21.13.01 · number-theory / dirichlet-l-functions-characters

Zero-Free Regions for Dirichlet L-Functions and Exceptional (Siegel) Zeros

shipped3 tiersLean: none

Anchor (Master): Davenport 2000 *Multiplicative Number Theory* 3e §14 (the full zero-free-region argument, the Deuring-Heilbronn phenomenon, Siegel's theorem stated); Iwaniec-Kowalski 2004 *Analytic Number Theory* (AMS Colloquium Publications 53) §5.9 (zero-free regions and the exceptional zero); Montgomery-Vaughan 2007 *Multiplicative Number Theory I* §11; Page 1935 *Proc. London Math. Soc.* (2) 39; Landau 1918 *Math. Z.* 1 (the real-zero repulsion); Siegel 1935 *Acta Arithmetica* 1 (the ineffective lower bound $L(1, \chi) \gg_\varepsilon q^{-\varepsilon}$)

Intuition Beginner

The prime number theorem says the primes thin out at a predictable rate, and the proof rests on a single geometric fact: the zeros of the Riemann zeta function all sit strictly to the left of a certain vertical line. The further left they stay, the better the prime count is controlled. A zero-free region is a band, hugging the line where the real part equals one, inside which you can promise there are no zeros at all. The wider the band, the sharper the error term in the prime count.

Dirichlet's $L$ -functions are the character-twisted cousins of zeta, and they govern primes sorted into arithmetic progressions — primes that leave a fixed remainder when divided by a fixed number. Each one comes with its own zero-free band, and for almost all of them the band looks exactly like zeta's: a clean strip with no zeros. The proof uses a small trick. A certain sum of cosines, $3 + 4 cos θ + cos 2 θ$ , is never negative; feeding this positivity into the $L$ -functions pushes the zeros away from the edge.

There is one stubborn exception. For the special $L$ -functions built from a real-valued character — the ones that only ever output $+ 1$ , $- 1$ , or $0$ — the cosine trick can fail to rule out a single real zero sitting alarmingly close to one. This lonely intruder is the exceptional zero, or Siegel zero. Nobody has ever found one, and most experts believe none exists, but no proof rules it out. Its possible presence is the reason our control over primes in progressions is weaker, and less honest, than the prime number theorem itself.

Visual Beginner

Picture the strip of the complex plane between the vertical lines "real part $= 0$ " and "real part $= 1$ ". The zeros of every $L$ -function live inside this strip. The zero-free region is the curved sliver carved out of the right edge: it touches the line "real part $= 1$ " and bows leftward as you climb, narrowing the higher you go, following the shape "real part $> 1 - c / lo g (height)$ ".

character type	zero-free band near $s = 1$	exceptional real zero?
complex $χ$	full band, no exceptions	impossible
real $χ$ , generic	full band	none found
real $χ$ , worst case	band minus one point	one possible $β$ near $1$

The single dot labelled $β$ on the real axis, just inside the right edge, is the whole story. Every other zero is safely pushed back; this one might creep up toward $1$ , and that is the gap between what we can prove and what we believe.

Worked example Beginner

See why the number $3 + 4 cos θ + cos 2 θ$ is never negative — the inequality that drives the whole zero-free region.

Step 1. Rewrite the last term using the double-angle rule $cos 2 θ = 2 cos^{2} θ - 1$ . Substituting, $$ 3 + 4\cos\theta + \cos 2\theta = 3 + 4\cos\theta + 2\cos^2\theta - 1 = 2 + 4\cos\theta + 2\cos^2\theta. $$

Step 2. Factor out the $2$ and recognise a perfect square: $$ 2 + 4\cos\theta + 2\cos^2\theta = 2\big(1 + 2\cos\theta + \cos^2\theta\big) = 2\big(1 + \cos\theta\big)^2. $$

Step 3. A real number squared is never negative, and $2$ times a non-negative number stays non-negative. So $3 + 4 cos θ + cos 2 θ = 2 (1 + cos θ)^{2} \geq 0$ for every angle $θ$ , with equality exactly when $cos θ = - 1$ .

What this tells us: this one line of algebra is the engine of the zero-free region. The weights $3, 4, 1$ are chosen so that the combination collapses to a perfect square. When the same weights are applied to the logarithms of three $L$ -functions, the non-negativity says a zero too close to the line "real part $= 1$ " would force an impossible negative quantity — and that contradiction is what keeps the zeros away.

Check your understanding Beginner

Exercise (easy, true-false).

True or false: the exceptional (Siegel) zero, if it exists, can occur for a Dirichlet $L$ -function built from a complex-valued character.

Hint

The cosine trick rules out near-line zeros completely for complex characters; the loophole is only for real-valued characters, where a character equals its own conjugate.

Answer

False. The exceptional zero can only arise for a real character — one whose values are $+ 1$ , $- 1$ , or $0$ . For a complex character the zero-free band has no exceptions, because the conjugate character supplies an extra factor that the cosine argument uses to push every near-line zero away. Feedback-correct: the real-character case is exactly where the standard argument loses its grip, leaving a single possible real zero. Feedback-wrong: if you answered true, recall that a complex $χ$ and its conjugate $\overline{χ}$ are different characters, and that pairing is what closes the loophole.

Exercise (easy, multiple choice).

A wider zero-free region for an $L$ -function translates directly into which improvement?

A. More primes overall
B. A smaller error term in counting primes in arithmetic progressions
C. A proof of the Riemann hypothesis
D. Faster multiplication of large numbers

Hint

Zeros contribute oscillating corrections to the prime count; the further left they are forced, the smaller those corrections.

Answer

B. Each zero contributes an oscillating term whose size grows with its real part. A wider zero-free band forces every zero further left, shrinking those corrections and tightening the error term in the count of primes in each residue class. Feedback-correct: this is precisely the mechanism behind the prime number theorem for arithmetic progressions and its error estimate. Feedback-wrong: the Riemann hypothesis would put all zeros on a single line — far stronger than any known zero-free region, which only carves out a thin curved sliver.

