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number_theory.zeta_function

Definition of the Riemann zeta function #

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Main definitions: #

riemann_zeta: the Riemann zeta function ζ : ℂ → ℂ.
riemann_completed_zeta: the completed zeta function Λ : ℂ → ℂ, which satisfies Λ(s) = π ^ (-s / 2) Γ(s / 2) ζ(s) (away from the poles of Γ(s / 2)).
riemann_completed_zeta₀: the entire function Λ₀ satisfying Λ₀(s) = Λ(s) + 1 / (s - 1) - 1 / s wherever the RHS is defined.

Note that mathematically ζ(s) is undefined at s = 1, while Λ(s) is undefined at both s = 0 and s = 1. Our construction assigns some values at these points (which are not arbitrary, but I haven't checked exactly what they are).

Main results: #

differentiable_completed_zeta₀ : the function Λ₀(s) is entire.
differentiable_at_completed_zeta : the function Λ(s) is differentiable away from s = 0 and s = 1.
differentiable_at_riemann_zeta : the function ζ(s) is differentiable away from s = 1.
zeta_eq_tsum_of_one_lt_re : for 1 < re s, we have ζ(s) = ∑' (n : ℕ), 1 / (n + 1) ^ s.
riemann_completed_zeta₀_one_sub, riemann_completed_zeta_one_sub, and riemann_zeta_one_sub : functional equation relating values at s and 1 - s
riemann_zeta_neg_nat_eq_bernoulli : for any k ∈ ℕ we have the formula riemann_zeta (-k) = (-1) ^ k * bernoulli (k + 1) / (k + 1)
riemann_zeta_two_mul_nat: formula for ζ(2 * k) for k ∈ ℕ, k ≠ 0 in terms of Bernoulli numbers

Outline of proofs: #

We define two related functions on the reals, zeta_kernel₁ and zeta_kernel₂. The first is (θ (t * I) - 1) / 2, where θ is Jacobi's theta function; its Mellin transform is exactly the completed zeta function. The second is obtained by subtracting a linear combination of powers on the interval Ioc 0 1 to give a function with exponential decay at both 0 and ∞. We then define riemann_completed_zeta₀ as the Mellin transform of the second zeta kernel, and define riemann_completed_zeta and riemann_zeta from this.

Since zeta_kernel₂ has rapid decay and satisfies a functional equation relating its values at t and 1 / t, we deduce the analyticity of riemann_completed_zeta₀ and the functional equation relating its values at s and 1 - s. On the other hand, since zeta_kernel₁ can be expanded in powers of exp (-π * t) and the Mellin transform integrated term-by-term, we obtain the relation to the naive Dirichlet series ∑' (n : ℕ), 1 / (n + 1) ^ s.

noncomputable def zeta_kernel₁ (t : ℝ) :

Function whose Mellin transform is π ^ (-s) * Γ(s) * zeta (2 * s), for 1 / 2 < Re s.

Equations

zeta_kernel₁ t = ∑' (n : ℕ), ↑(rexp (-real.pi * t * (↑n + 1) ^ 2))

noncomputable def zeta_kernel₂ :

Modified zeta kernel, whose Mellin transform is entire. -

Equations

zeta_kernel₂ = zeta_kernel₁ + (set.Ioc 0 1).indicator (λ (t : ℝ), (1 - 1 / ↑(√ t)) / 2)

noncomputable def riemann_completed_zeta₀ (s : ℂ) :

The completed Riemann zeta function with its poles removed, Λ(s) + 1 / s - 1 / (s - 1).

Equations

riemann_completed_zeta₀ s = mellin zeta_kernel₂ (s / 2)

noncomputable def riemann_completed_zeta (s : ℂ) :

The completed Riemann zeta function, Λ(s), which satisfies Λ(s) = π ^ (-s / 2) Γ(s / 2) ζ(s) (up to a minor correction at s = 0).

Equations

riemann_completed_zeta s = riemann_completed_zeta₀ s - 1 / s + 1 / (s - 1)

@[irreducible]

noncomputable def riemann_zeta (a : ℂ) :

The Riemann zeta function ζ(s). We set this to be irreducible to hide messy implementation details.

Equations

riemann_zeta = function.update (λ (s : ℂ), ↑real.pi ^ (s / 2) * riemann_completed_zeta s / complex.Gamma (s / 2)) 0 ((-1) / 2)

theorem riemann_zeta_zero :

riemann_zeta 0 = (-1) / 2

We have ζ(0) = -1 / 2.

