Siegel Zeros and Siegel–Walfisz

Every Dirichlet L-function L(s,\chi) comes with its own zero-free region, and — just as for \zeta — the size of that region is exactly what controls how evenly the primes fall into arithmetic progressions. For almost every character the region is clean. But there is one nagging possibility the theory has never been able to rule out: a single stray zero, sitting on the real axis, so close to s = 1 that it spoils the best bounds. This is the Siegel zero — also called the exceptional or Landau–Siegel zero — and the whole story of this page is our uneasy truce with it.

The truce has a name: the Siegel–Walfisz theorem, the strongest unconditional prime-number theorem for arithmetic progressions we possess. Its power and its frustrating weakness both trace back to a single phantom zero we cannot prove doesn't exist — and cannot, even in principle, locate if it does.

Zero-free regions, and the one gap that won't close

Recall the classical zero-free region: there is a constant c > 0 such that \zeta(s) has no zeros with

\sigma > 1 - \frac{c}{\log(|t| + 2)}, \qquad s = \sigma + it.

The proof — the famous "3 + 4\cos\theta + \cos 2\theta \ge 0" trick — carries over to L(s,\chi) almost verbatim. For a modulus q one gets: there is a constant c > 0 so that L(s,\chi) has no zeros in

\sigma > 1 - \frac{c}{\log\!\big(q(|t| + 2)\big)}

with one possible exception. The exception can only occur when t is essentially zero and \chi is a real (quadratic) primitive character — and then it is a single, simple, real zero \beta \in (0,1), extremely close to 1:

\beta > 1 - \frac{c}{\log q}.

That is the Siegel zero. If it exists, it lies just a whisker inside the line \Re(s) = 1, closer than the ordinary zero-free region would allow.

Why only real characters can misbehave

The lone-exception clause is not an accident of the proof — it is forced. The key inequality behind every zero-free region compares a character to its own square. Suppose L(\beta,\chi)=0 for a real \beta close to 1. The positivity argument controls this against the behaviour of \chi^2 at the point 2\beta - 1, near s = 1.

Siegel's theorem: pushing the zero back — ineffectively

The zero-free region only keeps \beta below 1 - c/\log q, which is agonisingly close to 1. Carl Ludwig Siegel proved, in 1935, a far stronger repulsion:

Notice the trade: q^{-\varepsilon} is a much bigger gap than 1/\log q, so the zero is pushed much further from 1. This looks like it should demolish the Siegel zero altogether. It does not — and the reason is the single most notorious feature of this whole subject.

The constant c(\varepsilon) is ineffective: Siegel's proof is a dichotomy. It argues, roughly, "either the Generalised Riemann Hypothesis-style bound already holds for all small moduli, or there exists some single exceptional character \chi_1 \pmod{q_1} with a very close zero — and if that one exists, use it as leverage to bound everyone else." Either branch yields the theorem, but the proof never tells you which branch is true, so it never reveals q_1, and the resulting c(\varepsilon) cannot be computed. We know a valid constant exists; we cannot write one down.

A worked example: how a Siegel zero fights the main term

Why does a zero near s = 1 hurt so much? Look at the explicit formula for the prime-counting sum of a real character. Writing \psi(x,\chi) = \sum_{n \le x} \Lambda(n)\chi(n), the explicit formula sums a term -x^{\rho}/\rho over the non-trivial zeros \rho of L(s,\chi):

\psi(x,\chi) = -\sum_{\rho} \frac{x^{\rho}}{\rho} + (\text{smaller terms}).

For a non-principal \chi there is no main term x (that only appears for \chi_0), so every zero contributes pure error. Now suppose there is a Siegel zero \beta = 1 - \delta with \delta tiny. Its single term is

-\frac{x^{\beta}}{\beta} \;=\; -\frac{x^{\,1-\delta}}{\,1-\delta\,} \;\approx\; -\,x\cdot x^{-\delta} \;=\; -\,x\,e^{-\delta \log x}.

Read off the sizes. If \delta \log x is small — which happens precisely when q is large relative to x, so that \delta \approx c/\log q is not much bigger than 1/\log x — then x^{-\delta} \approx 1, and this one term is of size nearly x. A single zero produces an error term as large as the entire main term of the principal character. That is the "conspiracy": the Siegel zero manufactures a term -x that biases \psi(x,\chi) systematically, and through it biases the count of primes in the progressions attached to \chi.

Numbers. Take x = 10^{12}, so \log x \approx 27.6. If \delta = 10^{-3} then x^{-\delta} = e^{-0.0276} \approx 0.973 — the rogue term is 0.973\,x, almost undiminished. Only once \delta grows to around 1/\log x \approx 0.036 does x^{-\delta} drop to e^{-1} \approx 0.37 and the term finally start to decay. This is exactly why the strength of the zero-free region — how far \beta is kept below 1 — is the strength of the prime-number theorem.