Formal definition Intermediate+

Throughout, $s = σ + i t$ is a complex variable, $χ$ is a Dirichlet character modulo $q$ as in 21.03.02, $L (s, χ) = \sum_{n} χ (n) n^{- s}$ is its Dirichlet $L$ -function with Euler product $\prod_{p} (1 - χ (p) p^{- s})^{- 1}$ for $σ > 1$ , and $Λ$ is the von Mangoldt function of 21.12.01. We follow Davenport ^{[Davenport §14]}.

Definition (zero-free region). A zero-free region for the family ${L (s, χ) : χ mod q}$ is a subset $R \subset C$ , depending on $q$ , such that $L (s, χ) \neq = 0$ for every non-principal $χ$ modulo $q$ and every $s \in R$ . The classical (de la Vallée Poussin) region is, for an absolute constant $c > 0$ , $$ R_c = \Big{ s = \sigma + it : \sigma > 1 - \frac{c}{\log!\big(q(|t| + 2)\big)} \Big}, $$ with the single possible exception, when $χ$ is real, of one real zero in $R_{c}$ . The factor $q (∣ t ∣ + 2)$ is the analytic conductor: it bundles the modulus $q$ and the height $∣ t ∣$ into the one quantity controlling the local density of zeros.

Definition (logarithmic derivative). For $σ > 1$ , twisting the identity of 21.12.01, $$ -\frac{L'(s, \chi)}{L(s, \chi)} = \sum_{n=1}^{\infty} \frac{\Lambda(n)\chi(n)}{n^{s}}, $$ the Dirichlet series whose real part the cosine inequality will be applied to. Its real part for real $s = σ$ is $\sum_{n} Λ (n) χ (n) n^{- σ}$ , with terms of mixed sign once $χ$ is non-principal.

Definition (exceptional / Siegel zero). Fix the constant $c$ of $R_{c}$ . A real non-principal character $χ$ modulo $q$ is said to have an exceptional zero (or Siegel zero, or Landau-Siegel zero) if $L (s, χ)$ has a real zero $β$ with $$ 1 - \frac{c}{\log q} < \beta < 1. $$ Such a $β$ is necessarily simple, real, and unique for the given $χ$ (proved below). The quantity $1 - β > 0$ measures how far $β$ sits from the edge; the smaller it is, the more damage $β$ does to prime estimates.

The notation $L (s, χ)$ , the character $χ$ , the conductor $q$ , the variable $s = σ + i t$ , the von Mangoldt $Λ$ , the exceptional zero $β$ , and the symbol $ℜ$ for real part are recorded in _meta/NOTATION.md. A character is real if $χ = \overline{χ}$ , equivalently $χ^{2} = χ_{0}$ ; otherwise it is complex.

Counterexamples to common slips

"Every Dirichlet $L$ -function can have an exceptional zero." Only real characters can. For complex $χ$ the conjugate factor $L (s, \overline{χ})$ enters the cosine argument symmetrically and forces a genuine zero-free region with no exception. The loophole is strictly a feature of $χ = \overline{χ}$ .
"An exceptional zero would contradict the non-vanishing $L (1, χ) \neq = 0$ of 21.03.02." It would not: $β < 1$ strictly, so $L (β, χ) = 0$ is consistent with $L (1, χ) \neq = 0$ . The exceptional zero sits just inside the edge, not on it; what it threatens is the width of the zero-free band, not the boundary non-vanishing.
"Page's bound $β < 1 - c / q$ rules the exceptional zero out." It only bounds $β$ away from $1$ by an amount shrinking like $q^{- 1/2}$ , far weaker than the $q^{- ε}$ of Siegel. Page's bound is effective but feeble; Siegel's is strong but ineffective. Neither eliminates $β$ .

Key theorem with proof Intermediate+

The signature result is the de la Vallée Poussin zero-free region for $L (s, χ)$ , obtained by the same $3 + 4 cos θ + cos 2 θ$ device that handles $ζ$ , now applied to the product $L (s, χ_{0}) L (s, χ) L (s, χ^{2})$ with exponents $3, 4, 1$ .

Theorem (zero-free region for $L (s, χ)$ ). There is an absolute constant $c > 0$ such that for every Dirichlet character $χ$ modulo $q$ , the function $L (s, χ)$ has no zero in the region $σ > 1 - c / lo g (q (∣ t ∣ + 2))$ , with the single exception, when $χ$ is real, of at most one real zero $β$ .

Proof. For $σ > 1$ and any character $ψ$ modulo $q$ , expand the logarithmic derivative: $$ -\Re\frac{L'(\sigma + it, \psi)}{L(\sigma + it, \psi)} = \sum_{n} \frac{\Lambda(n)\Re\big(\psi(n) n^{-it}\big)}{n^{\sigma}} = \sum_{n} \frac{\Lambda(n)}{n^{\sigma}}\Re\big(\psi(n) e^{-it\log n}\big). $$ Write $ψ (n) e^{- i t l o g n} = e^{i θ_{n}}$ in modulus-one cases; for principal arguments set $θ_{n} = - t lo g n$ . Apply the pointwise inequality $3 + 4 cos θ + cos 2 θ \geq 0$ , established as $2 (1 + cos θ)^{2} \geq 0$ , with the three characters $χ_{0}$ , $χ$ , $χ^{2}$ at the arguments $0$ , $t$ , $2 t$ . Concretely, since $ℜ (χ_{0} (n)) = 1$ , $ℜ (χ (n) n^{- i t}) = cos θ_{n}$ , and $ℜ (χ^{2} (n) n^{- 2 i t}) = cos 2 θ_{n}$ for the natural choice $θ_{n} = ar g χ (n) - t lo g n$ at prime powers coprime to $q$ , $$ 3\Big(-\Re\tfrac{L'}{L}(\sigma, \chi_0)\Big) + 4\Big(-\Re\tfrac{L'}{L}(\sigma + it, \chi)\Big) + \Big(-\Re\tfrac{L'}{L}(\sigma + 2it, \chi^2)\Big) = \sum_n \frac{\Lambda(n)}{n^\sigma}\big(3 + 4\cos\theta_n + \cos 2\theta_n\big) \geq 0. $$

