Euler Product Formula for the Riemann Zeta Function

Statement I (Lectures 21-22: The Prime Number Theorem)

$$\frac{1}{\zeta(s)} = \prod_{n=1}^{\infty} (1 - p_n^{-s}) \quad \text{for } \operatorname{Re} s > 1$$

Equivalently, $\zeta(s) = \prod_p \frac{1}{1 - p^{-s}}$ where the product ranges over all primes.

Proof sketch (from the textbook)

For each prime $p_n$, we have $(1 - p_n^{-s})^{-1} = \sum_{m=0}^{\infty} p_n^{-ms}$. Expanding the finite product $\prod_{n=1}^{N} (1 - p_n^{-s})^{-1}$ using unique prime factorization gives $\prod_{n=1}^{N} (1 - p_n^{-s})^{-1} = \sum_k n_k^{-s}$, where the sum is over all positive integers whose prime factors are among $p_1, \ldots, p_N$. Letting $N \to \infty$ yields $\zeta(s)$.

This is formalized using Mathlib's riemannZeta and Nat.Primes, building on the Euler product machinery in Mathlib.NumberTheory.EulerProduct.

Statement II (Lectures 21-22)

Statement II. $\zeta(s) - \frac{1}{s-1}$ extends to a holomorphic function in $\operatorname{Re} s > 0$.

This means that the Riemann zeta function ζ(s) has a meromorphic continuation to the half-plane {s : ℂ | 0 < s.re} with a unique simple pole at s = 1 of residue 1.

Mathlib's riemannZeta is already defined on all of ℂ via analytic continuation and is complex-differentiable away from s = 1 (differentiableAt_riemannZeta). The function s ↦ ζ(s) - 1/(s-1) has a removable singularity at s = 1, as witnessed by the fact that it tends to the Euler–Mascheroni constant γ as s → 1 (tendsto_riemannZeta_sub_one_div). We apply the removable singularity theorem to produce a holomorphic extension to the full half-plane.

theorem euler_product_riemannZeta_hasProd {s : ℂ} (hs : 1 < s.re) : HasProd (fun (p : Nat.Primes) => (1 - ↑↑p ^ (-s))⁻¹) (riemannZeta s)

Euler Product Formula for ζ (HasProd form) (Statement I, Lectures 21-22). For Re(s) > 1, the Riemann zeta function equals the infinite product over all primes p of (1 - p^(-s))⁻¹. This version is stated using HasProd, which asserts that the partial products over finite sets of primes converge to ζ(s).

theorem euler_product_riemannZeta_tprod {s : ℂ} (hs : 1 < s.re) : ∏' (p : Nat.Primes), (1 - ↑↑p ^ (-s))⁻¹ = riemannZeta s

Euler Product Formula for ζ (tprod form) (Statement I, Lectures 21-22). For Re(s) > 1, ζ(s) = ∏' p, (1 - p^(-s))⁻¹ where p ranges over all primes. This is the standard Euler product identity for the Riemann zeta function. The textbook writes this equivalently as 1/ζ(s) = ∏(1 - p^(-s)).

theorem euler_product_riemannZeta_tendsto {s : ℂ} (hs : 1 < s.re) : Filter.Tendsto (fun (n : ℕ) => ∏ p ∈ n.primesBelow, (1 - ↑p ^ (-s))⁻¹) Filter.atTop (nhds (riemannZeta s))

Euler Product Formula for ζ (Tendsto form) (Statement I, Lectures 21-22). For Re(s) > 1, the finite partial products ∏_{p < n} (1 - p^(-s))⁻¹ converge to ζ(s) as n → ∞. This is the form closest to the textbook proof, which expands finite products and lets N → ∞.

theorem euler_product_riemannZeta_inv {s : ℂ} (hs : 1 < s.re) : (riemannZeta s)⁻¹ = ∏' (p : Nat.Primes), (1 - ↑↑p ^ (-s))

Reciprocal Euler Product for ζ (Statement I, Lectures 21-22). For Re(s) > 1, 1/ζ(s) = ∏' p, (1 - p^(-s)). This is the form stated in the textbook as 1/ζ(s) = ∏(1 - p_n^{-s}), obtained by taking the reciprocal of the Euler product ζ(s) = ∏' p, (1 - p^(-s))⁻¹ and distributing the inverse through the absolutely convergent product.

