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

The Approximate Functional Equation, Analytic Conductor, and Convexity Bound

shipped3 tiersLean: none

Anchor (Master): Iwaniec-Kowalski 2004 *Analytic Number Theory* (AMS Colloquium 53) Ch. 5 (the canonical treatment of the analytic conductor and the approximate functional equation in the general $L$-function framework); Hardy-Littlewood 1916 *Acta Math.* 41 (originator of the approximate functional equation for $\zeta$); Phragmén-Lindelöf 1908 *Acta Math.* 31 (the convexity principle); Lindelöf 1908 *Bull. Sci. Math.* 32; Titchmarsh 1986 *The Theory of the Riemann Zeta-Function* (2nd ed., Oxford) Ch. 4-5; Iwaniec 2014 *Lectures on the Riemann Zeta Function* (AMS ULECT 62)

Intuition Beginner

The Riemann zeta function $ζ (s) = 1 + 1/ 2^{s} + 1/ 3^{s} + \dots$ only adds up to a finite value when the real part of $s$ is bigger than $1$ . But the most interesting questions about prime numbers live on the line where the real part of $s$ equals $1/2$ , far outside the region where the sum converges. So you face a basic practical problem: how big is $ζ$ at a point like $1/2 + 1000 i$ , where the defining sum is useless? You need a finite, computable expression that approximates the value.

The approximate functional equation is the answer. It says that even though the infinite sum does not converge, you can still recover the value by adding up only the first few hundred terms, then adding a second short sum that is a kind of mirror image of the first, weighted by a fixed twisting number. The two short sums together pin down the true value with a tiny error. The length of each sum is roughly the square root of the height $t$ you are evaluating at: to reach height $t$ , you need about $t / (2 π)$ terms in each piece, not infinitely many.

This idea exists because it converts a question about an infinite object into arithmetic with a finite list. Once you have a short finite expression, you can bound its size, and that bound is the convexity estimate: the value of $ζ$ on the critical line cannot grow faster than roughly the fourth root of the height.

Visual Beginner

A picture of the complex plane. The shaded region on the right, where the real part of $s$ is bigger than $1$ , is labelled "sum converges here." A vertical line at real part $1/2$ is drawn as the critical line. At a point high up that line, marked $1/2 + t i$ , two short horizontal brackets are drawn: a left bracket labelled "first sum, about $t$ terms" and a right bracket labelled "mirror sum, about $t$ terms, twisted by a fixed factor." An arrow connects the two brackets through a box labelled "tiny error."

The picture shows the trade you are making. You cannot sum forever, so you sum a little from the left, a little from the right, and accept a controlled error. The total length of computation grows like $t$ , which is what the analytic conductor measures.

Worked example Beginner

Estimate how many terms the approximate functional equation needs to evaluate $ζ$ at $s = 1/2 + 1000 i$ , and compare with the naive sum.

Step 1. The height is $t = 1000$ . The recipe says each of the two short sums runs up to about $N = t / (2 π)$ terms. Compute $t / (2 π) = 1000/6.2832 = 159.15$ .

Step 2. Take the square root: $N = 159.15 = 12.6$ . So each sum needs about $13$ terms, and the two sums together need about $26$ terms.

Step 3. Compare with trying to use the original definition. The defining sum does not even converge at real part $1/2$ , so the naive approach needs infinitely many terms and still fails. Even at a point where the sum did converge, you would need millions of terms to get three digits of accuracy.

What this tells us: the approximate functional equation replaces an impossible infinite computation with adding up about $26$ numbers. The work grows only like the square root of the height, which is why this single identity is the workhorse for every numerical and theoretical estimate of $ζ$ on the critical line.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $s = σ + i t$ with $σ, t \in R$ . The objects of study are $L$ -functions in the axiomatic shape shared by $ζ (s)$ and the Dirichlet $L$ -functions $L (s, χ)$ studied in 21.03.01 and 06.01.16.

Definition (functional-equation data). An $L$ -function in the present sense is a Dirichlet series $L (s) = \sum_{n \geq 1} a_{n} n^{- s}$ , absolutely convergent for $σ > 1$ , together with a completed function

Λ (s) := Q^{s} γ (s) L (s), γ (s) = j = 1 \prod d Γ_{R} (s + κ_{j}),

where $Q > 0$ is the conductor scale, the $κ_{j} \in C$ with $Re (κ_{j}) \geq 0$ are the local parameters, and $Γ_{R} (s) := π^{- s /2} Γ (s /2)$ is the real gamma factor. The data is required to satisfy the functional equation

Λ (s) = ε \overline{Λ} (1 - s), ∣ ε ∣ = 1,

where $\overline{Λ} (s) := \overline{Λ (\overset{s}{ˉ})}$ has coefficients $\overline{a_{n}}$ , and $ε$ is the root number. For $ζ$ one has $d = 1$ , $κ_{1} = 0$ , $Q = π^{- 1/2}$ absorbed into $γ$ , $a_{n} \equiv 1$ , and $ε = 1$ ; the completed function is $Λ (s) = π^{- s /2} Γ (s /2) ζ (s)$ , entire after removal of the pole.

Definition (analytic conductor). The analytic conductor of $L$ at $s$ is

q (s) := Q^{2} j = 1 \prod d (∣ s + κ_{j} ∣ + 3) .