Now suppose, toward a contradiction, that $L (β + iγ, χ) = 0$ with $β$ close to $1$ and $γ \neq = 0$ (the complex case). Near a zero $ρ = β + iγ$ , the partial-fraction expansion of $L^{'} / L$ inherited from the Hadamard product gives $- ℜ \frac{L ^{'}}{L} (σ + iγ, χ) \leq - \frac{1}{σ - β} + C lo g (q (∣ γ ∣ + 2))$ for an absolute $C$ , the negative term coming from the pole at $ρ$ . The principal factor contributes $- ℜ \frac{L ^{'}}{L} (σ, χ_{0}) \leq \frac{1}{σ - 1} + C lo g q$ from the pole of $L (s, χ_{0})$ at $s = 1$ . The third factor $L (s, χ^{2})$ contributes only $\leq C lo g (q (∣ γ ∣ + 2))$ (no pole, since $χ^{2}$ may be principal but the pole there is harmless after accounting). Substituting into the non-negativity, $$ 0 \leq \frac{3}{\sigma - 1} - \frac{4}{\sigma - \beta} + C'\log!\big(q(|\gamma| + 2)\big). $$ Set $σ = 1 + δ / lo g (q (∣ γ ∣ + 2))$ for a small constant $δ$ . The inequality reads $\frac{4}{σ - β} \leq \frac{3}{σ - 1} + C^{'} lo g (\dots)$ . If $β$ were closer to $1$ than $1 - c / lo g (q (∣ γ ∣ + 2))$ , the term $4/ (σ - β)$ would exceed the right side for suitable $δ$ and small $c$ , a contradiction. Optimising $δ$ yields $β < 1 - c / lo g (q (∣ γ ∣ + 2))$ , the region.

The argument needs $γ \neq = 0$ to place $χ$ at the displaced point $σ + i t$ with $t = γ$ while keeping $χ_{0}$ at the real pole. When $χ$ is complex and $γ = 0$ , the conjugate $\overline{χ} \neq = χ$ supplies an independent factor: applying the device to $L (σ, χ_{0})^{3} L (σ, χ)^{4} L (σ, χ^{2})$ with $χ^{2}$ non-principal still produces the bound, so even real zeros are excluded. When $χ$ is real and $γ = 0$ , the character $χ^{2} = χ_{0}$ is principal, and the third factor $L (σ, χ_{0})$ contributes a second pole at $s = 1$ rather than a bounded term. The inequality degrades to $0 \leq \frac{3 + 1}{σ - 1} - \frac{4}{σ - β} + C^{'} lo g q$ , whose pole budget no longer forces $β$ away from $1$ . This single failure is the exceptional real zero. $□$

Bridge. This zero-free region builds toward the prime number theorem in arithmetic progressions and appears again in the Siegel-Walfisz theorem of 21.13.03, where the de la Vallée Poussin error $ψ (x; q, a) = x / φ (q) + O (x exp (- c lo g x))$ is exactly the analytic payoff of the band proved here. The foundational reason the argument works is the perfect square $2 (1 + cos θ)^{2}$ : the weights $3, 4, 1$ are forced by the requirement that the constant term dominate, and this is exactly the device that gave $ζ (1 + i t) \neq = 0$ in 21.12.01, now twisted by $χ$ . The central insight is that complex characters carry their own conjugate partner, so the pole budget closes; real characters do not, so one pole leaks and the method generalises only up to a single possible real zero. Putting these together, the zero-free region is the bridge from the boundary non-vanishing $L (1, χ) \neq = 0$ of 21.03.02 to a quantitative interior statement, and the exceptional zero is precisely the place where that bridge has a missing plank.

Exercises Intermediate+

Exercise 3 (medium, symbolic).

Twist the identity of 21.12.01 to show that for $σ > 1$ , $- L^{'} (s, χ) / L (s, χ) = \sum_{n} Λ (n) χ (n) n^{- s}$ , where $Λ$ is the von Mangoldt function.

Hint

Take $lo g$ of the Euler product $L (s, χ) = \prod_{p} (1 - χ (p) p^{- s})^{- 1}$ , expand $- lo g (1 - z)$ , and differentiate term by term.

Answer

For $σ > 1$ the Euler product converges with every factor non-zero, so $$ \log L(s, \chi) = -\sum_p \log\big(1 - \chi(p) p^{-s}\big) = \sum_p \sum_{m \geq 1} \frac{\chi(p)^m p^{-ms}}{m} = \sum_p \sum_{m \geq 1} \frac{\chi(p^m) p^{-ms}}{m}, $$ using complete multiplicativity $χ (p)^{m} = χ (p^{m})$ . Differentiating term by term (justified by local uniform absolute convergence, dominated by $\sum_{n} (lo g n) n^{- σ} < \infty$ ), $$ \frac{L'(s, \chi)}{L(s, \chi)} = -\sum_p \sum_{m \geq 1} (\log p)\chi(p^m) p^{-ms}. $$ Reindex by $n = p^{m}$ : the coefficient of $n^{- s}$ is $(lo g p) χ (n) = Λ (n) χ (n)$ at prime powers and $0$ elsewhere. Negating gives $- L^{'} / L = \sum_{n} Λ (n) χ (n) n^{- s}$ . Rubric: full credit for the log-Euler expansion, term-by-term differentiation, and the reindexing collecting $Λ (n) χ (n)$ .

Exercise 4 (medium, symbolic).

Explain precisely why the trigonometric argument fails to exclude a real zero $β$ of $L (s, χ)$ when $χ$ is real. Identify the factor whose behaviour changes.

Hint

When $χ$ is real, $χ^{2} = χ_{0}$ . Track what happens to the third factor $L (s + 2 i t, χ^{2})$ when also $t = 0$ .

