ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- ArithmeticFunction.LSeriesSummable_mul
lemma LSeriesSummable_mul {f g : ArithmeticFunction ℂ} {s : ℂ} (hf : LSeriesSummable ↗f s)
    (hg : LSeriesSummable ↗g s) :
    LSeriesSummable ↗(f * g) s 
Arguments: {f g : ArithmeticFunction ℂ} {s : ℂ} (hf : LSeriesSummable (fun n => f n) s) (hg : LSeriesSummable (fun n => g n) s)
Goal: LSeriesSummable (fun n => (f * g) n) s
-- hasFDerivAt_jacobiTheta₂_term
lemma hasFDerivAt_jacobiTheta₂_term (n : ℤ) (z τ : ℂ) :
    HasFDerivAt (fun p : ℂ × ℂ ↦ jacobiTheta₂_term n p.1 p.2)
    (jacobiTheta₂_term_fderiv n z τ) (z, τ) 
Arguments: (n : ℤ) (z τ : ℂ)
Goal: HasFDerivAt (fun p => jacobiTheta₂_term n p.1 p.2) (jacobiTheta₂_term_fderiv n z τ) (z, τ)
-- norm_jacobiTheta₂_term
lemma norm_jacobiTheta₂_term (n : ℤ) (z τ : ℂ) :
    ‖jacobiTheta₂_term n z τ‖ = rexp (-π * n ^ 2 * τ.im - 2 * π * n * z.im) 
Arguments: (n : ℤ) (z τ : ℂ)
Goal: ‖jacobiTheta₂_term n z τ‖ = rexp (-π * ↑n ^ 2 * τ.im - 2 * π * ↑n * z.im)
-- norm_jacobiTheta₂_term_le
lemma norm_jacobiTheta₂_term_le {S T : ℝ} (hT : 0 < T) {z τ : ℂ}
    (hz : |im z| ≤ S) (hτ : T ≤ im τ) (n : ℤ) :
    ‖jacobiTheta₂_term n z τ‖ ≤ rexp (-π * (T * n ^ 2 - 2 * S * |n|)) 
Arguments: {S T : ℝ} (hT : 0 < T) {z τ : ℂ} (hz : |z.im| ≤ S) (hτ : T ≤ τ.im) (n : ℤ)
Goal: ‖jacobiTheta₂_term n z τ‖ ≤ rexp (-π * (T * ↑n ^ 2 - 2 * S * ↑|n|))
-- norm_jacobiTheta₂'_term_le
lemma norm_jacobiTheta₂'_term_le {S T : ℝ} (hT : 0 < T) {z τ : ℂ}
    (hz : |im z| ≤ S) (hτ : T ≤ im τ) (n : ℤ) :
    ‖jacobiTheta₂'_term n z τ‖ ≤ 2 * π * |n| * rexp (-π * (T * n ^ 2 - 2 * S * |n|)) 
Arguments: {S T : ℝ} (hT : 0 < T) {z τ : ℂ} (hz : |z.im| ≤ S) (hτ : T ≤ τ.im) (n : ℤ)
Goal: ‖jacobiTheta₂'_term n z τ‖ ≤ 2 * π * ↑|n| * rexp (-π * (T * ↑n ^ 2 - 2 * S * ↑|n|))
-- summable_pow_mul_jacobiTheta₂_term_bound
lemma summable_pow_mul_jacobiTheta₂_term_bound (S : ℝ) {T : ℝ} (hT : 0 < T) (k : ℕ) :
    Summable (fun n : ℤ ↦ (|n| ^ k : ℝ) * Real.exp (-π * (T * n ^ 2 - 2 * S * |n|))) 
Arguments: (S : ℝ) {T : ℝ} (hT : 0 < T) (k : ℕ)
Goal: Summable fun n => ↑|n| ^ k * rexp (-π * (T * ↑n ^ 2 - 2 * S * ↑|n|))
-- summable_jacobiTheta₂_term_iff
lemma summable_jacobiTheta₂_term_iff (z τ : ℂ) : Summable (jacobiTheta₂_term · z τ) ↔ 0 < im τ 
Arguments: (z τ : ℂ)
Goal: (Summable fun x => jacobiTheta₂_term x z τ) ↔ 0 < τ.im
-- norm_jacobiTheta₂_term_fderiv_le
lemma norm_jacobiTheta₂_term_fderiv_le (n : ℤ) (z τ : ℂ) :
    ‖jacobiTheta₂_term_fderiv n z τ‖ ≤ 3 * π * |n| ^ 2 * ‖jacobiTheta₂_term n z τ‖ 
Arguments: (n : ℤ) (z τ : ℂ)
Goal: ‖jacobiTheta₂_term_fderiv n z τ‖ ≤ 3 * π * ↑|n| ^ 2 * ‖jacobiTheta₂_term n z τ‖
-- norm_jacobiTheta₂_term_fderiv_ge
lemma norm_jacobiTheta₂_term_fderiv_ge (n : ℤ) (z τ : ℂ) :
