors of the form ${(z_2 - z_2^0) U'(z_1,z_2)}$ . We conclude that the resultant ${\mathrm{Res}_{z_1}(P, \partial_{z_1} P)}$ is divisible by ${(z_2 - z_2^0)^{J - O(1)}}$ .

On the other hand, a straightforward degree count shows that this resultant is a polynomial of ${z_2}$ of degree {O(RS)} . The hypothesis that {P} is irreducible and ${\partial_{z_1} \not \equiv 0}$ implies that the resultant is non-zero. So if {P} vanished to order at least {N} at each of the ${N^2}$ points ${(z_1^0, z_2^0)}$ of {E} , then the resultant would have to be divisible by ${N^2}$ different factors of ${(z_2 - z_2^0)^{J - O(1)}}$ (here we use the fact that the ${z_2^0}$ are all distinct), leading to the inequality

$\displaystyle O(RS) \geq N^2 (J - O(1))$

which is false for ${\varepsilon}$ small enough by the choice of {N} , {J} , {R} , and {S} . This proves Lemma 1.

— 3. Archimedean bound —

The Archimedean bound is straightforward. For any basis ${v_1,\dots,v_{RS}}$ of ${T\Lambda}$ , the quantity {M} is simply the square of the {RS} -dimensional parallelepiped with edges ${v_1,\dots,v_{RS}}$ , viewed as elements of ${{\bf C}^{J \times N^2} = {\bf R}^{2J \times N^2}}$ . By the Hadamard’s inequality, we thus have

$\displaystyle |M| \leq \prod_{i=1}^{RS} \| v_i \|^2$

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

$\displaystyle \log |M| \leq 2 \sum_{i=1}^{RS} \log \|v_i\|.$

Applying the sampling operator {T} to the monomial basis ${z_1^i z_2^j}$ , we thus have

$\displaystyle \log |M| \leq 2 \sum_{0 \leq i < R} \sum_{0 \leq j < S} \log \|T (z_1^i z_2^j) \|.$

Each point in the grid {E} is of size ${O( N \sqrt{D} ) = O( N^{1+O(\varepsilon)})}$ by choice of {N} . Every component of ${T(z_1^i z_2^j)}$ then has size ${O( R^J O(N^{1+O(\varepsilon)})^{i+j})}$ , and so

$\displaystyle \|T (z_1^i z_2^j) \| \ll N^{O(1)} R^J O(N^{1+O(\varepsilon)})^{i+j}$

or on taking logs and using the construction of {R}

$\displaystyle \log \|T (z_1^i z_2^j) \| \leq (1 + O(\varepsilon)) (i+j) \log N + O( J \log N ).$

Summing the arithmetic series in {i,j} we obtain

$\displaystyle \log |M| \leq (1 + O(\varepsilon)) (R+S) RS \log N + O( R S J \log N ).$

By construction we have ${R+S = (1+O(\varepsilon)) R}$ and ${J = O(\varepsilon R)}$ , so Theorem 2 follows.

— 4. Non-Archimedean bound —

Let {p} be a prime with ${\varepsilon^{-2} \leq p \leq \varepsilon R}$ and ${\left(\frac{D}{p}\right) = -1}$ . As discussed in the high-level overview, the polynomial ${z_1^p - z_2}$ takes values in ${p {\bf Z}[\sqrt{D}]}$ on the entire grid {E} . This has the following consequence: any polynomial ${P(z_1,z_2)}$ which is divisible by ${(z_1^p-z_2)^k}$ for some {k} will take values in ${p^k {\bf Z}[\sqrt{D}]}$ . Furthermore, the same is true for any iterated ${z_1}$ -derivative of {P} : note from the chain rule that applying ${\partial_{z_1}}$ to ${(z_1^p-z_2)^k}$ will lower the exponent {k} by {1} , thus removing one factor of {p} , but also produce an additional factor of ${p z_1^{p-1}}$ which restores that missing factor. We conclude that every coefficient of {TP} then lies in ${p^k {\bf Z}[\sqrt{D}]}$ .

To use this, we replace the standard monomial basis ${z_1^i z_2^j}$ , ${0 \leq i < R}$ , ${0 \leq j < S}$ of ${\Lambda}$ with the modified basis

$\displaystyle z_1^{i - p k_{i,j}} (z_1^p - z_2)^{k_{i,j}} z_2^j$

for ${0 \leq i < R}$ , ${0 \leq j < S}$ , where

$\displaystyle k_{i,j} := \min ( \lfloor \frac{i}{p} \rfloor, S-1 - j ).$

Note that such a basis element still lies in ${\Lambda}$ , and is of the form ${z_1^i z_2^j}$ plus a linear combination of monomials that are lower than this in the lexicographical ordering. So this modified basis remains a basis of ${\Lambda}$ . Applying {T} , we see from the previous discussion that each of the basis elements is divisible by ${p^{k_{i,j}}}$ . Computing the Gram matrix in this basis, we conclude that

$\displaystyle v_p(M) \geq 2 \sum_{0 \leq i < R} \sum_{0 \leq j < S} k_{i,j}.$

For ${j \leq (1 - \varepsilon) S}$ , one has ${k_{i,j} = (1 + O(\varepsilon)) \frac{i}{p} - O(1)}$ by the hypotheses on {p} and the choice of {R,S} . We thus have

$\displaystyle v_p(M) \geq 2 (1 - O(\varepsilon)) S \sum_{0 \leq i < R} \frac{i}{p} - O(RS)$

and Theorem 3 then follows by summing the arithmetic series and noting that the second error term is negligible.

注意转载授权 · 阅读原文

Terence Tao 梳理 Green 对无 Siegel 零点证明的简化版本

摘要

正文 · 原文

引用原文