Answer

For a putative real zero one sets $t = γ = 0$ , so the three factors are evaluated on the real axis: $L (σ, χ_{0})^{3} L (σ, χ)^{4} L (σ, χ^{2})$ . When $χ$ is real, $χ^{2} = χ_{0}$ , so the third factor is also a principal-character $L$ -function with a pole at $s = 1$ . The pole budget becomes $\frac{3}{σ - 1}$ from the cube plus $\frac{1}{σ - 1}$ from the third factor, total $\frac{4}{σ - 1}$ , exactly matching the $\frac{4}{σ - β}$ deficit from the zero of $L (σ, χ)^{4}$ . The inequality $0 \leq \frac{4}{σ - 1} - \frac{4}{σ - β} + C lo g q$ no longer forces $β$ bounded away from $1$ , because the two singular terms can balance. For complex $χ$ , the third factor $χ^{2} \neq = χ_{0}$ has no pole, leaving $\frac{3}{σ - 1}$ against $\frac{4}{σ - β}$ , which does force the gap. Rubric: full credit for identifying $χ^{2} = χ_{0}$ , the second pole, and the balanced pole budget.

Exercise 5 (medium, symbolic).

Show that an exceptional zero $β$ of a real character $χ$ is simple (multiplicity one), using the non-negativity of the coefficients $Λ (n) (1 + χ (n)) \geq 0$ .

Hint

Consider the product $ζ (s) L (s, χ)$ , whose log-derivative has coefficients $Λ (n) (1 + χ (n)) \geq 0$ . A double zero at $β$ would force two pole-strength units against a limited budget.

Answer

The Dirichlet series $- \frac{d}{d s} lo g (ζ (s) L (s, χ)) = \sum_{n} Λ (n) (1 + χ (n)) n^{- s}$ has non-negative coefficients, since $χ (n) \in {- 1, 0, + 1}$ gives $1 + χ (n) \in {0, 1, 2}$ . Suppose $β$ is a zero of $L (s, χ)$ of multiplicity $k \geq 1$ . Then $ζ (s) L (s, χ)$ has a zero of multiplicity $k$ at $β$ (since $ζ (β) \neq = 0$ for $β$ real in $(0, 1)$ near $1$ ), so $- ℜ \frac{( ζ L ) ^{'}}{ζ L} (σ) \leq - \frac{k}{σ - β} + \frac{1}{σ - 1} + C lo g q$ . But the left side is $\geq 0$ by coefficient non-negativity. Letting $σ \to β^{+}$ with $β$ within $c / lo g q$ of $1$ , a multiplicity $k \geq 2$ makes $- \frac{k}{σ - β}$ dominate the single $+ \frac{1}{σ - 1}$ , forcing the right side negative — a contradiction. Hence $k = 1$ : $β$ is simple. Rubric: full credit for the non-negative-coefficient product, the pole comparison, and the contradiction for $k \geq 2$ .

Exercise 6 (hard, symbolic).

Prove Landau's repulsion in its simplest form: if $χ_{1}, χ_{2}$ are distinct real non-principal characters modulo $q_{1}, q_{2}$ with respective real zeros $β_{1}, β_{2}$ , then $min (β_{1}, β_{2}) < 1 - c / lo g (q_{1} q_{2})$ for an absolute $c$ . Conclude at most one exceptional zero per dyadic range of moduli.

Hint

Apply the non-negative-coefficient argument to $ζ (s) L (s, χ_{1}) L (s, χ_{2}) L (s, χ_{1} χ_{2})$ , whose log-derivative coefficients are $Λ (n) (1 + χ_{1} (n)) (1 + χ_{2} (n)) \geq 0$ .

Answer

Consider $F (s) = ζ (s) L (s, χ_{1}) L (s, χ_{2}) L (s, χ_{1} χ_{2})$ . Its log-derivative has coefficients $$ \Lambda(n)\big(1 + \chi_1(n)\big)\big(1 + \chi_2(n)\big) = \Lambda(n)\big(1 + \chi_1(n) + \chi_2(n) + \chi_1\chi_2(n)\big) \geq 0, $$ non-negative because each factor $1 + χ_{i} (n) \geq 0$ for real $χ_{i}$ . So $- ℜ \frac{F ^{'}}{F} (σ) = \sum_{n} Λ (n) (1 + χ_{1} (n)) (1 + χ_{2} (n)) n^{- σ} \geq 0$ for $σ > 1$ . Now $F$ has a simple pole at $s = 1$ (from $ζ$ ) and zeros at $β_{1}$ (from $L (s, χ_{1})$ ) and $β_{2}$ (from $L (s, χ_{2})$ ); when $χ_{1} χ_{2}$ is non-principal it contributes no pole. The pole-zero bookkeeping gives $$ 0 \leq -\Re\frac{F'}{F}(\sigma) \leq \frac{1}{\sigma - 1} - \frac{1}{\sigma - \beta_1} - \frac{1}{\sigma - \beta_2} + C\log(q_1 q_2). $$ If both $β_{1}, β_{2}$ exceeded $1 - c / lo g (q_{1} q_{2})$ , the two negative pole terms would each be at least comparable to $\frac{1}{σ - 1}$ for $σ = 1 + c / lo g (q_{1} q_{2})$ , so their sum exceeds the single positive $\frac{1}{σ - 1}$ plus the bounded $C lo g (q_{1} q_{2})$ , forcing the right side negative — contradiction. Hence at most one of $β_{1}, β_{2}$ can lie that close to $1$ : exceptional zeros repel. Summing the bound over a dyadic block $Q < q \leq 2 Q$ shows at most one modulus in the block carries an exceptional zero. Rubric: full credit for the non-negative product $F$ , the pole-zero inequality, and the two-pole contradiction.

Exercise 7 (hard, symbolic).

Using the Dirichlet class-number formula $L (1, χ_{D}) = π h (D) / (w ∣ D ∣)$ for the imaginary quadratic field of discriminant $D < 0$ (real character $χ_{D}$ , class number $h (D)$ , $w$ roots of unity), translate Siegel's lower bound $L (1, χ) ≫_{ε} q^{- ε}$ into a lower bound on $h (D)$ , and explain the word "ineffective".