It is a smooth, polynomially varying positive quantity that measures the local complexity of $L$ at $s$ : it combines the arithmetic conductor (through $Q$ ) with the archimedean data (through the shifted gamma parameters). For $ζ$ at $s = 1/2 + i t$ one finds $q (s) ≍ ∣ t ∣ + 3$ , so $q^{1/4} ≍ t^{1/4}$ on the critical line. The bracket constant $3$ keeps $q (s) \geq 1$ and is a normalisation convention; any fixed constant $\geq 1$ gives an equivalent quantity.

Definition (root number and sign). The root number $ε$ is the unimodular constant relating $Λ (s)$ to $\overline{Λ} (1 - s)$ . When the coefficients are real ( $\overline{a_{n}} = a_{n}$ ) and the gamma data is self-conjugate, $Λ$ is self-dual and $ε \in {+ 1, - 1}$ ; the case $ε = - 1$ forces $Λ (1/2) = 0$ , a central zero of arithmetic significance.

Counterexamples to common slips

"The analytic conductor is the arithmetic conductor $q$ of the character." For a Dirichlet $L$ -function $L (s, χ)$ of modulus $q$ , the analytic conductor at $s = 1/2 + i t$ is $q (s) ≍ q (∣ t ∣ + 3)$ , not $q$ . It blends the modulus $q$ with the height $t$ ; the two enter on the same footing, which is the entire point of packaging them into one quantity. Bounds stated as $q^{1/4 + ε}$ then specialise to the $t$ -aspect ( $q$ fixed, $t \to \infty$ ) and the $q$ -aspect ( $t$ fixed, $q \to \infty$ ) uniformly.
"The approximate functional equation has $X$ free to choose, so it is not really an identity." The balancing length is free, but for any admissible $X > 0$ the identity holds exactly (with its error term). The optimal choice $X ≍ q$ equalises the two sums; other choices lengthen one sum and shorten the other but never break the identity. Freedom in $X$ is a feature exploited in the moment method, not a defect.
"Convexity gives the truth on the critical line." The convexity bound $L (1/2) ≪ q^{1/4 + ε}$ is an upper bound, not an asymptotic. It is provably non-optimal under the Lindelöf hypothesis, which predicts $L (1/2) ≪ q^{ε}$ . Any power saving $q^{1/4 - δ}$ with $δ > 0$ is a theorem of subconvexity type and is genuinely hard.

Key theorem with proof Intermediate+

The signature result is the convexity bound, derived from the functional equation by the Phragmén-Lindelöf principle. We first record the approximate functional equation, then prove convexity, since the convexity argument is the cleaner consequence of the functional-equation data and the maximum-modulus circle of ideas from 06.01.12.

Theorem (approximate functional equation). Let $L (s) = \sum_{n} a_{n} n^{- s}$ have functional-equation data $(Q, γ, ε)$ as above, with $L$ entire (or with the polar contribution subtracted). Fix a holomorphic even test function $G (u)$ , normalised by $G (0) = 1$ , with rapid decay in vertical strips. For $X > 0$ and $0 \leq σ \leq 1$ ,

L (s) = n \geq 1 \sum \frac{a _{n}}{n ^{s}} V_{s} (\frac{n}{X}) + ε \frac{Q ^{1 - 2 s} γ ( 1 - s )}{γ ( s )} n \geq 1 \sum \frac{a _{n}}{n ^{1 - s}} W_{s} (\frac{n X}{q}) + (error),

where $V_{s}, W_{s}$ are smooth cutoff functions, each $\approx 1$ for argument $≪ 1$ and rapidly decaying beyond, defined by the inverse Mellin transforms

V_{s} (y) = \frac{1}{2 π i} \int_{(2)} y^{- u} \frac{γ ( s + u )}{γ ( s )} Q^{u} G (u) \frac{d u}{u} .

The optimal balance $X ≍ q (s)$ truncates both sums at length $≪ q (s) (lo g q)^{O (1)}$ .

Proof. Consider the contour integral

I (s) := \frac{1}{2 π i} \int_{(2)} Λ (s + u) X^{u} G (u) \frac{d u}{u},

where $Λ (s) = Q^{s} γ (s) L (s)$ . On the line $Re (u) = 2$ the Dirichlet series for $L (s + u)$ converges absolutely; inserting it and integrating term by term gives, after dividing by $Q^{s} γ (s)$ ,

\frac{I ( s )}{Q ^{s} γ ( s )} = n \geq 1 \sum \frac{a _{n}}{n ^{s}} V_{s} (\frac{n}{X}),

since the $u$ -integral of $n^{- u} X^{u} (γ (s + u) / γ (s)) G (u) / u$ is exactly $V_{s} (n / X)$ . Now shift the contour from $Re (u) = 2$ to $Re (u) = - 2$ . The integrand has a simple pole at $u = 0$ from the factor $1/ u$ , with residue $Λ (s) X^{0} G (0) = Λ (s)$ . (If $L$ has a pole, the polar terms of $Λ$ contribute additional explicit residues, which is the parenthetical error/main-term adjustment.) Hence

I (s) = Λ (s) + \frac{1}{2 π i} \int_{(- 2)} Λ (s + u) X^{u} G (u) \frac{d u}{u} .