Seeing the near-cancellation

The curves below plot, against \log x, the size of the main term (1, in units of x) and the size of the Siegel-zero term x^{-\delta} = e^{-\delta \log x}, for a small \delta. For a long stretch the exceptional term shadows the main term almost exactly — the near-cancellation — and only peels away once \delta \log x becomes an appreciable fraction of 1. The smaller the zero's distance \delta from 1, the longer that shadow lasts.

Drag \delta toward 0 (a zero ever closer to 1) and watch the exceptional curve refuse to fall — the conspiracy persists across a wider and wider range of x.

The Siegel–Walfisz theorem

Feed Siegel's (ineffective) bound into the explicit formula and you get the best unconditional prime-number theorem for arithmetic progressions. For \gcd(a,q) = 1, let \pi(x;q,a) count primes p \le x with p \equiv a \pmod q.

Read it as: the primes up to x split equitably among the \varphi(q) allowed residue classes — each class gets its fair 1/\varphi(q) share of \operatorname{Li}(x) — with an error that beats every fixed power of \log x. The error x\exp(-c\sqrt{\log x}) is smaller than x/(\log x)^{B} for every B, which is exactly the strength needed to make the leading term dominate.

But look at the range of uniformity: q \le (\log x)^{A}. The modulus is allowed to grow only as slowly as a power of a logarithm of x. That is a punishingly small range — for x = 10^{100} and A = 10, it caps q at only about (230)^{10} \approx 4\times 10^{23}, laughably small next to x. This tiny range is the price of ineffectivity.

The error term from the explicit formula is controlled by x^{\beta}, and to make x^{\beta} = x\cdot x^{-\delta} genuinely smaller than the main term you need \delta \log x to be large — say larger than (A+1)\log\log x. Siegel guarantees \delta = 1 - \beta > c(\varepsilon)\,q^{-\varepsilon}, so

\delta \log x > c(\varepsilon)\,q^{-\varepsilon}\log x.

To force the right-hand side above the threshold you must keep q small enough that q^{\varepsilon} stays below \sim \log x — i.e. q \le (\log x)^{A} for a suitable A. And because c(\varepsilon) is unknown, you can't even name the threshold precisely; you can only say a bound exists for each fixed A. Push q larger — say q \le x^{1/2-\varepsilon}, which is what you'd want for real applications — and unconditional methods fail. That larger range is exactly what the Bombieri–Vinogradov theorem recovers on average over q, and what the Generalised Riemann Hypothesis would give for every individual q at a stroke.

Two traps live here. First, ineffective does not mean the constant is unknown "for now, until someone computes harder." Siegel's proof is structurally non-constructive: it splits into two cases and proves the theorem in each, without ever deciding which case we are in. So c(\varepsilon) — and hence the constant in Siegel–Walfisz — cannot be extracted from the proof at all. This is why you may never silently extend the range past q \le (\log x)^{A}: there is no known number to plug in.

Second — and this is the part students most often get backwards — Siegel zeros are not known to be impossible. The Generalised Riemann Hypothesis would rule them out instantly (a zero at \beta real and near 1 is off the critical line \Re(s) = 1/2). But GRH is unproven. As of today no one has shown that a single real primitive L-function is free of an exceptional zero. We conjecture Siegel zeros don't exist; we cannot prove it. All the ineffectivity above is the cost of hedging against a phantom we cannot exorcise.

The illusory world where a Siegel zero exists

Here is the strangest twist in the subject. Instead of assuming Siegel zeros away, ask: what if one really existed? A close real zero \beta = 1 - \delta would mean L(1,\chi) is tiny, which forces the character sum \chi to be biased in a very rigid way — the primes would "know about" \chi and arrange themselves to keep L(1,\chi) small. This is a real conspiracy: a pathological correlation between the primes and one quadratic character.

Astonishingly, such a conspiracy is so rigid that it would imply new theorems. Roger Heath-Brown showed in 1983 that if Siegel zeros exist (infinitely often, with \delta small enough) then the twin prime conjecture is true — there would be infinitely many primes p with p + 2 also prime. The illusory world with an exceptional zero is, in several respects, better behaved than the world we believe we live in: the bias forced by the zero over-regularises the primes into cooperating.

It is one of the eeriest situations in analytic number theory: a hypothetical object nobody believes exists, whose existence would nonetheless hand us results we cannot otherwise reach. The Siegel zero is the problem's ghost and its secret benefactor at once.

A small L(1,\chi) for the quadratic character \chi_d = \big(\tfrac{d}{\cdot}\big) is not just an abstract nuisance — by Dirichlet's class number formula it means the imaginary quadratic field \mathbb{Q}(\sqrt{d}) has a surprisingly small class number h(d). Ruling out Siegel zeros is therefore intimately tied to effective lower bounds on class numbers — the Gauss class number problem. Deuring, Heilbronn, Landau and Siegel's chain of arguments in the 1930s was precisely about showing h(d) \to \infty, and the ineffectivity of Siegel's constant is the same ineffectivity that long blocked a computable list of all imaginary quadratic fields of a given class number. Different corners of number theory, one phantom zero.