    π * |n| ^ 2 * ‖jacobiTheta₂_term n z τ‖ ≤ ‖jacobiTheta₂_term_fderiv n z τ‖ 
Arguments: (n : ℤ) (z τ : ℂ)
Goal: π * ↑|n| ^ 2 * ‖jacobiTheta₂_term n z τ‖ ≤ ‖jacobiTheta₂_term_fderiv n z τ‖
-- summable_jacobiTheta₂_term_fderiv_iff
lemma summable_jacobiTheta₂_term_fderiv_iff (z τ : ℂ) :
    Summable (jacobiTheta₂_term_fderiv · z τ) ↔ 0 < im τ 
Arguments: (z τ : ℂ)
Goal: (Summable fun x => jacobiTheta₂_term_fderiv x z τ) ↔ 0 < τ.im
-- summable_jacobiTheta₂'_term_iff
lemma summable_jacobiTheta₂'_term_iff (z τ : ℂ) :
    Summable (jacobiTheta₂'_term · z τ) ↔ 0 < im τ 
Arguments: (z τ : ℂ)
Goal: (Summable fun x => jacobiTheta₂'_term x z τ) ↔ 0 < τ.im
-- hasSum_jacobiTheta₂_term
lemma hasSum_jacobiTheta₂_term (z : ℂ) {τ : ℂ} (hτ : 0 < im τ) :
    HasSum (fun n ↦ jacobiTheta₂_term n z τ) (jacobiTheta₂ z τ) 
Arguments: (z : ℂ) {τ : ℂ} (hτ : 0 < τ.im)
Goal: HasSum (fun n => jacobiTheta₂_term n z τ) (jacobiTheta₂ z τ)
-- hasSum_jacobiTheta₂_term_fderiv
lemma hasSum_jacobiTheta₂_term_fderiv (z : ℂ) {τ : ℂ} (hτ : 0 < im τ) :
    HasSum (fun n ↦ jacobiTheta₂_term_fderiv n z τ) (jacobiTheta₂_fderiv z τ) 
Arguments: (z : ℂ) {τ : ℂ} (hτ : 0 < τ.im)
Goal: HasSum (fun n => jacobiTheta₂_term_fderiv n z τ) (jacobiTheta₂_fderiv z τ)
-- hasSum_jacobiTheta₂'_term
lemma hasSum_jacobiTheta₂'_term (z : ℂ) {τ : ℂ} (hτ : 0 < im τ) :
    HasSum (fun n ↦ jacobiTheta₂'_term n z τ) (jacobiTheta₂' z τ) 
Arguments: (z : ℂ) {τ : ℂ} (hτ : 0 < τ.im)
Goal: HasSum (fun n => jacobiTheta₂'_term n z τ) (jacobiTheta₂' z τ)
-- jacobiTheta₂_undef
lemma jacobiTheta₂_undef (z : ℂ) {τ : ℂ} (hτ : im τ ≤ 0) : jacobiTheta₂ z τ = 0 
Arguments: (z : ℂ) {τ : ℂ} (hτ : τ.im ≤ 0)
Goal: jacobiTheta₂ z τ = 0
-- jacobiTheta₂_fderiv_undef
lemma jacobiTheta₂_fderiv_undef (z : ℂ) {τ : ℂ} (hτ : im τ ≤ 0) : jacobiTheta₂_fderiv z τ = 0 
Arguments: (z : ℂ) {τ : ℂ} (hτ : τ.im ≤ 0)
Goal: jacobiTheta₂_fderiv z τ = 0
-- jacobiTheta₂'_undef
lemma jacobiTheta₂'_undef (z : ℂ) {τ : ℂ} (hτ : im τ ≤ 0) : jacobiTheta₂' z τ = 0 
Arguments: (z : ℂ) {τ : ℂ} (hτ : τ.im ≤ 0)
Goal: jacobiTheta₂' z τ = 0
-- hasFDerivAt_jacobiTheta₂
lemma hasFDerivAt_jacobiTheta₂ (z : ℂ) {τ : ℂ} (hτ : 0 < im τ) :
    HasFDerivAt (fun p : ℂ × ℂ ↦ jacobiTheta₂ p.1 p.2) (jacobiTheta₂_fderiv z τ) (z, τ) 
Arguments: (z : ℂ) {τ : ℂ} (hτ : 0 < τ.im)
Goal: HasFDerivAt (fun p => jacobiTheta₂ p.1 p.2) (jacobiTheta₂_fderiv z τ) (z, τ)
-- continuousAt_jacobiTheta₂
lemma continuousAt_jacobiTheta₂ (z : ℂ) {τ : ℂ} (hτ : 0 < im τ) :
    ContinuousAt (fun p : ℂ × ℂ ↦ jacobiTheta₂ p.1 p.2) (z, τ) 
Arguments: (z : ℂ) {τ : ℂ} (hτ : 0 < τ.im)
Goal: ContinuousAt (fun p => jacobiTheta₂ p.1 p.2) (z, τ)
-- differentiableAt_jacobiTheta₂_fst
lemma differentiableAt_jacobiTheta₂_fst (z : ℂ) {τ : ℂ} (hτ : 0 < im τ) :