In the shifted integral substitute $u \mapsto - u$ and apply the functional equation $Λ (s - u) = ε \overline{Λ} (1 - s + u)$ . Writing out $\overline{Λ} (1 - s + u) = Q^{1 - s + u} γ (1 - s + u) \overline{L} (1 - s + u)$ and expanding $\overline{L}$ as its Dirichlet series on $Re (u) = 2$ gives, after dividing through by $Q^{s} γ (s)$ , the second sum with its smooth cutoff $W_{s}$ and the explicit prefactor $ε Q^{1 - 2 s} γ (1 - s) / γ (s)$ . Equating the two evaluations of $I (s) / (Q^{s} γ (s))$ yields the stated identity. The cutoffs $V_{s} (y), W_{s} (y)$ are $1 + O (y^{A})$ for small $y$ and $O (y^{- A})$ for large $y$ , for every $A$ , by moving the $u$ -contour in their Mellin definitions and using the rapid decay of $G$ together with Stirling's bound on $γ (s + u) / γ (s)$ . Choosing $X = q (s)$ makes the effective ranges of both sums $n ≪ q (s)$ up to logarithmic losses. $□$

Theorem (convexity bound). With the same data, for $0 \leq σ \leq 1$ ,

L (σ + i t) ≪_{ε} q (s)^{\frac{1 - σ}{2} + ε}, in particular L (\frac{1}{2} + i t) ≪_{ε} q (\frac{1}{2} + i t)^{\frac{1}{4} + ε} .

Proof. Define $μ (σ) := lim sup_{∣ t ∣ \to \infty} lo g ∣ L (σ + i t) ∣/ lo g ∣ t ∣$ , the order of growth on the vertical line $Re (s) = σ$ . For $σ > 1$ the Dirichlet series is bounded, so $μ (σ) = 0$ . The functional equation $Λ (s) = ε \overline{Λ} (1 - s)$ rearranges to $L (s) = ε Q^{1 - 2 s} \frac{γ ( 1 - s )}{γ ( s )} \overline{L} (1 - s)$ . By Stirling's formula, $∣ γ (1 - s) / γ (s) ∣ ≍ q (s)^{1/2 - σ}$ as $∣ t ∣ \to \infty$ , and $q (s) ≍ ∣ t ∣^{d}$ in the $t$ -aspect. Therefore on $σ < 0$ we read off $μ (σ) = \frac{d}{2} (1 - 2 σ) \cdot \frac{1}{d} = \frac{1}{2} - σ$ in the normalised height $∣ t ∣$ , matching $μ (σ) = μ (1 - σ) + (1/2 - σ)$ from the reflection. Concretely, $μ (1 - σ) = (1 - σ) - 1/2 + μ (σ)$ supplies the two anchor values $μ (σ) = 0$ for $σ \geq 1$ and $μ (σ) = 1/2 - σ$ for $σ \leq 0$ .

The function $μ$ is convex. This is the Phragmén-Lindelöf theorem: $Λ (s)$ is holomorphic and of finite order in the strip $0 \leq σ \leq 1$ , with at most polynomial growth on the two boundary lines, so by the maximum-modulus principle applied to $e^{P (s)} Λ (s)$ for a suitable linear $P$ (the unbounded-strip extension of the principle in 06.01.12), $lo g ∣Λ∣$ is a convex function of $σ$ on each fixed- $∣ t ∣$ scale. Convexity of $μ$ between the anchors $μ (1) = 0$ and $μ (0) = 1/2$ forces, by linear interpolation,

μ (σ) \leq \frac{1 - σ}{2} (0 \leq σ \leq 1) .

At $σ = 1/2$ this is $μ (1/2) \leq 1/4$ , which is exactly $L (1/2 + i t) ≪ ∣ t ∣^{1/4 + ε} ≍ q^{1/4 + ε}$ . The same interpolation, run with the analytic conductor in place of $∣ t ∣$ to keep the $q$ -aspect uniform, gives the stated $q$ -uniform bound. $□$

Bridge. The convexity bound builds toward the subconvexity programme and appears again in every moment computation for $L$ -functions: the foundational reason a convexity exponent of $1/4$ is even available is that the functional equation pairs the line $σ = 1/2$ with itself, so the two boundary anchors $μ (0)$ and $μ (1)$ sit symmetrically about the critical line, and Phragmén-Lindelöf interpolates between them. This is exactly the maximum-modulus principle of 06.01.12 deployed on an unbounded strip, and the central insight is that the gamma factors, through Stirling, convert the reflection $s \mapsto 1 - s$ into the power $q^{1/2 - σ}$ that fixes the slope of $μ$ . Putting these together, the approximate functional equation and the convexity bound are two faces of the same contour shift — one keeps the sums, the other keeps only their size — and both generalise from $ζ$ in 21.03.01 to the full Selberg-class framework that organises 21.13.04 and the Langlands $L$ -functions of 21.10.01.

Exercises Intermediate+

Exercise 4 (medium, symbolic).

Show that for $ζ$ the gamma-factor ratio in the asymmetric functional equation produces the growth power $q^{1/2 - σ}$ . Use the asymmetric form $ζ (s) = χ (s) ζ (1 - s)$ with $χ (s) = 2^{s} π^{s - 1} sin (π s /2) Γ (1 - s)$ , and the Stirling asymptotic $∣Γ (σ + i t) ∣ ≍ ∣ t ∣^{σ - 1/2} e^{- π ∣ t ∣/2}$ .

Hint

Combine the polynomial part $2^{s} π^{s - 1}$ , the exponential decay of $sin (π s /2) ≍ e^{π ∣ t ∣/2}$ , and Stirling for $Γ (1 - s)$ . The exponential factors cancel, leaving a power of $∣ t ∣$ .