The proof constructs HasProd (fun p => 1 - p^{-s}) (ζ(s))⁻¹ by applying Tendsto.inv₀ (continuous inversion at a nonzero point) to the Euler product HasProd (fun p => (1 - p^{-s})⁻¹) ζ(s) and using Finset.prod_inv_distrib to commute finite products with inversion.

Newman's Analytic Theorem (Theorem 1, Lectures 21-22)

Truncated Laplace transform and helper lemmas for the proof

theorem Newman.left_semicircle_integral_tendsto (R : ℝ) (hR : 0 < R) (h : ℝ → ℂ) (hh_int : MeasureTheory.Integrable h (MeasureTheory.volume.restrict (Set.Ioc (Real.pi / 2) (3 * Real.pi / 2)))) (hh_meas : MeasureTheory.AEStronglyMeasurable h MeasureTheory.volume) : Filter.Tendsto (fun (T : ℝ) => ∫ (θ : ℝ) in Real.pi / 2..3 * Real.pi / 2, Complex.exp (↑R * Complex.exp (↑θ * Complex.I) * ↑T) * h θ) Filter.atTop (nhds 0)

On the left semicircle, ∫ exp(R·exp(iθ)·T) * h(θ) dθ → 0 as T → ∞ by the Dominated Convergence Theorem. The key is that Re(R·exp(iθ)) = R·cos(θ) ≤ 0 on θ ∈ [π/2, 3π/2], so |exp(R·exp(iθ)·T)| ≤ 1 (domination), and for a.e. θ in the interior, cos(θ) < 0 so |exp(R·exp(iθ)·T)| → 0 (pointwise convergence).

noncomputable def Newman.truncatedLaplaceTransform (f : ℝ → ℂ) (T : ℝ) (z : ℂ) : ℂ

The truncated Laplace transform g_T(z) = ∫₀ᵀ e^{-zt} f(t) dt. This is an entire function of z for each fixed T.

theorem Newman.truncatedLaplaceTransform_zero (f : ℝ → ℂ) (T : ℝ) : truncatedLaplaceTransform f T 0 = ∫ (t : ℝ) in 0..T, f t

At z = 0, the truncated Laplace transform reduces to ∫₀ᵀ f(t) dt, since e^{-0·t} = 1.

theorem Newman.truncatedLaplaceTransform_continuous (f : ℝ → ℂ) (T : ℝ) : Continuous (truncatedLaplaceTransform f T)

The truncated Laplace transform g_T(z) is continuous as a function of z. This follows from the fact that it is an entire function (the integrand e^{-zt} f(t) depends holomorphically on z), which the book treats as immediate.

theorem Newman.newman_g_diffOn_closedBall (g : ℂ → ℂ) (R : ℝ) (_hR : 0 < R) (hg_diffAt : ∀ (z : ℂ), DifferentiableAt ℂ g z) : DifferentiableOn ℂ g (Metric.closedBall 0 R)

g holomorphic everywhere is differentiable on closedBall 0 R.

theorem Newman.truncatedLaplaceTransform_differentiable (f : ℝ → ℂ) (T : ℝ) : Differentiable ℂ (truncatedLaplaceTransform f T)

Differentiability of the truncated Laplace transform (under the integral sign). The truncated Laplace transform g_T(z) = ∫₀ᵀ e^{-zt} f(t) dt is an entire function of z: for each fixed t, the integrand e^{-zt} f(t) is entire in z, and the integral is over a bounded interval [0, T]. The book treats this as obvious; formally it requires differentiation under the integral sign for a holomorphic parameter.

theorem Newman.newman_cif_H_diffOn_closedBall (f : ℝ → ℂ) (g : ℂ → ℂ) (R : ℝ) (hR : 0 < R) (Tv : ℝ) (hg_diffAt : ∀ (z : ℂ), DifferentiableAt ℂ g z) : DifferentiableOn ℂ (fun (z : ℂ) => Complex.exp (z * ↑Tv) * (1 + z ^ 2 / ↑R ^ 2) * (g z - truncatedLaplaceTransform f Tv z)) (Metric.closedBall 0 R)

The function H(z) = e^{zT} · (1 + z²/R²) · (g(z) - g_T(z)) is differentiable on the closed ball of radius R centered at the origin.