    DifferentiableAt ℂ (jacobiTheta₂ · τ) z 
Arguments: (z : ℂ) {τ : ℂ} (hτ : 0 < τ.im)
Goal: DifferentiableAt ℂ (fun x => jacobiTheta₂ x τ) z
-- differentiableAt_jacobiTheta₂_snd
lemma differentiableAt_jacobiTheta₂_snd (z : ℂ) {τ : ℂ} (hτ : 0 < im τ) :
    DifferentiableAt ℂ (jacobiTheta₂ z) τ 
Arguments: (z : ℂ) {τ : ℂ} (hτ : 0 < τ.im)
Goal: DifferentiableAt ℂ (jacobiTheta₂ z) τ
-- hasDerivAt_jacobiTheta₂_fst
lemma hasDerivAt_jacobiTheta₂_fst (z : ℂ) {τ : ℂ} (hτ : 0 < im τ) :
    HasDerivAt (jacobiTheta₂ · τ) (jacobiTheta₂' z τ) z 
Arguments: (z : ℂ) {τ : ℂ} (hτ : 0 < τ.im)
Goal: HasDerivAt (fun x => jacobiTheta₂ x τ) (jacobiTheta₂' z τ) z
-- continuousAt_jacobiTheta₂'
lemma continuousAt_jacobiTheta₂' (z : ℂ) {τ : ℂ} (hτ : 0 < im τ) :
    ContinuousAt (fun p : ℂ × ℂ ↦ jacobiTheta₂' p.1 p.2) (z, τ) 
Arguments: (z : ℂ) {τ : ℂ} (hτ : 0 < τ.im)
Goal: ContinuousAt (fun p => jacobiTheta₂' p.1 p.2) (z, τ)
-- jacobiTheta₂_add_right
lemma jacobiTheta₂_add_right (z τ : ℂ) : jacobiTheta₂ z (τ + 2) = jacobiTheta₂ z τ 
Arguments: (z τ : ℂ)
Goal: jacobiTheta₂ z (τ + 2) = jacobiTheta₂ z τ
-- jacobiTheta₂_add_left
lemma jacobiTheta₂_add_left (z τ : ℂ) : jacobiTheta₂ (z + 1) τ = jacobiTheta₂ z τ 
Arguments: (z τ : ℂ)
Goal: jacobiTheta₂ (z + 1) τ = jacobiTheta₂ z τ
-- jacobiTheta₂_add_left'
lemma jacobiTheta₂_add_left' (z τ : ℂ) :
    jacobiTheta₂ (z + τ) τ = cexp (-π * I * (τ + 2 * z)) * jacobiTheta₂ z τ 
Arguments: (z τ : ℂ)
Goal: jacobiTheta₂ (z + τ) τ = cexp (-↑π * Complex.I * (τ + 2 * z)) * jacobiTheta₂ z τ
-- jacobiTheta₂_neg_left
lemma jacobiTheta₂_neg_left (z τ : ℂ) : jacobiTheta₂ (-z) τ = jacobiTheta₂ z τ 
Arguments: (z τ : ℂ)
Goal: jacobiTheta₂ (-z) τ = jacobiTheta₂ z τ
-- jacobiTheta₂_conj
lemma jacobiTheta₂_conj (z τ : ℂ) :
    conj (jacobiTheta₂ z τ) = jacobiTheta₂ (conj z) (-conj τ) 
Arguments: (z τ : ℂ)
Goal: (starRingEnd ℂ) (jacobiTheta₂ z τ) = jacobiTheta₂ ((starRingEnd ℂ) z) (-(starRingEnd ℂ) τ)
-- jacobiTheta₂'_add_right
lemma jacobiTheta₂'_add_right (z τ : ℂ) : jacobiTheta₂' z (τ + 2) = jacobiTheta₂' z τ 
Arguments: (z τ : ℂ)
Goal: jacobiTheta₂' z (τ + 2) = jacobiTheta₂' z τ
-- jacobiTheta₂'_add_left
lemma jacobiTheta₂'_add_left (z τ : ℂ) : jacobiTheta₂' (z + 1) τ = jacobiTheta₂' z τ 
Arguments: (z τ : ℂ)
Goal: jacobiTheta₂' (z + 1) τ = jacobiTheta₂' z τ
-- jacobiTheta₂'_add_left'
lemma jacobiTheta₂'_add_left' (z τ : ℂ) :
    jacobiTheta₂' (z + τ) τ =
      cexp (-π * I * (τ + 2 * z)) * (jacobiTheta₂' z τ - 2 * π * I * jacobiTheta₂ z τ) 
Arguments: (z τ : ℂ)
Goal: jacobiTheta₂' (z + τ) τ =
  cexp (-↑π * Complex.I * (τ + 2 * z)) * (jacobiTheta₂' z τ - 2 * ↑π * Complex.I * jacobiTheta₂ z τ)
-- jacobiTheta₂'_neg_left
lemma jacobiTheta₂'_neg_left (z τ : ℂ) : jacobiTheta₂' (-z) τ = -jacobiTheta₂' z τ 