Answer

Write $s = σ + i t$ with $t \to + \infty$ . The factor $∣ 2^{s} π^{s - 1} ∣ = 2^{σ} π^{σ - 1}$ is bounded (no $t$ -growth). For $sin (π s /2)$ , as $t \to \infty$ , $∣ sin (π s /2) ∣ ≍ \frac{1}{2} e^{π t /2}$ . For $Γ (1 - s) = Γ (1 - σ - i t)$ , Stirling gives $∣Γ (1 - σ - i t) ∣ ≍ ∣ t ∣^{(1 - σ) - 1/2} e^{- π ∣ t ∣/2} = ∣ t ∣^{1/2 - σ} e^{- π t /2}$ . Multiplying, the exponential factors $e^{π t /2} \cdot e^{- π t /2} = 1$ cancel, leaving

∣ χ (σ + i t) ∣ ≍ ∣ t ∣^{1/2 - σ} .

Since $q (s) ≍ ∣ t ∣$ for $ζ$ , this is $q^{1/2 - σ}$ . The slope $\partial_{σ} lo g ∣ χ ∣ = - lo g ∣ t ∣$ is what makes $μ (σ)$ decrease at unit rate per unit $σ$ in the reflected region, fixing the anchor $μ (0) = 1/2$ . $□$

Exercise 5 (medium, numeric).

The fourth-moment bound $\int_{0}^{T} ∣ ζ (1/2 + i t) ∣^{4} d t ≪ T (lo g T)^{4}$ is consistent with Lindelöf but not with convexity used pointwise. If $∣ ζ (1/2 + i t) ∣$ were always at its convexity ceiling $t^{1/4}$ , what power of $T$ would the fourth moment grow like? Compare with $T (lo g T)^{4}$ .

Hint

If $∣ ζ (1/2 + i t) ∣ ≍ t^{1/4}$ then $∣ ζ ∣^{4} ≍ t$ , and $\int_{0}^{T} t d t ≍ T^{2}$ .

Answer

$T^{2}$ . If the convexity ceiling were attained for all $t$ , then $∣ ζ (1/2 + i t) ∣^{4} ≍ (t^{1/4})^{4} = t$ , so $\int_{0}^{T} ∣ ζ ∣^{4} d t ≍ \int_{0}^{T} t d t = T^{2} /2 ≍ T^{2}$ . The actual fourth moment is $≪ T (lo g T)^{4}$ , only barely above $T^{1}$ , which is far below $T^{2}$ . This shows the convexity bound is wildly pessimistic on average: the large values predicted by convexity occur on a set of measure too small to influence the moment. This averaging gap is the engine behind subconvexity proofs.

Exercise 6 (medium, multiple choice).

The subconvexity problem for an $L$ -function asks to prove a bound of which shape?

A. $L (1/2) ≪ q^{1/4 + ε}$
B. $L (1/2) ≪ q^{1/4 - δ}$ for some fixed $δ > 0$
C. $L (1/2) ≪ q^{ε}$
D. $L (1/2) = 0$

Hint

Subconvexity means any fixed power saving below the convexity exponent $1/4$ ; the full $q^{ε}$ is the stronger Lindelöf prediction.

Answer

B. Subconvexity is any bound $L (1/2) ≪ q^{1/4 - δ}$ with a fixed $δ > 0$ — a power saving over convexity. Option A is exactly the convexity bound (no saving). Option C is the Lindelöf-on-average / Lindelöf hypothesis bound, far stronger than what "subconvexity" requires. Option D is a vanishing statement, unrelated to size bounds. The Weyl exponent $1/6$ for $ζ$ (Hardy-Littlewood, Weyl) and the Burgess exponent $3/16$ for Dirichlet $L$ -functions are classical subconvexity results.

Exercise 7 (hard, symbolic).

Prove the Phragmén-Lindelöf convexity step in detail: let $F$ be holomorphic in the strip $a \leq σ \leq b$ , of finite order (bounded by $exp (∣ t ∣^{A})$ for some $A$ , uniformly), with $∣ F (a + i t) ∣ \leq C_{a} (1 + ∣ t ∣)^{α}$ and $∣ F (b + i t) ∣ \leq C_{b} (1 + ∣ t ∣)^{β}$ . Show that $∣ F (σ + i t) ∣ ≪ (1 + ∣ t ∣)^{ℓ (σ)}$ where $ℓ$ is the linear function interpolating $α$ at $a$ and $β$ at $b$ .

Hint

Apply the Phragmén-Lindelöf principle to $G (s) = F (s) (s + λ)^{- ℓ (s)}$ for a large constant $λ$ making the branch single-valued, or use the three-lines lemma after dividing out the boundary growth. The finite-order hypothesis kills the exponential escape that would otherwise break the maximum principle on an unbounded region.

Answer

Let $ℓ (σ)$ be the linear function with $ℓ (a) = α$ , $ℓ (b) = β$ . Choose a branch of $(s + λ)^{- ℓ (s)}$ with $λ$ large enough that $Re (s + λ) > 0$ throughout the strip; since $ℓ$ is affine, $lo g (s + λ) \cdot ℓ (s)$ is holomorphic, so $H (s) := F (s) (s + λ)^{- ℓ (s)}$ is holomorphic in the strip. On the boundary lines, $∣ (s + λ)^{- ℓ (s)} ∣ ≍ (1 + ∣ t ∣)^{- ℓ (σ)}$ , so $∣ H (a + i t) ∣ ≪ C_{a} (1 + ∣ t ∣)^{α - α} = C_{a}$ and likewise $∣ H (b + i t) ∣ ≪ C_{b}$ ; $H$ is bounded on both edges.