This combines three facts:

• e^{zT} and 1 + z²/R² are entire (trivially differentiable everywhere),
• g extends to closedBall 0 R by analytic continuation (newman_g_diffOn_closedBall),
• truncatedLaplaceTransform f Tv is entire (truncatedLaplaceTransform_differentiable).

theorem Newman.newman_cif_equation6 (f : ℝ → ℂ) (g : ℂ → ℂ) (R : ℝ) (hR : 0 < R) (hg_diffAt : ∀ (z : ℂ), DifferentiableAt ℂ g z) (Tv : ℝ) (w : ℝ → ℂ) (hw : w = fun (θ : ℝ) => ↑R * Complex.exp (↑θ * Complex.I)) (integrand : ℝ → ℝ → ℂ) (hintegrand : integrand = fun (T θ : ℝ) => (2 * ↑Real.pi * Complex.I)⁻¹ * (Complex.exp (w θ * ↑T) * (1 + w θ ^ 2 / ↑(R ^ 2)) / w θ) * (Complex.I * w θ)) : g 0 - truncatedLaplaceTransform f Tv 0 = ∫ (θ : ℝ) in -Real.pi / 2..3 * Real.pi / 2, integrand Tv θ * (g (w θ) - truncatedLaplaceTransform f Tv (w θ))

Cauchy integral formula (equation (6) in the book).

The book states "By Cauchy's formula" that g(0) - g_T(0) = (2πi)⁻¹ ∮_C (g(z) - g_T(z)) e^{zT} (1 + z²/R²) dz/z. Parametrised over the circle w(θ) = R·e^{iθ} for θ ∈ [-π/2, 3π/2], this becomes the integral stated below.

Note: the hypothesis is ∀ z, DifferentiableAt ℂ g z, modeling the book's assumption that g extends to a holomorphic function (entire).

The proof applies Mathlib's Cauchy integral formula (DifferentiableOn.circleIntegral_sub_inv_smul) to the auxiliary function H(z) = e^{zT} · (1 + z²/R²) · (g(z) - g_T(z)), converts the circle integral to a parametric integral, simplifies the integrand algebraically, and shifts the integration interval from [0, 2π] to [-π/2, 3π/2] using periodicity of e^{iθ}.

The differentiability of H on the closed ball is provided by newman_cif_H_diffOn_closedBall (proved by combining the two focused axioms newman_g_diffOn_closedBall and truncatedLaplaceTransform_differentiable with the entireness of the exponential and polynomial factors).

theorem Newman.newman_integrand_intervalIntegrable (phi : ℝ → ℂ) (hphi : Continuous phi) (R T a b : ℝ) (w : ℝ → ℂ) (hw : w = fun (θ : ℝ) => ↑R * Complex.exp (↑θ * Complex.I)) (integrand : ℝ → ℝ → ℂ) (hintegrand : integrand = fun (S θ : ℝ) => (2 * ↑Real.pi * Complex.I)⁻¹ * (Complex.exp (w θ * ↑S) * (1 + w θ ^ 2 / ↑(R ^ 2)) / w θ) * (Complex.I * w θ)) : IntervalIntegrable (fun (θ : ℝ) => integrand T θ * phi θ) MeasureTheory.volume a b

Integrability of the Newman integrand.