Hint

Solve the class-number formula for $h (D)$ and substitute Siegel's bound with $q = ∣ D ∣$ . The ineffectivity is inherited from the proof, which assumes the existence of some exceptional zero to bound others.

Answer

From $L (1, χ_{D}) = π h (D) / (w ∣ D ∣)$ with $w \in {2, 4, 6}$ bounded, solving for the class number gives $h (D) = \frac{w ∣ D ∣}{π} L (1, χ_{D}) ≫ ∣ D ∣ L (1, χ_{D})$ . Substituting Siegel's bound $L (1, χ_{D}) ≫_{ε} ∣ D ∣^{- ε}$ (taking $q = ∣ D ∣$ ), $$ h(D) \gg_\varepsilon |D|^{1/2 - \varepsilon}, $$ the Siegel lower bound on class numbers of imaginary quadratic fields: the class number grows nearly like $∣ D ∣$ . Ineffective means the implied constant in $≫_{ε}$ cannot be computed from the proof for $ε < 1/2$ . Siegel's argument splits on whether some real character has an exceptional zero: if none does, one bound holds with explicit constants; if one does, that zero is used to bound $L (1, χ)$ for all other characters — but since we cannot decide which case holds (no exceptional zero is known), the constant is not extractable. The bound exists; the constant is invisible. Rubric: full credit for the class-number substitution, the exponent $1/2 - ε$ , and a correct account of ineffectivity as a case-split on an unknown zero.

Exercise 8 (hard, symbolic).

State Page's bound $β < 1 - c / (q lo g^{2} q)$ for the exceptional zero and contrast its effectivity with Siegel's bound. Which is stronger, and which can actually be used to compute?

Hint

Compare the gap $1 - β$ : Page gives $≫ q^{- 1/2} / lo g^{2} q$ , Siegel gives $≫_{ε} q^{- ε}$ . Larger gap is stronger. Effectivity is the opposite story.

Answer

Page's bound (1935). For a real non-principal $χ$ modulo $q$ , any exceptional zero satisfies $β < 1 - c / (q lo g^{2} q)$ with an absolute, computable $c$ ; equivalently $1 - β ≫ q^{- 1/2} / lo g^{2} q$ . **Siegel's bound (1935).** For every $ε > 0$ , $1 - β ≫_{ε} q^{- ε}$ , but the constant is ineffective for $ε < 1/2$ . Strength. Siegel is stronger: $q^{- ε}$ exceeds $q^{- 1/2}$ for any $ε < 1/2$ , so Siegel pushes $β$ much further from $1$ . Effectivity. Page is effective — its constant is explicit, so it can be plugged into a computation (for example, ruling out exceptional zeros for small $q$ numerically). Siegel cannot, because its proof case-splits on the unknown existence of an exceptional zero. The standard situation in analytic number theory: one has a weak effective bound (Page) and a strong ineffective bound (Siegel), and for genuinely uniform results one must often retreat to the effective but feeble Page-type estimate. Rubric: full credit for both statements, the correct strength comparison, and the correct effectivity contrast.

Advanced results Master

Theorem 1 (de la Vallée Poussin region, off-line case). For an absolute $c > 0$ and every non-principal $χ$ modulo $q$ , $L (σ + i t, χ) \neq = 0$ whenever $σ > 1 - c / lo g (q (∣ t ∣ + 2))$ and $∣ t ∣ \geq 1/ lo g q$ ^{[Davenport §14]}. The argument is the weighted log-derivative inequality with weights $3, 4, 1$ ; the displaced height $t$ keeps the three factors at distinct points, so all three contribute bounded-or-pole terms in the correct pattern and the perfect square $2 (1 + cos θ)^{2}$ forces the gap. The complex-character case ( $χ^{2} \neq = χ_{0}$ ) extends the conclusion to $t = 0$ , excluding real zeros entirely.

Theorem 2 (exceptional zero: existence dichotomy). For real non-principal $χ$ modulo $q$ , exactly one of two situations holds ^{[Iwaniec-Kowalski §5.9]}: either $L (s, χ)$ has no zero in $σ > 1 - c / lo g q$ , or it has a single simple real zero $β$ there and no other zero in the region. The zero $β$ is real because a non-real zero would come with its conjugate, and the pair would overspend the pole budget exactly as two real zeros do. Its simplicity is the non-negative-coefficient argument applied to $ζ (s) L (s, χ)$ .

Theorem 3 (Landau repulsion / Deuring-Heilbronn). If $L (s, χ_{1})$ has an exceptional zero $β_{1}$ very close to $1$ , then every other $L (s, χ)$ in a wide range of moduli has its zeros pushed further from $s = 1$ than the standard region guarantees, by a factor logarithmic in $1 - β_{1}$ ^{[Davenport §14]}. Quantitatively, no second zero $ρ$ of any $L (s, ψ)$ with $ψ$ modulo $q^{'} \leq q^{A}$ lies in $σ > 1 - c lo g (\frac{1}{( 1 - β _{1} ) l o g q}) / lo g (q^{'} (∣ t ∣ + 2))$ . The closer the exceptional zero approaches $1$ , the cleaner every other $L$ -function becomes — exceptional zeros are gregarious in their solitude.

Theorem 4 (Siegel's lower bound). For every $ε > 0$ there is an ineffective constant $C (ε) > 0$ with $L (1, χ) \geq C (ε) q^{- ε}$ for every real primitive $χ$ modulo $q$ ^{[Siegel 1935]}, equivalently $1 - β \geq C (ε) q^{- ε}$ . The proof, fully developed in 21.13.02, picks an auxiliary real character $χ_{1}$ with an exceptional zero as close to $1$ as the hypothesis permits and uses the non-negativity of the coefficients of $ζ (s) L (s, χ) L (s, χ_{1}) L (s, χ χ_{1})$ to bound $L (1, χ)$ from below — the ineffectivity entering because the choice of $χ_{1}$ depends on an existence that cannot be witnessed.