First properties of the zeta kernels #

theorem summable_exp_neg_pi_mul_nat_sq {t : ℝ} (ht : 0 < t) :

summable (λ (n : ℕ), rexp (-real.pi * t * (↑n + 1) ^ 2))

The sum defining zeta_kernel₁ is convergent.

theorem zeta_kernel₁_eq_jacobi_theta {t : ℝ} (ht : 0 < t) :

zeta_kernel₁ t = (jacobi_theta (↑t * complex.I) - 1) / 2

Relate zeta_kernel₁ to the Jacobi theta function on ℍ. (We don't use this as the definition of zeta_kernel₁, since the sum over ℕ rather than ℤ is more convenient for relating zeta to the Dirichlet series for re s > 1.)

theorem continuous_at_zeta_kernel₁ {t : ℝ} (ht : 0 < t) :

continuous_at zeta_kernel₁ t

Continuity of zeta_kernel₁.

theorem locally_integrable_zeta_kernel₁ :

measure_theory.locally_integrable_on zeta_kernel₁ (set.Ioi 0) measure_theory.measure_space.volume

Local integrability of zeta_kernel₁.

theorem locally_integrable_zeta_kernel₂ :

measure_theory.locally_integrable_on zeta_kernel₂ (set.Ioi 0) measure_theory.measure_space.volume

Local integrability of zeta_kernel₂.

theorem zeta_kernel₂_one_div {t : ℝ} (ht : 0 < t) :

zeta_kernel₂ (1 / t) = ↑(√ t) * zeta_kernel₂ t

Functional equation for zeta_kernel₂.

Bounds for zeta kernels #

We now establish asymptotic bounds for the zeta kernels as t → ∞ and t → 0, and use these to show holomorphy of their Mellin transforms (for 1 / 2 < re s for zeta_kernel₁, and all s for zeta_kernel₂).

theorem is_O_at_top_zeta_kernel₁ :

zeta_kernel₁ =O[filter.at_top ] λ (t : ℝ), rexp (-real.pi * t)

Bound for zeta_kernel₁ for large t.

theorem is_O_at_top_zeta_kernel₂ :

zeta_kernel₂ =O[filter.at_top ] λ (t : ℝ), rexp (-real.pi * t)

Bound for zeta_kernel₂ for large t.

theorem is_O_zero_zeta_kernel₂ :

zeta_kernel₂ =O[nhds_within 0 (set.Ioi 0)] λ (t : ℝ), rexp (-real.pi / t) / √ t

Precise but awkward-to-use bound for zeta_kernel₂ for t → 0.

theorem is_O_zero_zeta_kernel₂_rpow (a : ℝ) :

zeta_kernel₂ =O[nhds_within 0 (set.Ioi 0)] λ (t : ℝ), t ^ a

Weaker but more usable bound for zeta_kernel₂ for t → 0.

theorem is_O_zero_zeta_kernel₁ :

zeta_kernel₁ =O[nhds_within 0 (set.Ioi 0)] λ (t : ℝ), t ^ -(1 / 2)

Bound for zeta_kernel₁ for t → 0.

Differentiability of the completed zeta function #

theorem differentiable_at_mellin_zeta_kernel₁ {s : ℂ} (hs : 1 / 2 < s.re) :

differentiable_at ℂ (mellin zeta_kernel₁) s

The Mellin transform of the first zeta kernel is holomorphic for 1 / 2 < re s.

theorem differentiable_mellin_zeta_kernel₂ :

differentiable ℂ (mellin zeta_kernel₂)

The Mellin transform of the second zeta kernel is entire.

theorem differentiable_completed_zeta₀ :

differentiable ℂ riemann_completed_zeta₀

The modified completed Riemann zeta function Λ(s) + 1 / s - 1 / (s - 1) is entire.

theorem differentiable_at_completed_zeta {s : ℂ} (hs : s ≠ 0) (hs' : s ≠ 1) :

differentiable_at ℂ riemann_completed_zeta s

The completed Riemann zeta function Λ(s) is differentiable away from s = 0 and s = 1 (where it has simple poles).

theorem differentiable_at_riemann_zeta {s : ℂ} (hs' : s ≠ 1) :

differentiable_at ℂ riemann_zeta s

The Riemann zeta function is differentiable away from s = 1.

theorem riemann_zeta_neg_two_mul_nat_add_one (n : ℕ) :

riemann_zeta ((-2) * (↑n + 1)) = 0

The trivial zeroes of the zeta function.

def riemann_hypothesis :

Prop

A formal statement of the Riemann hypothesis – constructing a term of this type is worth a million dollars.

Equations

riemann_hypothesis = ∀ (s : ℂ), riemann_completed_zeta s = 0 → (¬∃ (n : ℕ), s = (-2) * (↑n + 1)) → s.re = 1 / 2

Relating the Mellin transforms of the two zeta kernels #

theorem has_mellin_one_div_sqrt_Ioc {s : ℂ} (hs : 1 / 2 < s.re) :

has_mellin ((set.Ioc 0 1).indicator (λ (t : ℝ), 1 / ↑(√ t))) s (1 / (s - 1 / 2))

theorem has_mellin_one_div_sqrt_sub_one_div_two_Ioc {s : ℂ} (hs : 1 / 2 < s.re) :

has_mellin ((set.Ioc 0 1).indicator (λ (t : ℝ), (1 - 1 / ↑(√ t)) / 2)) s (1 / (2 * s) - 1 / (2 * s - 1))