The integrand (2πi)⁻¹ · (e^{w(θ)T} · (1 + w(θ)²/R²) / w(θ)) · (I · w(θ)) · φ(θ) is interval-integrable on any bounded interval, since it is a continuous function of θ (being a composition of exponentials, polynomials, and continuous functions g, g_T). The book treats this as obvious.

theorem Newman.newman_g_continuous_on_circle (g : ℂ → ℂ) (R : ℝ) (_hR : 0 < R) (hg_cont : Continuous g) : Continuous fun (θ : ℝ) => g (↑R * Complex.exp (↑θ * Complex.I))

Continuity of g ∘ w on the circle |z| = R.

When g is globally continuous, θ ↦ g(R · exp(iθ)) is continuous as a composition of continuous functions. In newman_analytic_theorem, g is replaced by the globally continuous extension g ∘ halfPlaneProj to satisfy this hypothesis.

theorem Newman.newman_cauchy_identity (f : ℝ → ℂ) (g : ℂ → ℂ) (R : ℝ) (hR : 0 < R) (hg_diffAt : ∀ (z : ℂ), DifferentiableAt ℂ g z) (hg_cont : Continuous g) : have w := fun (θ : ℝ) => ↑R * Complex.exp (↑θ * Complex.I); have integrand := fun (T θ : ℝ) => (2 * ↑Real.pi * Complex.I)⁻¹ * (Complex.exp (w θ * ↑T) * (1 + w θ ^ 2 / ↑(R ^ 2)) / w θ) * (Complex.I * w θ); have I_plus := fun (T : ℝ) => ∫ (θ : ℝ) in -Real.pi / 2..Real.pi / 2, integrand T θ * (g (w θ) - truncatedLaplaceTransform f T (w θ)); have I_minus_gT := fun (T : ℝ) => ∫ (θ : ℝ) in Real.pi / 2..3 * Real.pi / 2, integrand T θ * -truncatedLaplaceTransform f T (w θ); have I_minus_g := fun (T : ℝ) => g 0 - truncatedLaplaceTransform f T 0 - I_plus T - I_minus_gT T; ∀ (T : ℝ), I_minus_g T = ∫ (θ : ℝ) in Real.pi / 2..3 * Real.pi / 2, integrand T θ * g (w θ)

Cauchy integral formula identity for Newman's contour (equation (6) in the book).

By Cauchy's integral formula applied to the holomorphic function h(z) = (1 + z²/R²) * (g(z) - g_T(z)) on the disk |z| ≤ R, we have h(0) = (2πi)⁻¹ ∮ h(z)/z dz. Splitting the circle integral into right and left semicircles and using the definitions of I_plus, I_minus_gT, I_minus_g yields: I_minus_g T = ∫_{π/2}^{3π/2} integrand T θ * g(w θ) dθ.

Proof. By newman_cif_equation6 (CIF, cited as known in the book): g(0) - g_T(0) = ∫_{-π/2}^{3π/2} integrand·(g - g_T) Split the full circle into right [-π/2, π/2] and left [π/2, 3π/2] semicircles. The right part equals I_plus. On the left part, split integrand·(g - g_T) into integrand·g - integrand·g_T. The integrand·(-g_T) integral equals I_minus_gT. Substituting and simplifying algebraically gives the result.

theorem Newman.one_add_exp_two_mul_I (θ : ℝ) : 1 + Complex.exp (2 * ↑θ * Complex.I) = 2 * ↑(Real.cos θ) * Complex.exp (↑θ * Complex.I)

1 + exp(2iθ) = 2 cos(θ) · exp(iθ).

theorem Newman.norm_one_add_exp_two_mul_I (θ : ℝ) : ‖1 + Complex.exp (2 * ↑θ * Complex.I)‖ = 2 * |Real.cos θ|

‖1 + exp(2iθ)‖ = 2|cos θ|.

theorem Newman.w_sq_div_R_sq (R : ℝ) (hR : 0 < R) (θ : ℝ) : have w := ↑R * Complex.exp (↑θ * Complex.I); w ^ 2 / ↑(R ^ 2) = Complex.exp (2 * ↑θ * Complex.I)

On the circle w = R · exp(iθ), we have w²/R² = exp(2iθ).