Theorem 5 (class number of imaginary quadratic fields). For the fundamental discriminant $D < 0$ with associated real character $χ_{D}$ (the Kronecker symbol), the Dirichlet class-number formula $h (D) = w ∣ D ∣ L (1, χ_{D}) / (2 π)$ converts Siegel's bound into $h (D) ≫_{ε} ∣ D ∣^{1/2 - ε}$ ^{[Davenport §14]}. This was the original motivation: Gauss conjectured $h (D) \to \infty$ , and the exceptional-zero machinery is precisely what governs how slowly the smallest class numbers can grow. The list of imaginary quadratic fields with $h (D) = 1$ is finite (the nine Heegner discriminants), but Siegel's ineffectivity is why determining that list required separate effective methods (Heegner, Baker, Stark).

Synthesis. The zero-free region is the foundational reason the prime number theorem in arithmetic progressions is a theorem with an error term rather than a bare asymptotic, and the exceptional zero is the single structural flaw running through the whole edifice. The central insight is that the $3 + 4 cos θ + cos 2 θ = 2 (1 + cos θ)^{2}$ device, which gave $ζ (1 + i t) \neq = 0$ in 21.12.01, generalises to $L (s, χ)$ verbatim for complex characters but leaves one plank missing for real characters: putting these together, the conjugate symmetry $χ \leftrightarrow \overline{χ}$ that closes the pole budget for complex $χ$ has no analogue when $χ = \overline{χ}$ , and this is exactly the gap the Siegel zero inhabits. The bridge is everywhere the pole budget: the principal factor's pole at $s = 1$ is the positive currency, each zero is a debit, and the cosine weights $3, 4, 1$ balance the books — except in the one configuration where $χ^{2} = χ_{0}$ doubles the credit and lets a single real zero slip through.

This is dual to the class-number story: the same $L (1, χ)$ that measures the width of the zero-free band measures the size of $h (D)$ , so the exceptional zero near $1$ is precisely a small class number, and Siegel's ineffective bound is the analytic shadow of Gauss's effectively-open class-number problem. The central insight recurs across the theory: Landau's repulsion, the Deuring-Heilbronn phenomenon, and the Siegel-Walfisz uniformity all turn on the same fact — that exceptional zeros, if they exist at all, must be rare and lonely, and the entire quantitative theory of primes in progressions is written in the conditional mood they impose.

Full proof set Master

Proposition 1 (the engine inequality). For all real $θ$ , $3 + 4 cos θ + cos 2 θ = 2 (1 + cos θ)^{2} \geq 0$ .

Proof. By the double-angle identity $cos 2 θ = 2 cos^{2} θ - 1$ , so $3 + 4 cos θ + cos 2 θ = 2 cos^{2} θ + 4 cos θ + 2 = 2 (cos θ + 1)^{2}$ . A real square is non-negative; multiplying by $2$ preserves this. Equality holds iff $cos θ = - 1$ . $□$

Proposition 2 (twisted log-derivative identity). For $Re (s) > 1$ and any Dirichlet character $χ$ modulo $q$ , $- L^{'} (s, χ) / L (s, χ) = \sum_{n} Λ (n) χ (n) n^{- s}$ .

Proof. The Euler product $L (s, χ) = \prod_{p} (1 - χ (p) p^{- s})^{- 1}$ converges absolutely with non-zero factors for $σ > 1$ , so $lo g L (s, χ) = \sum_{p} \sum_{m \geq 1} χ (p^{m}) p^{- m s} / m$ using $χ (p)^{m} = χ (p^{m})$ . The differentiated series $\sum_{p} \sum_{m} (lo g p) χ (p^{m}) p^{- m s}$ converges locally uniformly, dominated in modulus by $\sum_{n} (lo g n) n^{- σ} < \infty$ , so term-by-term differentiation gives $L^{'} / L = - \sum_{p} \sum_{m} (lo g p) χ (p^{m}) p^{- m s}$ . Reindexing $n = p^{m}$ collects the coefficient $Λ (n) χ (n)$ , and negating yields the claim. $□$

Proposition 3 (exceptional zero is real and simple). If a real non-principal $χ$ modulo $q$ has a zero $β + iγ$ with $β > 1 - c / lo g q$ and $c$ small, then $γ = 0$ and the zero is simple.

Proof. The function $ζ (s) L (s, χ)$ has log-derivative coefficients $Λ (n) (1 + χ (n)) \geq 0$ for real $χ$ , since $1 + χ (n) \in {0, 1, 2}$ . Thus $- ℜ \frac{( ζ L ) ^{'}}{ζ L} (σ + i y) = \sum_{n} Λ (n) (1 + χ (n)) cos (y lo g n) n^{- σ}$ , and at $y = 0$ this equals $\sum_{n} Λ (n) (1 + χ (n)) n^{- σ} \geq 0$ . Suppose a non-real zero $β + iγ$ ( $γ \neq = 0$ ) existed in the region; then $β - iγ$ is also a zero (real $χ$ forces conjugate-symmetric zeros), and the pair contributes $- \frac{2 ( σ - β )}{( σ - β ) ^{2} + γ ^{2}}$ to $- ℜ \frac{( ζ L ) ^{'}}{ζ L} (σ)$ , which combined with the single pole $\frac{1}{σ - 1}$ of $ζ$ overspends the budget as $σ \to 1^{+}$ , contradicting non-negativity. Hence $γ = 0$ . For simplicity, a double real zero contributes $- \frac{2}{σ - β}$ , again exceeding $\frac{1}{σ - 1} + C lo g q$ for $β$ within $c / lo g q$ of $1$ , a contradiction; so the zero is simple. $□$

Proposition 4 (at most one exceptional modulus per range). There is an absolute $c$ such that for each $Q$ , at most one real primitive character of modulus $q \in (Q, 2 Q]$ has a zero $β > 1 - c / lo g Q$ .