Arguments: (z τ : ℂ)
Goal: jacobiTheta₂' (-z) τ = -jacobiTheta₂' z τ
-- jacobiTheta₂'_conj
lemma jacobiTheta₂'_conj (z τ : ℂ) :
    conj (jacobiTheta₂' z τ) = jacobiTheta₂' (conj z) (-conj τ) 
Arguments: (z τ : ℂ)
Goal: (starRingEnd ℂ) (jacobiTheta₂' z τ) = jacobiTheta₂' ((starRingEnd ℂ) z) (-(starRingEnd ℂ) τ)
-- jacobiTheta₂_functional_equation
theorem jacobiTheta₂_functional_equation (z τ : ℂ) : jacobiTheta₂ z τ =
    1 / (-I * τ) ^ (1 / 2 : ℂ) * cexp (-π * I * z ^ 2 / τ) * jacobiTheta₂ (z / τ) (-1 / τ) 
Arguments: (z τ : ℂ)
Goal: jacobiTheta₂ z τ = 1 / (-Complex.I * τ) ^ (1 / 2) * cexp (-↑π * Complex.I * z ^ 2 / τ) * jacobiTheta₂ (z / τ) (-1 / τ)
-- jacobiTheta₂'_functional_equation
theorem jacobiTheta₂'_functional_equation (z τ : ℂ) :
    jacobiTheta₂' z τ = 1 / (-I * τ) ^ (1 / 2 : ℂ) * cexp (-π * I * z ^ 2 / τ) / τ *
      (jacobiTheta₂' (z / τ) (-1 / τ) - 2 * π * I * z * jacobiTheta₂ (z / τ) (-1 / τ)) 
Arguments: (z τ : ℂ)
Goal: jacobiTheta₂' z τ =
  1 / (-Complex.I * τ) ^ (1 / 2) * cexp (-↑π * Complex.I * z ^ 2 / τ) / τ *
    (jacobiTheta₂' (z / τ) (-1 / τ) - 2 * ↑π * Complex.I * z * jacobiTheta₂ (z / τ) (-1 / τ))
-- bernoulliFun_eval_zero
theorem bernoulliFun_eval_zero (k : ℕ) : bernoulliFun k 0 = bernoulli k 
Arguments: (k : ℕ)
Goal: bernoulliFun k 0 = ↑(bernoulli k)
-- bernoulliFun_endpoints_eq_of_ne_one
theorem bernoulliFun_endpoints_eq_of_ne_one {k : ℕ} (hk : k ≠ 1) :
    bernoulliFun k 1 = bernoulliFun k 0 
Arguments: {k : ℕ} (hk : k ≠ 1)
Goal: bernoulliFun k 1 = bernoulliFun k 0
-- bernoulliFun_eval_one
theorem bernoulliFun_eval_one (k : ℕ) : bernoulliFun k 1 = bernoulliFun k 0 + ite (k = 1) 1 0 
Arguments: (k : ℕ)
Goal: bernoulliFun k 1 = bernoulliFun k 0 + if k = 1 then 1 else 0
-- hasDerivAt_bernoulliFun
theorem hasDerivAt_bernoulliFun (k : ℕ) (x : ℝ) :
    HasDerivAt (bernoulliFun k) (k * bernoulliFun (k - 1) x) x 
Arguments: (k : ℕ) (x : ℝ)
Goal: HasDerivAt (bernoulliFun k) (↑k * bernoulliFun (k - 1) x) x
-- antideriv_bernoulliFun
theorem antideriv_bernoulliFun (k : ℕ) (x : ℝ) :
    HasDerivAt (fun x => bernoulliFun (k + 1) x / (k + 1)) (bernoulliFun k x) x 
Arguments: (k : ℕ) (x : ℝ)
Goal: HasDerivAt (fun x => bernoulliFun (k + 1) x / (↑k + 1)) (bernoulliFun k x) x
-- integral_bernoulliFun_eq_zero
theorem integral_bernoulliFun_eq_zero {k : ℕ} (hk : k ≠ 0) :
    ∫ x : ℝ in (0)..1, bernoulliFun k x = 0 
Arguments: {k : ℕ} (hk : k ≠ 0)
Goal: ∫ (x : ℝ) in 0 ..1, bernoulliFun k x = 0
-- bernoulliFourierCoeff_recurrence
theorem bernoulliFourierCoeff_recurrence (k : ℕ) {n : ℤ} (hn : n ≠ 0) :
    bernoulliFourierCoeff k n =
      1 / (-2 * π * I * n) * (ite (k = 1) 1 0 - k * bernoulliFourierCoeff (k - 1) n) 
Arguments: (k : ℕ) {n : ℤ} (hn : n ≠ 0)