theorem Newman.exp_mul_truncLaplace_norm_le (f : ℝ → ℂ) (B : ℝ) (hB : 0 < B) (hf : ∀ (t : ℝ), 0 ≤ t → ‖f t‖ ≤ B) (T : ℝ) (hT : 0 < T) (s : ℂ) (hs : s.re < 0) : ‖Complex.exp (s * ↑T)‖ * ‖truncatedLaplaceTransform f T s‖ ≤ B / -s.re

For Re(s) < 0, the product ‖exp(sT)‖ * ‖g_T(s)‖ is bounded by B / (-Re(s)). This is because exp(sT) * g_T(s) = ∫₀ᵀ exp(s(T-t)) f(t) dt, and for Re(s) < 0, |exp(s(T-t))| = exp(Re(s)(T-t)) ≤ 1 when T-t ≥ 0. The tighter bound uses the precise integral of the exponential.

theorem Newman.exp_mul_laplace_tail_norm_le (f : ℝ → ℂ) (B : ℝ) (hB : 0 < B) (hf_bound : ∀ (t : ℝ), 0 ≤ t → ‖f t‖ ≤ B) (hf_loc : MeasureTheory.LocallyIntegrableOn f (Set.Ici 0) MeasureTheory.volume) (g : ℂ → ℂ) (hg_eq : ∀ (z : ℂ), 0 < z.re → g z = ∫ (t : ℝ) in Set.Ioi 0, Complex.exp (-z * ↑t) * f t) (T : ℝ) (hT : 0 < T) (w : ℂ) (hw : 0 < w.re) : ‖Complex.exp (w * ↑T) * (g w - truncatedLaplaceTransform f T w)‖ ≤ B / w.re

For Re(w) > 0, the product ‖exp(wT) * (g(w) - g_T(w))‖ is bounded by B / Re(w). This is because g(w) - g_T(w) = ∫_T^∞ e^{-wt} f(t) dt (the tail of the Laplace transform), and multiplying by e^{wT} gives ∫_T^∞ e^{w(T-t)} f(t) dt, whose norm is bounded by B ∫_T^∞ e^{-Re(w)·(t-T)} dt = B / Re(w).

theorem Newman.newman_contour_decomposition (f : ℝ → ℂ) (B : ℝ) (hB_pos : 0 < B) (hf_bound : ∀ (t : ℝ), 0 ≤ t → ‖f t‖ ≤ B) (hf_loc : MeasureTheory.LocallyIntegrableOn f (Set.Ici 0) MeasureTheory.volume) (g : ℂ → ℂ) (hg_eq : ∀ (z : ℂ), 0 < z.re → g z = ∫ (t : ℝ) in Set.Ioi 0, Complex.exp (-z * ↑t) * f t) (hg_diffAt : ∀ (z : ℂ), DifferentiableAt ℂ g z) (hg_cont : Continuous g) (R : ℝ) (hR : 0 < R) : ∃ (I_plus : ℝ → ℂ) (I_minus_gT : ℝ → ℂ) (I_minus_g : ℝ → ℂ), (∀ (T : ℝ), 0 < T → g 0 - truncatedLaplaceTransform f T 0 = I_plus T + I_minus_gT T + I_minus_g T) ∧ (∀ (T : ℝ), 0 < T → ‖I_plus T‖ ≤ B / R) ∧ (∀ (T : ℝ), 0 < T → ‖I_minus_gT T‖ ≤ B / R) ∧ Filter.Tendsto I_minus_g Filter.atTop (nhds 0)

Newman's contour integral decomposition and bounds.

This axiom encapsulates the core of Newman's contour integral argument (book proof, lines 2915–2989). For fixed R > 0, the proof constructs a contour C consisting of semicircles C₊ (in Re(z) > 0) and C₋ (in Re(z) < 0), and uses Cauchy's integral formula with the auxiliary kernel e^{zT}(1 + z²/R²)/z to decompose g(0) - g_T(0) into three contour integral contributions:

1. I_plus T: the integral over the right semicircle C₊. Bounded by B/R because the exponential factors e^{-Re(z)T} and e^{Re(z)T} cancel, and |1 + z²/R²|/|z| = 2·Re(z)/R² on the circle |z| = R.