Proof. Suppose $χ_{1}, χ_{2}$ are distinct real primitive characters of moduli $q_{1}, q_{2} \in (Q, 2 Q]$ with zeros $β_{1}, β_{2} > 1 - c / lo g Q$ . The product $F (s) = ζ (s) L (s, χ_{1}) L (s, χ_{2}) L (s, χ_{1} χ_{2})$ has non-negative log-derivative coefficients $Λ (n) (1 + χ_{1} (n)) (1 + χ_{2} (n)) \geq 0$ , and $χ_{1} χ_{2}$ is non-principal (distinct primitive real characters), so $F$ has a single pole at $s = 1$ and zeros at $β_{1}, β_{2}$ . Then $0 \leq - ℜ \frac{F ^{'}}{F} (σ) \leq \frac{1}{σ - 1} - \frac{1}{σ - β _{1}} - \frac{1}{σ - β _{2}} + C lo g Q$ . Choosing $σ = 1 + c / lo g Q$ , each $\frac{1}{σ - β _{i}} \geq \frac{1}{2} \frac{l o g Q}{c}$ while $\frac{1}{σ - 1} = \frac{l o g Q}{c}$ , so the two negative terms sum to at least $\frac{l o g Q}{c}$ , cancelling the positive pole and leaving $\leq C lo g Q$ , which is negative for $c$ small enough relative to the budget — a contradiction. Hence at most one such modulus exists in the range. $□$

Connections Master

The trigonometric engine $3 + 4 cos θ + cos 2 θ$ and the pole-budget method are imported directly from 21.12.01, where the same weights gave the zero-free line $ζ (1 + i t) \neq = 0$ for the Riemann zeta function; this unit twists that argument by a Dirichlet character, so the analytic mechanism is identical and only the conjugate-symmetry bookkeeping changes, opening the exceptional-zero loophole for real characters.
The boundary non-vanishing $L (1, χ) \neq = 0$ of 21.03.02 is the qualitative statement that this unit makes quantitative: where 21.03.02 proves the $L$ -function is non-zero on the line $σ = 1$ , the zero-free region proves it is non-zero in a band inside the line, and the exceptional zero is precisely the obstruction to making that band uniform across real characters.
The analytic continuation, functional equation, and Hadamard-product zero structure of $ζ$ from 06.01.16 supply the partial-fraction expansion of $L^{'} / L$ that converts a zero near $s = 1$ into a pole term in the log-derivative inequality; the completed $L$ -function and its zeros inherit exactly the entire-function order-one theory developed there for $ξ$ .
The effective zero-free region for $ζ$ of 21.12.03 is the sibling input: the constant $c$ and the analytic-conductor shape $1 - c / lo g (q (∣ t ∣ + 2))$ specialise to $ζ$ at $q = 1$ , and the present unit is the character-twisted generalisation feeding the Siegel-Walfisz theorem of 21.13.03 on PNT in arithmetic progressions uniform up to $q \leq (lo g x)^{A}$ .
Siegel's ineffective lower bound, stated here as Theorem 4, is proved in full in the sibling unit 21.13.02, which develops the auxiliary-character argument and the class-number formula in detail; the present unit supplies the zero-free-region and repulsion infrastructure that Siegel's proof consumes.

Historical & philosophical context Master

The zero-free region for the Riemann zeta function originates with Charles-Jean de la Vallée Poussin's 1896 proof of the prime number theorem, which isolated the inequality $3 + 4 cos θ + cos 2 θ \geq 0$ as the device guaranteeing $ζ (1 + i t) \neq = 0$ . The extension to Dirichlet $L$ -functions, and with it the prime number theorem for arithmetic progressions, followed the same year, but the real-character obstruction was understood only gradually. Edmund Landau, in a 1918 pair of papers ^{[Landau 1918]}, proved that real zeros of distinct real-character $L$ -functions repel one another, establishing that exceptional zeros — if they occur — must be rare; his work tied the question directly to the class numbers of imaginary quadratic fields, reviving a problem Gauss had posed in the Disquisitiones Arithmeticae of 1801.

Arnold Page's 1935 paper ^{[Page 1935]} gave the first effective bound on the exceptional zero, $β < 1 - c / q$ up to logarithmic factors, and in the same year Carl Ludwig Siegel ^{[Siegel 1935]} proved the stronger but ineffective $L (1, χ) ≫_{ε} q^{- ε}$ , yielding $h (D) ≫_{ε} ∣ D ∣^{1/2 - ε}$ for imaginary quadratic class numbers. The ineffectivity is structural: Siegel's argument bounds all $L (1, χ)$ in terms of a single character with the most extreme exceptional zero, whose existence cannot be decided. The contrast between Page's weak-but-computable and Siegel's strong-but-invisible bounds remains the defining tension of the subject, and the elimination of the exceptional zero — the so-called Landau-Siegel zero problem — is among the most resistant open questions in analytic number theory, equivalent in strength to deep statements about the distribution of primes in short intervals and the class numbers of quadratic fields. Davenport's Multiplicative Number Theory and Iwaniec-Kowalski's Analytic Number Theory are the canonical modern treatments.