Goal: bernoulliFourierCoeff k n =
  1 / (-2 * ↑π * Complex.I * ↑n) * ((if k = 1 then 1 else 0) - ↑k * bernoulliFourierCoeff (k - 1) n)
-- bernoulli_zero_fourier_coeff
theorem bernoulli_zero_fourier_coeff {n : ℤ} (hn : n ≠ 0) : bernoulliFourierCoeff 0 n = 0 
Arguments: {n : ℤ} (hn : n ≠ 0)
Goal: bernoulliFourierCoeff 0 n = 0
-- bernoulliFourierCoeff_zero
theorem bernoulliFourierCoeff_zero {k : ℕ} (hk : k ≠ 0) : bernoulliFourierCoeff k 0 = 0 
Arguments: {k : ℕ} (hk : k ≠ 0)
Goal: bernoulliFourierCoeff k 0 = 0
-- bernoulliFourierCoeff_eq
theorem bernoulliFourierCoeff_eq {k : ℕ} (hk : k ≠ 0) (n : ℤ) :
    bernoulliFourierCoeff k n = -k ! / (2 * π * I * n) ^ k 
Arguments: {k : ℕ} (hk : k ≠ 0) (n : ℤ)
Goal: bernoulliFourierCoeff k n = -↑k ! / (2 * ↑π * Complex.I * ↑n) ^ k
-- periodizedBernoulli.continuous
theorem periodizedBernoulli.continuous {k : ℕ} (hk : k ≠ 1) : Continuous (periodizedBernoulli k) 
Arguments: {k : ℕ} (hk : k ≠ 1)
Goal: Continuous (periodizedBernoulli k)
-- fourierCoeff_bernoulli_eq
theorem fourierCoeff_bernoulli_eq {k : ℕ} (hk : k ≠ 0) (n : ℤ) :
    fourierCoeff ((↑) ∘ periodizedBernoulli k : 𝕌 → ℂ) n = -k ! / (2 * π * I * n) ^ k 
Arguments: {k : ℕ} (hk : k ≠ 0) (n : ℤ)
Goal: fourierCoeff (Complex.ofReal' ∘ periodizedBernoulli k) n = -↑k ! / (2 * ↑π * Complex.I * ↑n) ^ k
-- summable_bernoulli_fourier
theorem summable_bernoulli_fourier {k : ℕ} (hk : 2 ≤ k) :
    Summable (fun n => -k ! / (2 * π * I * n) ^ k : ℤ → ℂ) 
Arguments: {k : ℕ} (hk : 2 ≤ k)
Goal: Summable fun n => -↑k ! / (2 * ↑π * Complex.I * ↑n) ^ k
-- hasSum_one_div_pow_mul_fourier_mul_bernoulliFun
theorem hasSum_one_div_pow_mul_fourier_mul_bernoulliFun {k : ℕ} (hk : 2 ≤ k) {x : ℝ}
    (hx : x ∈ Icc (0 : ℝ) 1) :
    HasSum (fun n : ℤ => 1 / (n : ℂ) ^ k * fourier n (x : 𝕌))
      (-(2 * π * I) ^ k / k ! * bernoulliFun k x) 
Arguments: {k : ℕ} (hk : 2 ≤ k) {x : ℝ} (hx : x ∈ Set.Icc 0 1)
Goal: HasSum (fun n => 1 / ↑n ^ k * (fourier n) ↑x) (-(2 * ↑π * Complex.I) ^ k / ↑k ! * ↑(bernoulliFun k x))
-- hasSum_one_div_nat_pow_mul_fourier
theorem hasSum_one_div_nat_pow_mul_fourier {k : ℕ} (hk : 2 ≤ k) {x : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) :
    HasSum
      (fun n : ℕ =>
        (1 : ℂ) / (n : ℂ) ^ k * (fourier n (x : 𝕌) + (-1 : ℂ) ^ k * fourier (-n) (x : 𝕌)))
      (-(2 * π * I) ^ k / k ! * bernoulliFun k x) 
Arguments: {k : ℕ} (hk : 2 ≤ k) {x : ℝ} (hx : x ∈ Set.Icc 0 1)
Goal: HasSum (fun n => 1 / ↑n ^ k * ((fourier ↑n) ↑x + (-1) ^ k * (fourier (-↑n)) ↑x))
  (-(2 * ↑π * Complex.I) ^ k / ↑k ! * ↑(bernoulliFun k x))
-- hasSum_one_div_nat_pow_mul_cos
theorem hasSum_one_div_nat_pow_mul_cos {k : ℕ} (hk : k ≠ 0) {x : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) :
    HasSum (fun n : ℕ => 1 / (n : ℝ) ^ (2 * k) * Real.cos (2 * π * n * x))