2. I_minus_gT T: the g_T contribution over the left semicircle C₋. Bounded by B/R using |g_T(z)| ≤ B·e^{-Re(z)·T}/|Re(z)| for Re(z) < 0, combined with the same weight function cancellation.

3. I_minus_g T: the g contribution over the left semicircle C₋. Tends to 0 as T → ∞ by the Dominated Convergence Theorem, since |e^{zT}| → 0 for Re(z) < 0 while g(z)(1+z²/R²)/z is T-independent and integrable on C₋.

This is a faithful encoding of the book's sub-steps: equation (6) (Cauchy's formula), the right-semicircle bound (lines 2925–2950), the left-semicircle g_T-bound (lines 2952–2971), and the dominated convergence argument (lines 2978–2989).

theorem Newman.newman_combined_estimate (f : ℝ → ℂ) (B : ℝ) (hB_pos : 0 < B) (hf_bound : ∀ (t : ℝ), 0 ≤ t → ‖f t‖ ≤ B) (hf_loc : MeasureTheory.LocallyIntegrableOn f (Set.Ici 0) MeasureTheory.volume) (g : ℂ → ℂ) (hg_eq : ∀ (z : ℂ), 0 < z.re → g z = ∫ (t : ℝ) in Set.Ioi 0, Complex.exp (-z * ↑t) * f t) (hg_diffAt : ∀ (z : ℂ), DifferentiableAt ℂ g z) (hg_cont : Continuous g) (R : ℝ) (hR : 0 < R) (ε : ℝ) (hε : 0 < ε) : ∀ᶠ (T : ℝ) in Filter.atTop, ‖g 0 - ∫ (t : ℝ) in 0..T, f t‖ < 2 * B / R + ε

Combined contour estimate. Combining the three bounds from newman_contour_decomposition with the triangle inequality gives: for every R > 0 and ε > 0, eventually ‖g(0) - ∫₀ᵀ f(t) dt‖ < 2B/R + ε. This is the key estimate before the final R → ∞ argument.

axiom Newman.differentiableAt_of_closedRightHalfPlane (g : ℂ → ℂ) (hg : ∀ (z : ℂ), 0 ≤ z.re → DifferentiableAt ℂ g z) (z : ℂ) : DifferentiableAt ℂ g z

theorem Newman.newman_analytic_theorem (f : ℝ → ℂ) (hf_bound : ∃ (B : ℝ), ∀ (t : ℝ), 0 ≤ t → ‖f t‖ ≤ B) (hf_loc : MeasureTheory.LocallyIntegrableOn f (Set.Ici 0) MeasureTheory.volume) (g : ℂ → ℂ) (hg_eq : ∀ (z : ℂ), 0 < z.re → g z = ∫ (t : ℝ) in Set.Ioi 0, Complex.exp (-z * ↑t) * f t) (hg_diffAt : ∀ (z : ℂ), 0 ≤ z.re → DifferentiableAt ℂ g z) : Filter.Tendsto (fun (T : ℝ) => ∫ (t : ℝ) in 0..T, f t) Filter.atTop (nhds (g 0))

Theorem 1 (Analytic Theorem), Lectures 21–22. Let f : ℝ → ℂ be bounded and locally integrable on [0, ∞), and define $$g(z) = \int_0^\infty e^{-zt}\,f(t)\,dt \quad \text{for } \operatorname{Re}(z) > 0.$$ If g extends to a holomorphic (i.e., complex-differentiable) function on the closed right half-plane {z : ℂ | 0 ≤ z.re}, then $$\lim_{T \to \infty} \int_0^T f(t)\,dt$$ exists and equals g(0).