Now $H$ is of finite order in the strip (the factor $(s + λ)^{- ℓ (s)}$ is of polynomial type, and $F$ has order $< \infty$ ). For $η > 0$ consider $H_{η} (s) := H (s) e^{η (s - c)^{2}}$ centred at $c = (a + b) /2$ ; the term $e^{η (s - c)^{2}}$ has modulus $e^{η ((σ - c)^{2} - t^{2})}$ , which decays like $e^{- η t^{2}}$ as $∣ t ∣ \to \infty$ uniformly in $σ \in [a, b]$ . This decay defeats the finite-order growth of $H$ , so $H_{η} \to 0$ as $∣ t ∣ \to \infty$ in the strip; combined with the boundary bound $∣ H_{η} ∣ \leq max (C_{a}, C_{b}) e^{η (b - a)^{2} /4}$ on the edges, the maximum-modulus principle on the resulting bounded region (truncate at $∣ t ∣ = T$ , let $T \to \infty$ ) gives $∣ H_{η} (s) ∣ \leq max (C_{a}, C_{b}) e^{η (b - a)^{2} /4}$ throughout. Let $η \to 0^{+}$ : $∣ H (s) ∣ \leq max (C_{a}, C_{b})$ in the whole strip. Unwinding, $∣ F (σ + i t) ∣ \leq max (C_{a}, C_{b}) ∣ (s + λ)^{ℓ (s)} ∣ ≪ (1 + ∣ t ∣)^{ℓ (σ)}$ . The finite-order hypothesis is essential: $F (s) = exp (e^{- i s})$ is bounded on the real axis edges of a horizontal strip yet unbounded inside, the standard counterexample when finite order fails. $□$

Exercise 8 (hard, symbolic).

State the Lindelöf hypothesis in terms of $μ (σ)$ and prove that it is equivalent to $μ (1/2) = 0$ . Then show Lindelöf follows from the Riemann hypothesis (you may cite the standard zero-density consequence).

Hint

Lindelöf is $μ (1/2) = 0$ . The convexity of $μ$ plus the anchors $μ (σ) = 0$ for $σ \geq 1$ and $μ (σ) = 1/2 - σ$ for $σ \leq 0$ pins down everything once you know $μ (1/2)$ . For RH $\Rightarrow$ Lindelöf, use $lo g ∣ ζ (s) ∣ ≪ (lo g t)^{2 - 2 σ + ε}$ under RH.

Answer

Statement. The Lindelöf hypothesis is $μ (1/2) = 0$ , equivalently $ζ (1/2 + i t) ≪_{ε} ∣ t ∣^{ε}$ for every $ε > 0$ .

Equivalence with the full Lindelöf profile. Since $μ$ is convex (Phragmén-Lindelöf) with $μ (σ) = 0$ for $σ \geq 1$ and $μ (σ) = 1/2 - σ$ for $σ \leq 0$ , the value $μ (1/2)$ controls the entire graph on $[0, 1]$ . Convexity forces $μ (σ) \leq max (0, 1/2 - σ) + (contribution from μ (1/2))$ . If $μ (1/2) = 0$ , then on $[1/2, 1]$ convexity between $μ (1/2) = 0$ and $μ (1) = 0$ forces $μ \equiv 0$ , and on $[0, 1/2]$ convexity between $μ (0) = 1/2$ and $μ (1/2) = 0$ forces $μ (σ) \leq 1 - 2 σ$ over the half-interval, with the functional-equation reflection $μ (σ) = μ (1 - σ) + 1/2 - σ$ pinning $μ (σ) = 1/2 - σ$ exactly there. Thus $μ (1/2) = 0$ gives the full profile $μ (σ) = max (0, 1/2 - σ)$ , and conversely that profile has $μ (1/2) = 0$ .

RH $\Rightarrow$ Lindelöf. Under the Riemann hypothesis, the standard bound $lo g ∣ ζ (σ + i t) ∣ ≪ (lo g t)^{2 - 2 σ} / lo g lo g t + O (lo g lo g t)$ holds for $1/2 \leq σ \leq 1$ (Littlewood 1924). At $σ = 1/2$ this gives $lo g ∣ ζ (1/2 + i t) ∣ ≪ lo g t / lo g lo g t$ , hence $∣ ζ (1/2 + i t) ∣ ≪ exp (c lo g t / lo g lo g t) ≪ t^{ε}$ for every $ε > 0$ . Therefore $μ (1/2) = 0$ , which is Lindelöf. The converse is false: Lindelöf does not imply RH (Lindelöf is a statement about size, RH about zero location), though Lindelöf does imply the zero-density bound $N (σ, T) = o (T)$ for every $σ > 1/2$ . $□$

Advanced results Master

The convexity exponent as a slope, and where it is beaten

The function $μ (σ)$ is the Lindelöf $μ$ -function: $μ (σ) = in f {c : ζ (σ + i t) ≪ ∣ t ∣^{c}}$ . It is non-increasing, convex, non-negative, and tends to $0$ as $σ \to \infty$ ^[Titchmarsh]. The functional equation gives the exact reflection $μ (σ) - μ (1 - σ) = 1/2 - σ$ , so the two anchors $μ (σ) = 0$ ( $σ \geq 1$ ) and $μ (σ) = 1/2 - σ$ ( $σ \leq 0$ ) determine $μ$ up to its convex interior. Convexity alone yields the convexity bound $μ (σ) \leq (1 - σ) /2$ on $[0, 1]$ , with the critical value $μ (1/2) \leq 1/4$ . Every theorem that lowers $μ (1/2)$ below $1/4$ is a subconvexity result.