      ((-1 : ℝ) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! *
        (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k))).eval x) 
Arguments: {k : ℕ} (hk : k ≠ 0) {x : ℝ} (hx : x ∈ Set.Icc 0 1)
Goal: HasSum (fun n => 1 / ↑n ^ (2 * k) * Real.cos (2 * π * ↑n * x))
  ((-1) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / ↑(2 * k)! *
    Polynomial.eval x (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k))))
-- hasSum_one_div_nat_pow_mul_sin
theorem hasSum_one_div_nat_pow_mul_sin {k : ℕ} (hk : k ≠ 0) {x : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) :
    HasSum (fun n : ℕ => 1 / (n : ℝ) ^ (2 * k + 1) * Real.sin (2 * π * n * x))
      ((-1 : ℝ) ^ (k + 1) * (2 * π) ^ (2 * k + 1) / 2 / (2 * k + 1)! *
        (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k + 1))).eval x) 
Arguments: {k : ℕ} (hk : k ≠ 0) {x : ℝ} (hx : x ∈ Set.Icc 0 1)
Goal: HasSum (fun n => 1 / ↑n ^ (2 * k + 1) * Real.sin (2 * π * ↑n * x))
  ((-1) ^ (k + 1) * (2 * π) ^ (2 * k + 1) / 2 / ↑(2 * k + 1)! *
    Polynomial.eval x (Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k + 1))))
-- hasSum_zeta_nat
theorem hasSum_zeta_nat {k : ℕ} (hk : k ≠ 0) :
    HasSum (fun n : ℕ => 1 / (n : ℝ) ^ (2 * k))
      ((-1 : ℝ) ^ (k + 1) * (2 : ℝ) ^ (2 * k - 1) * π ^ (2 * k) *
        bernoulli (2 * k) / (2 * k)!) 
Arguments: {k : ℕ} (hk : k ≠ 0)
Goal: HasSum (fun n => 1 / ↑n ^ (2 * k)) ((-1) ^ (k + 1) * 2 ^ (2 * k - 1) * π ^ (2 * k) * ↑(bernoulli (2 * k)) / ↑(2 * k)!)
-- hasSum_zeta_two
theorem hasSum_zeta_two : HasSum (fun n : ℕ => (1 : ℝ) / (n : ℝ) ^ 2) (π ^ 2 / 6) 
Goal: HasSum (fun n => 1 / ↑n ^ 2) (π ^ 2 / 6)
-- hasSum_zeta_four
theorem hasSum_zeta_four : HasSum (fun n : ℕ => (1 : ℝ) / (n : ℝ) ^ 4) (π ^ 4 / 90) 
Goal: HasSum (fun n => 1 / ↑n ^ 4) (π ^ 4 / 90)
-- Polynomial.bernoulli_three_eval_one_quarter
theorem Polynomial.bernoulli_three_eval_one_quarter :
    (Polynomial.bernoulli 3).eval (1 / 4) = 3 / 64 
Goal: Polynomial.eval (1 / 4) (Polynomial.bernoulli 3) = 3 / 64
-- hasSum_L_function_mod_four_eval_three
theorem hasSum_L_function_mod_four_eval_three :
    HasSum (fun n : ℕ => (1 : ℝ) / (n : ℝ) ^ 3 * Real.sin (π * n / 2)) (π ^ 3 / 32) 
Goal: HasSum (fun n => 1 / ↑n ^ 3 * Real.sin (π * ↑n / 2)) (π ^ 3 / 32)
-- HurwitzZeta.completedHurwitzZetaEven_zero
lemma HurwitzZeta.completedHurwitzZetaEven_zero (s : ℂ) :
    completedHurwitzZetaEven 0 s = completedRiemannZeta s 
Arguments: (s : ℂ)
Goal: HurwitzZeta.completedHurwitzZetaEven 0 s = completedRiemannZeta s
-- HurwitzZeta.completedHurwitzZetaEven₀_zero
lemma HurwitzZeta.completedHurwitzZetaEven₀_zero (s : ℂ) :
    completedHurwitzZetaEven₀ 0 s = completedRiemannZeta₀ s 
Arguments: (s : ℂ)