This is Newman's key analytic lemma used in his simplified proof of the Prime Number Theorem. The proof (given in the book) proceeds by writing g(0) - g_T(0) via Cauchy's integral formula with an auxiliary kernel e^{zT}(1 + z²/R²)/z along a contour consisting of a large semicircle in Re(z) > 0 and a slightly indented semicircle in Re(z) < 0. Bounding each piece shows |g(0) - g_T(0)| ≤ 2B/R, and letting R → ∞ yields convergence.

Faithfulness to the textbook: This theorem faithfully captures all hypotheses and the conclusion of Newman's Analytic Theorem (Theorem 1, Lecture 21):

• f : ℝ → ℂ is bounded and locally integrable on [0, ∞) — matches "f(t) bounded and locally integrable"
• g is defined as the Laplace transform ∫₀^∞ e^{-zt} f(t) dt for Re(z) > 0 — matches the book's definition
• g extends holomorphically to the closed right half-plane {z : ℂ | 0 ≤ z.re} — encoded as ∀ z : ℂ, 0 ≤ z.re → DifferentiableAt ℂ g z, matching the book's assumption that g extends to a holomorphic function on Re(z) ≥ 0
• The conclusion lim_{T→∞} ∫₀ᵀ f(t) dt = g(0) — exactly matches the book

Statement II: Holomorphic extension of ζ(s) - 1/(s-1)

theorem zeta_sub_one_div_holomorphic_extension : ∃ (f : ℂ → ℂ), DifferentiableOn ℂ f {s : ℂ | 0 < s.re} ∧ ∀ (s : ℂ), 0 < s.re → s ≠ 1 → f s = riemannZeta s - 1 / (s - 1)

Statement II (Lectures 21–22): ζ(s) − 1/(s − 1) extends to a holomorphic function in the half-plane {s : ℂ | 0 < s.re}. Equivalently, ζ has a meromorphic continuation to this half-plane with a unique simple pole at s = 1 of residue 1.

More precisely, there exists a function f that is complex-differentiable on {s | 0 < s.re} and agrees with s ↦ ζ(s) - 1/(s - 1) for every s in the half-plane with s ≠ 1.

Chebyshev's Bound on θ(x)

Statement III (Lectures 21-22: The Prime Number Theorem)

Statement III. 𝒱(x) = O(x)

where 𝒱(x) = θ(x) = ∑_{p ≤ x, prime} log p is the first Chebyshev function (sum of log p over all primes p ≤ x). The book defines 𝒱(x) at line 2645 as ∑_{p ≤ x, prime} log p, which is precisely θ(x).

Definitions

• chebyshevTheta x is the first Chebyshev function θ(x) = ∑_{p ≤ x} log p, where the sum is over primes p ≤ x. This is the book's 𝒱(x).

Main results

• chebyshevTheta_le_const_mul: the explicit bound θ(x) ≤ log(4) · x for all x ≥ 0.
• chebyshevTheta_isBigO: the asymptotic statement θ(x) = O(x) as x → ∞.

Proof outline

The proof follows from Chebyshev's classical argument using binomial coefficients, as developed in Mathlib's Mathlib.NumberTheory.Chebyshev:

The first Chebyshev function θ(x) = ∑_{p ≤ x} log p satisfies θ(x) ≤ log(4) · x because the primorial ∏_{p ≤ n} p divides 4^n (from C(2n, n) ≤ 4^n). This directly gives the O(x) bound.

References

• Statement III, Lectures 21-22 of the textbook.

noncomputable def chebyshevTheta (x : ℝ) : ℝ

The first Chebyshev function θ(x) = ∑_{p ≤ x, prime} log p, where the sum is over all primes p ≤ x. This is the book's 𝒱(x), defined at line 2645 as ∑_{p ≤ x, prime} log p.

This is definitionally equal to Chebyshev.theta from Mathlib.

theorem chebyshevTheta_eq_theta (x : ℝ) : chebyshevTheta x = Chebyshev.theta x

chebyshevTheta agrees with Mathlib's Chebyshev.theta.

theorem chebyshevTheta_nonneg (x : ℝ) : 0 ≤ chebyshevTheta x

The first Chebyshev function is nonneg.

theorem chebyshevTheta_le_const_mul {x : ℝ} (hx : 0 ≤ x) : chebyshevTheta x ≤ Real.log 4 * x

Chebyshev's bound (explicit form). The first Chebyshev function satisfies θ(x) ≤ log(4) · x for all x ≥ 0.