The first subconvexity bound for $ζ$ is the Weyl-Hardy-Littlewood exponent $μ (1/2) \leq 1/6$ , obtained by bounding the exponential sums $\sum_{n \leq N} n^{- i t}$ appearing in the approximate functional equation through Weyl differencing (van der Corput's method of exponent pairs). Successive refinements — van der Corput, Phillips, Min, Bombieri-Iwaniec 1986, Huxley's discrete-Hardy-Littlewood method — have lowered the exponent to Bourgain's $13/84$ (2017), the current record. The Lindelöf hypothesis is the assertion $μ (1/2) = 0$ , equivalent to $ζ (1/2 + i t) ≪_{ε} ∣ t ∣^{ε}$ ; it is implied by the Riemann hypothesis but is strictly weaker.

The general analytic conductor and uniform subconvexity

For a general $L$ -function $L (s, π)$ attached to an automorphic representation $π$ , the analytic conductor $q (π, s) = Q^{2} \prod_{j} (∣ s + κ_{j} ∣ + 3)$ unifies the level, the height $t$ , and the archimedean parameters of $π$ into a single quantity, and the convexity bound is the uniform statement $L (1/2, π) ≪ q (π, 1/2)^{1/4 + ε}$ ^{[Iwaniec-Kowalski 2004]}. The strength of the analytic-conductor formulation is that subconvexity in any aspect — the $t$ -aspect, the level (conductor) aspect, or the spectral (eigenvalue) aspect — is one and the same inequality $L (1/2, π) ≪ q^{1/4 - δ}$ , with the aspect determined by which parameter is sent to infinity. The Burgess bound $L (1/2, χ) ≪ q^{3/16 + ε}$ is subconvexity in the $q$ -aspect; the Weyl bound is the $t$ -aspect; Duke-Friedlander-Iwaniec's amplification method delivers subconvexity in the conductor aspect for $GL_{2}$ $L$ -functions.

Why convexity is the natural barrier

The convexity bound is exactly what the functional equation alone can supply: it uses no information about the coefficients $a_{n}$ beyond their being bounded on average, only the gamma factors, the root number, and the maximum-modulus principle. Subconvexity must therefore exploit cancellation among the $a_{n}$ — additive cancellation in the exponential sums (van der Corput, Weyl), multiplicative cancellation via the spectral theory of automorphic forms (Kuznetsov, Petersson), or cancellation through the amplification and moment methods. The square-root-of-the-conductor truncation in the approximate functional equation is precisely the threshold at which such cancellation becomes available: each sum has length $q$ , and beating the term-by-term bound $q \cdot q^{- 1/4}$ on each sum is the analytic content of subconvexity.

Synthesis. The approximate functional equation and the convexity bound are the two structural outputs of a single contour shift, and putting these together gives the working geometry of every $L$ -function on its critical line. The foundational reason the convexity exponent equals $1/4$ is that the functional equation reflects $σ = 1/2$ onto itself, so the Phragmén-Lindelöf interpolation between $μ (0) = 1/2$ and $μ (1) = 0$ deposits exactly $1/4$ at the centre; this is exactly the slope that Stirling's asymptotics for the gamma factors impose, and it generalises verbatim from $ζ$ to the Selberg class and the automorphic $L$ -functions of 21.10.01. The central insight is that the analytic conductor $q$ is the single scalar in which all of this is uniform: the truncation length is $q$ , the convexity ceiling is $q^{1/4}$ , and subconvexity is any power saving below it. The bridge is between the maximum-modulus principle of 06.01.12 — a statement in pure complex analysis — and the arithmetic of primes encoded in the coefficients $a_{n}$ , and the approximate functional equation is dual to the convexity bound in the precise sense that one retains the arithmetic sums while the other retains only their archimedean size. This is the foundational reason that subconvexity, not convexity, is where the arithmetic begins.

Full proof set Master

Proposition (analytic conductor is multiplicative-stable under shifts). For fixed $u$ with $∣ u ∣ \leq 1$ , $q (s + u) ≍ q (s)$ with implied constants depending only on the degree $d$ and the parameters $κ_{j}$ .

Proof. By definition $q (s) = Q^{2} \prod_{j = 1}^{d} (∣ s + κ_{j} ∣ + 3)$ . For each $j$ , the triangle inequality gives $∣ s + u + κ_{j} ∣ \leq ∣ s + κ_{j} ∣ + ∣ u ∣ \leq ∣ s + κ_{j} ∣ + 1$ , and likewise $∣ s + u + κ_{j} ∣ \geq ∣ s + κ_{j} ∣ - 1$ . Hence

\frac{∣ s + u + κ _{j} ∣ + 3}{∣ s + κ _{j} ∣ + 3} \in [\frac{∣ s + κ _{j} ∣ + 2}{∣ s + κ _{j} ∣ + 3}, \frac{∣ s + κ _{j} ∣ + 4}{∣ s + κ _{j} ∣ + 3}] \subseteq [\frac{1}{2}, 2],

since $(x + 2) / (x + 3) \geq 1/2$ and $(x + 4) / (x + 3) \leq 2$ for $x \geq 0$ . Taking the product over the $d$ factors, the ratio $q (s + u) / q (s)$ lies in $[2^{- d}, 2^{d}]$ . $□$

Proposition (cutoff functions are smooth and rapidly decaying). The function $V_{s} (y) = \frac{1}{2 π i} \int_{(2)} y^{- u} \frac{γ ( s + u )}{γ ( s )} Q^{u} G (u) \frac{d u}{u}$ satisfies $V_{s} (y) = 1 + O_{A} (y^{A})$ as $y \to 0$ and $V_{s} (y) = O_{A} (y^{- A})$ as $y \to \infty$ , for every $A > 0$ , uniformly for $s$ in the strip $0 \leq σ \leq 1$ with $X ≍ q$ .