Goal: HurwitzZeta.completedHurwitzZetaEven₀ 0 s = completedRiemannZeta₀ s
-- HurwitzZeta.completedCosZeta_zero
lemma HurwitzZeta.completedCosZeta_zero (s : ℂ) :
    completedCosZeta 0 s = completedRiemannZeta s 
Arguments: (s : ℂ)
Goal: HurwitzZeta.completedCosZeta 0 s = completedRiemannZeta s
-- HurwitzZeta.completedCosZeta₀_zero
lemma HurwitzZeta.completedCosZeta₀_zero (s : ℂ) :
    completedCosZeta₀ 0 s = completedRiemannZeta₀ s 
Arguments: (s : ℂ)
Goal: HurwitzZeta.completedCosZeta₀ 0 s = completedRiemannZeta₀ s
-- completedRiemannZeta_eq
lemma completedRiemannZeta_eq (s : ℂ) :
    completedRiemannZeta s = completedRiemannZeta₀ s - 1 / s - 1 / (1 - s) 
Arguments: (s : ℂ)
Goal: completedRiemannZeta s = completedRiemannZeta₀ s - 1 / s - 1 / (1 - s)
-- differentiable_completedZeta₀
theorem differentiable_completedZeta₀ : Differentiable ℂ completedRiemannZeta₀ 
Goal: Differentiable ℂ completedRiemannZeta₀
-- differentiableAt_completedZeta
theorem differentiableAt_completedZeta {s : ℂ} (hs : s ≠ 0) (hs' : s ≠ 1) :
    DifferentiableAt ℂ completedRiemannZeta s 
Arguments: {s : ℂ} (hs : s ≠ 0) (hs' : s ≠ 1)
Goal: DifferentiableAt ℂ completedRiemannZeta s
-- completedRiemannZeta₀_one_sub
theorem completedRiemannZeta₀_one_sub (s : ℂ) :
    completedRiemannZeta₀ (1 - s) = completedRiemannZeta₀ s 
Arguments: (s : ℂ)
Goal: completedRiemannZeta₀ (1 - s) = completedRiemannZeta₀ s
-- completedRiemannZeta_one_sub
theorem completedRiemannZeta_one_sub (s : ℂ) :
    completedRiemannZeta (1 - s) = completedRiemannZeta s 
Arguments: (s : ℂ)
Goal: completedRiemannZeta (1 - s) = completedRiemannZeta s
-- completedRiemannZeta_residue_one
lemma completedRiemannZeta_residue_one :
    Tendsto (fun s ↦ (s - 1) * completedRiemannZeta s) (𝓝[≠] 1) (𝓝 1) 
Goal: Filter.Tendsto (fun s => (s - 1) * completedRiemannZeta s) (𝓝[≠] 1) (𝓝 1)
-- HurwitzZeta.hurwitzZetaEven_zero
lemma HurwitzZeta.hurwitzZetaEven_zero : hurwitzZetaEven 0 = riemannZeta 
Goal: HurwitzZeta.hurwitzZetaEven 0 = riemannZeta
-- HurwitzZeta.cosZeta_zero
lemma HurwitzZeta.cosZeta_zero : cosZeta 0 = riemannZeta 
Goal: HurwitzZeta.cosZeta 0 = riemannZeta
-- HurwitzZeta.hurwitzZeta_zero
lemma HurwitzZeta.hurwitzZeta_zero : hurwitzZeta 0 = riemannZeta 
Goal: HurwitzZeta.hurwitzZeta 0 = riemannZeta
-- HurwitzZeta.expZeta_zero
lemma HurwitzZeta.expZeta_zero : expZeta 0 = riemannZeta 
Goal: HurwitzZeta.expZeta 0 = riemannZeta
-- differentiableAt_riemannZeta
theorem differentiableAt_riemannZeta {s : ℂ} (hs' : s ≠ 1) : DifferentiableAt ℂ riemannZeta s 
Arguments: {s : ℂ} (hs' : s ≠ 1)
Goal: DifferentiableAt ℂ riemannZeta s
-- riemannZeta_zero
theorem riemannZeta_zero : riemannZeta 0 = -1 / 2 
Goal: riemannZeta 0 = -1 / 2
-- riemannZeta_def_of_ne_zero