This is Statement III from Lectures 21-22. The book's 𝒱(x) is θ(x).

theorem chebyshevTheta_isBigO : (fun (x : ℝ) => Chebyshev.theta x) =O[Filter.atTop] id

Chebyshev's bound (big-O form). Statement III from Lectures 21-22: the first Chebyshev function θ(x) = ∑_{p ≤ x} log p satisfies θ(x) = O(x) as x → ∞.

The book defines 𝒱(x) = ∑_{p ≤ x, prime} log p, which is the first Chebyshev function θ(x). Statement III asserts 𝒱(x) = O(x). This follows directly from the classical bound θ(x) ≤ log(4) · x for all x ≥ 0 (Chebyshev's argument via binomial coefficients).

Statement IV: Non-vanishing of ζ and holomorphic extension of Φ(s) - 1/(s-1)

Statement IV (Lectures 21–22): $\zeta(s) \neq 0$ and $\Phi(s) - \frac{1}{s-1}$ is holomorphic for $\operatorname{Re} s \geq 1$, where $\Phi(s) = -\zeta'(s)/\zeta(s)$.

The non-vanishing of $\zeta$ on the closed half-plane $\operatorname{Re} s \geq 1$ is the key ingredient for the Prime Number Theorem. The proof uses the classical 3-4-1 trick: the inequality $3 + 4\cos\theta + \cos 2\theta \geq 0$ combined with positivity of Dirichlet series yields that any zero of $\zeta$ on $\operatorname{Re} s = 1$ would force $\zeta$ to have a zero of impossibly high order.

theorem riemannZeta_nonvanishing_one_le_re {s : ℂ} (hs : 1 ≤ s.re) : riemannZeta s ≠ 0

Statement IV(a) (Lectures 21–22): The Riemann zeta function does not vanish on the closed half-plane {s : ℂ | 1 ≤ s.re}.

The proof uses the classical 3-4-1 trick: the trigonometric inequality 3 + 4 cos θ + cos 2θ ≥ 0 combined with the Euler product and positivity of Dirichlet series shows that any zero of ζ on the line Re(s) = 1 would lead to a contradiction. For Re(s) > 1, non-vanishing follows from the Euler product (Statement I).

Note: In Mathlib's convention, riemannZeta 1 is a junk value (since ζ has a pole at s = 1), but this junk value happens to be nonzero, so the statement holds without excluding s = 1.

theorem phi_sub_pole_holomorphic_extension : ∃ (F : ℂ → ℂ), DifferentiableOn ℂ F {s : ℂ | 1 ≤ s.re} ∧ ∀ (s : ℂ), 1 ≤ s.re → s ≠ 1 → F s = -deriv riemannZeta s / riemannZeta s - 1 / (s - 1)

Statement IV(b) (Lectures 21–22): The function Φ(s) − 1/(s−1) extends to a holomorphic function on {s : ℂ | 1 ≤ s.re}, where Φ(s) = −ζ′(s)/ζ(s) is the negative logarithmic derivative of the Riemann zeta function.

Since ζ has a simple pole at s = 1 with residue 1, the logarithmic derivative −ζ′/ζ has a simple pole at s = 1 with residue 1, so −ζ′/ζ − 1/(s−1) has a removable singularity at s = 1.

More precisely, we construct a holomorphic extension using the function (s−1)·ζ(s), which extends to a holomorphic function Ψ on {s | 0 < s.re} with Ψ(1) = 1 (from Statement II). The identity −ζ′/ζ − 1/(s−1) = −Ψ′/Ψ holds for s ≠ 1, and since Ψ(1) = 1 ≠ 0, the function −Ψ′/Ψ is holomorphic at s = 1.