Evaluate the Mellin transform of the "fudge factor" in zeta_kernel₂

theorem mellin_zeta_kernel₂_eq_of_lt_re {s : ℂ} (hs : 1 / 2 < s.re) :

mellin zeta_kernel₂ s = mellin zeta_kernel₁ s + 1 / (2 * s) - 1 / (2 * s - 1)

theorem completed_zeta_eq_mellin_of_one_lt_re {s : ℂ} (hs : 1 < s.re) :

riemann_completed_zeta s = mellin zeta_kernel₁ (s / 2)

Relating the first zeta kernel to the Dirichlet series #

theorem integral_cpow_mul_exp_neg_pi_mul_sq {s : ℂ} (hs : 0 < s.re) (n : ℕ) :

∫ (t : ℝ) in set.Ioi 0, ↑t ^ (s - 1) * ↑(rexp (-real.pi * t * (↑n + 1) ^ 2)) = ↑real.pi ^ -s * complex.Gamma s * (1 / (↑n + 1) ^ (2 * s))

Auxiliary lemma for mellin_zeta_kernel₁_eq_tsum, computing the Mellin transform of an individual term in the series.

theorem mellin_zeta_kernel₁_eq_tsum {s : ℂ} (hs : 1 / 2 < s.re) :

mellin zeta_kernel₁ s = ↑real.pi ^ -s * complex.Gamma s * ∑' (n : ℕ), 1 / (↑n + 1) ^ (2 * s)

theorem completed_zeta_eq_tsum_of_one_lt_re {s : ℂ} (hs : 1 < s.re) :

riemann_completed_zeta s = ↑real.pi ^ (-s / 2) * complex.Gamma (s / 2) * ∑' (n : ℕ), 1 / (↑n + 1) ^ s

theorem zeta_eq_tsum_one_div_nat_add_one_cpow {s : ℂ} (hs : 1 < s.re) :

riemann_zeta s = ∑' (n : ℕ), 1 / (↑n + 1) ^ s

The Riemann zeta function agrees with the naive Dirichlet-series definition when the latter converges. (Note that this is false without the assumption: when re s ≤ 1 the sum is divergent, and we use a different definition to obtain the analytic continuation to all s.)

theorem zeta_eq_tsum_one_div_nat_cpow {s : ℂ} (hs : 1 < s.re) :

riemann_zeta s = ∑' (n : ℕ), 1 / ↑n ^ s

Alternate formulation of zeta_eq_tsum_one_div_nat_add_one_cpow without the + 1, using the fact that for s ≠ 0 we define 0 ^ s = 0.

theorem zeta_nat_eq_tsum_of_gt_one {k : ℕ} (hk : 1 < k) :

riemann_zeta ↑k = ∑' (n : ℕ), 1 / ↑n ^ k

Special case of zeta_eq_tsum_one_div_nat_cpow when the argument is in ℕ, so the power function can be expressed using naïve pow rather than cpow.

theorem riemann_zeta_two_mul_nat {k : ℕ} (hk : k ≠ 0) :

riemann_zeta (2 * ↑k) = (-1) ^ (k + 1) * 2 ^ (2 * k - 1) * ↑real.pi ^ (2 * k) * ↑(bernoulli (2 * k)) / ↑((2 * k).factorial)

Explicit formula for ζ (2 * k), for k ∈ ℕ with k ≠ 0: we have ζ (2 * k) = (-1) ^ (k + 1) * 2 ^ (2 * k - 1) * π ^ (2 * k) * bernoulli (2 * k) / (2 * k)!. Compare has_sum_zeta_nat for a version formulated explicitly as a sum, and riemann_zeta_neg_nat_eq_bernoulli for values at negative integers (equivalent to the above via the functional equation).

theorem riemann_zeta_two :

riemann_zeta 2 = ↑real.pi ^ 2 / 6

theorem riemann_zeta_four :

riemann_zeta 4 = ↑real.pi ^ 4 / 90

Functional equation #

theorem riemann_completed_zeta₀_one_sub (s : ℂ) :

riemann_completed_zeta₀ (1 - s) = riemann_completed_zeta₀ s

Riemann zeta functional equation, formulated for Λ₀: for any complex s we have Λ₀(1 - s) = Λ₀ s.

theorem riemann_completed_zeta_one_sub (s : ℂ) :

riemann_completed_zeta (1 - s) = riemann_completed_zeta s

Riemann zeta functional equation, formulated for Λ: for any complex s we have Λ (1 - s) = Λ s.

theorem riemann_zeta_one_sub {s : ℂ} (hs : ∀ (n : ℕ), s ≠ -↑n) (hs' : s ≠ 1) :

riemann_zeta (1 - s) = 2 ^ (1 - s) * ↑real.pi ^ -s * complex.Gamma s * complex.sin (↑real.pi * (1 - s) / 2) * riemann_zeta s

Riemann zeta functional equation, formulated for ζ: if 1 - s ∉ ℕ, then we have ζ (1 - s) = 2 ^ (1 - s) * π ^ (-s) * Γ s * sin (π * (1 - s) / 2) * ζ s.

theorem riemann_zeta_neg_nat_eq_bernoulli (k : ℕ) :

riemann_zeta (-↑k) = (-1) ^ k * ↑(bernoulli (k + 1)) / (↑k + 1)