Proof. For large $y$ , move the contour to $Re (u) = A$ . No pole is crossed (the integrand is holomorphic for $Re (u) > 0$ since $1/ u$ has its pole at $u = 0$ ), so $V_{s} (y) = \frac{1}{2 π i} \int_{(A)} y^{- u} \frac{γ ( s + u )}{γ ( s )} Q^{u} G (u) \frac{d u}{u}$ . On this line $∣ y^{- u} ∣ = y^{- A}$ , the ratio $γ (s + u) / γ (s)$ is bounded by $q (s)^{A /2}$ via Stirling (Proposition above plus the Stirling asymptotic for $Γ$ ), $Q^{u}$ contributes $Q^{A}$ , and $G (u)$ decays rapidly making the integral converge; the total is $O_{A} (y^{- A} q^{A /2} Q^{A})$ , and with $X ≍ q$ rescaling $y = n / X$ absorbs the $q^{A /2}$ into the truncation length, giving $O_{A} (y^{- A})$ in the effective variable. For small $y$ , move the contour to $Re (u) = - A$ , crossing the simple pole at $u = 0$ with residue $\frac{γ ( s )}{γ ( s )} Q^{0} G (0) = 1$ ; the remaining integral on $Re (u) = - A$ is $O_{A} (y^{A})$ by the same estimate with the sign of $A$ reversed. Hence $V_{s} (y) = 1 + O_{A} (y^{A})$ . The evenness and normalisation $G (0) = 1$ are what make the residue exactly $1$ . $□$

Proposition (convexity is sharp for the gamma factors). On the lines $σ = 0$ and $σ = 1$ the convexity bound is attained in order of magnitude: $∣ χ (σ + i t) ∣ ≍ q^{1/2 - σ}$ , so $μ (0) = 1/2$ and $μ (1) = 0$ are exact, not merely upper bounds.

Proof. From Exercise 4, $∣ χ (σ + i t) ∣ ≍ ∣ t ∣^{1/2 - σ}$ with $χ$ the asymmetric functional-equation factor, and $q ≍ ∣ t ∣$ for $ζ$ . On $σ = 1$ , $ζ (1 + i t)$ is bounded above and below by absolute constants times powers of $lo g ∣ t ∣$ (the Dirichlet series and its zero-free region), so $μ (1) = 0$ exactly. On $σ = 0$ , the functional equation $ζ (i t) = χ (i t) ζ (1 - i t)$ combined with $∣ χ (i t) ∣ ≍ ∣ t ∣^{1/2}$ and $∣ ζ (1 - i t) ∣ ≍ 1$ (up to logs) gives $∣ ζ (i t) ∣ ≍ ∣ t ∣^{1/2}$ , so $μ (0) = 1/2$ exactly. The interior bound $μ (1/2) \leq 1/4$ is therefore the genuine convexity ceiling, and the open question is whether it is attained — Lindelöf says it is not. $□$

Connections Master

The convexity bound is the maximum-modulus principle of 06.01.12 applied on an unbounded vertical strip: the Phragmén-Lindelöf extension converts the two boundary growth rates into the interior interpolation $μ (σ) \leq (1 - σ) /2$ , so the entire convexity exponent is a complex-analytic fact about holomorphic functions of finite order, with the arithmetic entering only through the gamma factors.
The approximate functional equation specialises the functional equation of $ζ$ from 21.03.01 and 06.01.16: the same completed function $Λ (s) = π^{- s /2} Γ (s /2) ζ (s)$ and its symmetry $Λ (s) = Λ (1 - s)$ are the input to the contour shift, so this unit is the quantitative companion to the qualitative functional equation proved there.
The analytic conductor and the convexity-subconvexity dichotomy extend to the automorphic $L$ -functions of the Langlands programme in 21.10.01: the degree- $d$ gamma factor $\prod_{j} Γ_{R} (s + κ_{j})$ is exactly the archimedean local factor of an automorphic representation, and subconvexity in the conductor aspect is a central tool in the arithmetic applications (equidistribution, the subconvexity-to-Duke-theorem pipeline).

Historical & philosophical context Master

The approximate functional equation originates with Hardy and Littlewood, who in their 1916 Acta Mathematica memoir ^{[Hardy-Littlewood 1916]} proved that $ζ (s)$ in the critical strip is well approximated by two truncated Dirichlet sums of lengths $x$ and $y$ with $x y = ∣ t ∣/ (2 π)$ , replacing Riemann-Siegel-type contour estimates with an explicit finite expression. The Riemann-Siegel formula, recovered by Siegel in 1932 from Riemann's unpublished Nachlass, refined the error term and remains the basis of large-scale numerical verification of the Riemann hypothesis.

The convexity bound rests on the Phragmén-Lindelöf principle, established by Phragmén and Lindelöf in their 1908 Acta Mathematica paper ^{[Phragmén-Lindelöf 1908]} as an extension of the maximum-modulus principle to unbounded regions under a growth hypothesis. In the same year Lindelöf ^{[Lindelöf 1908]} introduced the function $μ (σ)$ measuring the order of $ζ$ on vertical lines, proved its convexity, and formulated the hypothesis $μ (1/2) = 0$ now bearing his name. The convexity bound $μ (1/2) \leq 1/4$ is the immediate corollary; the gap between $1/4$ and the conjectured $0$ has driven a century of work in exponential sums and, since the 1990s, in the spectral theory of automorphic forms.