Bibliography Master

@article{landau1918klassenzahl,
  author  = {Landau, Edmund},
  title   = {\"{U}ber die Klassenzahl imagin\"{a}r-quadratischer Zahlk\"{o}rper},
  journal = {Mathematische Zeitschrift},
  volume  = {1},
  pages   = {213--219},
  year    = {1918}
}

@article{page1935,
  author  = {Page, Arnold},
  title   = {On the number of primes in an arithmetic progression},
  journal = {Proceedings of the London Mathematical Society (2)},
  volume  = {39},
  pages   = {116--141},
  year    = {1935}
}

@article{siegel1935,
  author  = {Siegel, Carl Ludwig},
  title   = {\"{U}ber die Classenzahl quadratischer Zahlk\"{o}rper},
  journal = {Acta Arithmetica},
  volume  = {1},
  pages   = {83--86},
  year    = {1935}
}

@book{davenport2000mult,
  author    = {Davenport, Harold},
  title     = {Multiplicative Number Theory},
  edition   = {3},
  series    = {Graduate Texts in Mathematics},
  volume    = {74},
  publisher = {Springer-Verlag},
  year      = {2000},
  note      = {Revised by H. L. Montgomery; \S14}
}

@book{iwaniec-kowalski2004,
  author    = {Iwaniec, Henryk and Kowalski, Emmanuel},
  title     = {Analytic Number Theory},
  series    = {American Mathematical Society Colloquium Publications},
  volume    = {53},
  publisher = {American Mathematical Society},
  year      = {2004},
  note      = {\S5.9--5.10: exceptional zeros and Deuring-Heilbronn}
}

@book{montgomery-vaughan2007zfr,
  author    = {Montgomery, Hugh L. and Vaughan, Robert C.},
  title     = {Multiplicative Number Theory I: Classical Theory},
  series    = {Cambridge Studies in Advanced Mathematics},
  volume    = {97},
  publisher = {Cambridge University Press},
  year      = {2007},
  note      = {\S11: zero-free regions and the Siegel-Walfisz theorem}
}

Prerequisites

21.12.01
21.03.02
06.01.16

Tier anchors

beginner: Derbyshire 2003 *Prime Obsession* (Joseph Henry Press) Ch. 21-22 (the zeros as the wandering correction to the prime count, told for $\zeta$ and informally extended to the character-twisted $L$-functions); Soundararajan's expository lecture notes on the Siegel zero as the 'one zero that refuses to move' for the simplest visual picture
intermediate: Davenport 2000 *Multiplicative Number Theory* 3e (Springer GTM 74) §14 (the zero-free region for $L(s, \chi)$ via $3 + 4\cos\theta + \cos 2\theta$, the exceptional real zero, Page's bound, Landau's repulsion theorem); Montgomery-Vaughan 2007 *Multiplicative Number Theory I* (Cambridge UP) §11
master: Davenport 2000 *Multiplicative Number Theory* 3e §14 (the full zero-free-region argument, the Deuring-Heilbronn phenomenon, Siegel's theorem stated); Iwaniec-Kowalski 2004 *Analytic Number Theory* (AMS Colloquium Publications 53) §5.9 (zero-free regions and the exceptional zero); Montgomery-Vaughan 2007 *Multiplicative Number Theory I* §11; Page 1935 *Proc. London Math. Soc.* (2) 39; Landau 1918 *Math. Z.* 1 (the real-zero repulsion); Siegel 1935 *Acta Arithmetica* 1 (the ineffective lower bound $L(1, \chi) \gg_\varepsilon q^{-\varepsilon}$)

References

Davenport, H. — Multiplicative Number Theory · Springer Graduate Texts in Mathematics 74, 3rd edition (2000), revised by H. L. Montgomery. §14. Derives the zero-free region $\sigma > 1 - c/\log(q(|t| + 2))$ for $L(s, \chi)$ from the inequality $3 + 4\cos\theta + \cos 2\theta \geq 0$ applied to $\log|L(s, \chi_0)^3 L(s+it, \chi)^4 L(s+2it, \chi^2)|$, isolates the possible exceptional real zero $\beta$ of $L(s, \chi)$ for real $\chi$, proves Page's bound $\beta < 1 - c/\sqrt{q}$ and Landau's theorem that the product over real characters mod $q$ has at most one such zero, and states Siegel's ineffective lower bound $L(1, \chi) \gg_\varepsilon q^{-\varepsilon}$.
Page, A. — On the number of primes in an arithmetic progression · Proceedings of the London Mathematical Society (2) 39 (1935), 116-141. Proves that for real non-principal $\chi$ modulo $q$ the function $L(s, \chi)$ has at most one real zero $\beta$ in the region $\sigma > 1 - c/\log q$, satisfying $\beta < 1 - c'/\sqrt{q}\log^2 q$ (Page's bound), and that across all moduli at most one such exceptional zero arises in a dyadic range.
Landau, E. — Über die Klassenzahl imaginär-quadratischer Zahlkörper · Mathematische Zeitschrift 1 (1918), 213-219, and Göttinger Nachrichten (1918), 285-295. Establishes the repulsion of exceptional zeros: among the real characters modulo $q$ (or in a range of moduli) at most one $L(s, \chi)$ can have a zero exceptionally close to $s = 1$, because two such zeros would force a contradiction with the non-negativity of the coefficients of a product of $L$-functions.
Siegel, C. L. — Über die Classenzahl quadratischer Zahlkörper · Acta Arithmetica 1 (1935), 83-86. Proves the ineffective lower bound $L(1, \chi) \gg_\varepsilon q^{-\varepsilon}$ for real primitive $\chi$ modulo $q$, equivalently $1 - \beta \gg_\varepsilon q^{-\varepsilon}$ for the exceptional zero, with an implied constant that is not effectively computable for $\varepsilon < 1/2$. The class-number lower bound $h(-q) \gg_\varepsilon q^{1/2 - \varepsilon}$ for imaginary quadratic fields follows by the Dirichlet class-number formula.
Iwaniec, H. & Kowalski, E. — Analytic Number Theory · AMS Colloquium Publications 53 (2004), §5.9-5.10. The exceptional zero (Landau-Siegel zero), the Deuring-Heilbronn repulsion phenomenon, the effect of the exceptional zero on PNT in arithmetic progressions, and the connection to the class number of imaginary quadratic fields.
Montgomery, H. L. & Vaughan, R. C. — Multiplicative Number Theory I: Classical Theory · Cambridge Studies in Advanced Mathematics 97 (2007), §11. The zero-free region for Dirichlet $L$-functions, the exceptional zero, the Page-Landau-Siegel chain, and the Siegel-Walfisz theorem on PNT in arithmetic progressions uniform up to $q \leq (\log x)^A$.

Estimated time

beginner: 18m
intermediate: 46m
master: 90m