The analytic-conductor formulation is due to Iwaniec and Kowalski, whose 2004 Colloquium volume ^{[Iwaniec-Kowalski 2004]} placed the conductor $q (s)$ at the centre of the analytic theory, unifying the $t$ -aspect, level aspect, and spectral aspect into a single scalar and stating convexity and subconvexity uniformly across all aspects. The reframing turned a collection of aspect-specific estimates into instances of one inequality.

Bibliography Master

@article{HardyLittlewood1916,
  author = {Hardy, G. H. and Littlewood, J. E.},
  title = {Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes},
  journal = {Acta Mathematica},
  volume = {41},
  year = {1916},
  pages = {119--196},
  note = {Originator of the approximate functional equation for zeta}
}

@article{PhragmenLindelof1908,
  author = {Phragm\'en, E. and Lindel\"of, E.},
  title = {Sur une extension d'un principe classique de l'analyse et sur quelques propri\'et\'es des fonctions monog\`enes dans le voisinage d'un point singulier},
  journal = {Acta Mathematica},
  volume = {31},
  year = {1908},
  pages = {381--406},
  note = {The maximum-modulus principle for unbounded strips}
}

@article{Lindelof1908,
  author = {Lindel\"of, E.},
  title = {Quelques remarques sur la croissance de la fonction $\zeta(s)$},
  journal = {Bulletin des Sciences Math\'ematiques},
  volume = {32},
  year = {1908},
  pages = {341--356},
  note = {The function mu(sigma) and the Lindelof hypothesis}
}

@book{Titchmarsh1986,
  author = {Titchmarsh, E. C.},
  title = {The Theory of the Riemann Zeta-Function},
  publisher = {Oxford University Press},
  edition = {2nd, revised by D. R. Heath-Brown},
  year = {1986},
  note = {Ch. 4 (approximate functional equation), Ch. 5 (order of zeta, Lindelof hypothesis)}
}

@book{IwaniecKowalski2004,
  author = {Iwaniec, Henryk and Kowalski, Emmanuel},
  title = {Analytic Number Theory},
  publisher = {American Mathematical Society},
  series = {Colloquium Publications},
  volume = {53},
  year = {2004},
  note = {Ch. 5: analytic conductor, approximate functional equation, convexity}
}

@book{Iwaniec2014,
  author = {Iwaniec, Henryk},
  title = {Lectures on the Riemann Zeta Function},
  publisher = {American Mathematical Society},
  series = {University Lecture Series},
  volume = {62},
  year = {2014},
  note = {Modern lecture treatment of the approximate functional equation and moments}
}

Prerequisites

21.03.01
06.01.16
06.01.12

Tier anchors

beginner: Iwaniec-Kowalski 2004 *Analytic Number Theory* (AMS Colloquium 53) §5.1-§5.2 informal opening; Stopple 2003 *A Primer of Analytic Number Theory* (Cambridge) Ch. 9 on the size of $\zeta$ in the critical strip
intermediate: Iwaniec-Kowalski 2004 *Analytic Number Theory* (AMS Colloquium 53) Ch. 5 §5.1-§5.3 (approximate functional equation, analytic conductor) and §5.7 (convexity); Montgomery-Vaughan 2007 *Multiplicative Number Theory I* (Cambridge Studies 97) Ch. 13
master: Iwaniec-Kowalski 2004 *Analytic Number Theory* (AMS Colloquium 53) Ch. 5 (the canonical treatment of the analytic conductor and the approximate functional equation in the general $L$-function framework); Hardy-Littlewood 1916 *Acta Math.* 41 (originator of the approximate functional equation for $\zeta$); Phragmén-Lindelöf 1908 *Acta Math.* 31 (the convexity principle); Lindelöf 1908 *Bull. Sci. Math.* 32; Titchmarsh 1986 *The Theory of the Riemann Zeta-Function* (2nd ed., Oxford) Ch. 4-5; Iwaniec 2014 *Lectures on the Riemann Zeta Function* (AMS ULECT 62)

References

Iwaniec, H. & Kowalski, E. — Analytic Number Theory · American Mathematical Society Colloquium Publications 53 (2004), Ch. 5. The analytic conductor $\mathfrak{q}(s)$, the general functional-equation framework with gamma factors and root number, the approximate functional equation via a smoothed contour shift, and the convexity bound $L(1/2 + it) \ll \mathfrak{q}^{1/4 + \varepsilon}$.
Hardy, G. H. & Littlewood, J. E. — Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes · *Acta Mathematica* 41 (1916), 119-196. The first approximate functional equation for $\zeta(s)$ with two truncated Dirichlet sums of lengths $\sqrt{|t|/2\pi}$.
Phragmén, E. & Lindelöf, E. — Sur une extension d'un principe classique de l'analyse et sur quelques propriétés des fonctions monogènes dans le voisinage d'un point singulier · *Acta Mathematica* 31 (1908), 381-406. The extension of the maximum-modulus principle to unbounded strips, the basis of the convexity bound.
Lindelöf, E. — Quelques remarques sur la croissance de la fonction $\zeta(s)$ · *Bulletin des Sciences Mathématiques* 32 (1908), 341-356. The convexity of the function $\mu(\sigma)$ measuring the order of growth of $\zeta$ on vertical lines, and the Lindelöf hypothesis.
Titchmarsh, E. C. — The Theory of the Riemann Zeta-Function · Oxford University Press, 2nd edition revised by D. R. Heath-Brown (1986). Ch. 4 (the approximate functional equation), Ch. 5 (the order of $\zeta(s)$, the function $\mu(\sigma)$, the Lindelöf hypothesis).

Estimated time

beginner: 20m
intermediate: 45m
master: 90m