lemma riemannZeta_def_of_ne_zero {s : ℂ} (hs : s ≠ 0) :
    riemannZeta s = completedRiemannZeta s / Gammaℝ s 
Arguments: {s : ℂ} (hs : s ≠ 0)
Goal: riemannZeta s = completedRiemannZeta s / s.Gammaℝ
-- riemannZeta_neg_two_mul_nat_add_one
theorem riemannZeta_neg_two_mul_nat_add_one (n : ℕ) : riemannZeta (-2 * (n + 1)) = 0 
Arguments: (n : ℕ)
Goal: riemannZeta (-2 * (↑n + 1)) = 0
-- riemannZeta_one_sub
theorem riemannZeta_one_sub {s : ℂ} (hs : ∀ n : ℕ, s ≠ -n) (hs' : s ≠ 1) :
    riemannZeta (1 - s) = 2 * (2 * π) ^ (-s) * Gamma s * cos (π * s / 2) * riemannZeta s 
Arguments: {s : ℂ} (hs : ∀ (n : ℕ), s ≠ -↑n) (hs' : s ≠ 1)
Goal: riemannZeta (1 - s) = 2 * (2 * ↑π) ^ (-s) * Complex.Gamma s * Complex.cos (↑π * s / 2) * riemannZeta s
-- completedZeta_eq_tsum_of_one_lt_re
theorem completedZeta_eq_tsum_of_one_lt_re {s : ℂ} (hs : 1 < re s) :
    completedRiemannZeta s =
      (π : ℂ) ^ (-s / 2) * Gamma (s / 2) * ∑' n : ℕ, 1 / (n : ℂ) ^ s 
Arguments: {s : ℂ} (hs : 1 < s.re)
Goal: completedRiemannZeta s = ↑π ^ (-s / 2) * Complex.Gamma (s / 2) * ∑' (n : ℕ), 1 / ↑n ^ s
-- zeta_eq_tsum_one_div_nat_cpow
theorem zeta_eq_tsum_one_div_nat_cpow {s : ℂ} (hs : 1 < re s) :
    riemannZeta s = ∑' n : ℕ, 1 / (n : ℂ) ^ s 
Arguments: {s : ℂ} (hs : 1 < s.re)
Goal: riemannZeta s = ∑' (n : ℕ), 1 / ↑n ^ s
-- zeta_eq_tsum_one_div_nat_add_one_cpow
theorem zeta_eq_tsum_one_div_nat_add_one_cpow {s : ℂ} (hs : 1 < re s) :
    riemannZeta s = ∑' n : ℕ, 1 / (n + 1 : ℂ) ^ s 
Arguments: {s : ℂ} (hs : 1 < s.re)
Goal: riemannZeta s = ∑' (n : ℕ), 1 / (↑n + 1) ^ s
-- zeta_nat_eq_tsum_of_gt_one
theorem zeta_nat_eq_tsum_of_gt_one {k : ℕ} (hk : 1 < k) :
    riemannZeta k = ∑' n : ℕ, 1 / (n : ℂ) ^ k 
Arguments: {k : ℕ} (hk : 1 < k)
Goal: riemannZeta ↑k = ∑' (n : ℕ), 1 / ↑n ^ k
-- riemannZeta_residue_one
lemma riemannZeta_residue_one : Tendsto (fun s ↦ (s - 1) * riemannZeta s) (𝓝[≠] 1) (𝓝 1) 
Goal: Filter.Tendsto (fun s => (s - 1) * riemannZeta s) (𝓝[≠] 1) (𝓝 1)
-- LSeries.term_add
lemma LSeries.term_add (f g : ℕ → ℂ) (s : ℂ) : term (f + g) s = term f s + term g s 
Arguments: (f g : ℕ → ℂ) (s : ℂ)
Goal: LSeries.term (f + g) s = LSeries.term f s + LSeries.term g s
-- LSeries.term_add_apply
lemma LSeries.term_add_apply (f g : ℕ → ℂ) (s : ℂ) (n : ℕ) :
    term (f + g) s n = term f s n + term g s n 
Arguments: (f g : ℕ → ℂ) (s : ℂ) (n : ℕ)
Goal: LSeries.term (f + g) s n = LSeries.term f s n + LSeries.term g s n
-- LSeriesHasSum.add
lemma LSeriesHasSum.add {f g : ℕ → ℂ} {s a b : ℂ} (hf : LSeriesHasSum f s a)
    (hg : LSeriesHasSum g s b) :
    LSeriesHasSum (f + g) s (a + b) 
Arguments: {f g : ℕ → ℂ} {s a b : ℂ} (hf : LSeriesHasSum f s a) (hg : LSeriesHasSum g s b)
Goal: LSeriesHasSum (f + g) s (a + b)
-- LSeriesSummable.add
lemma LSeriesSummable.add {f g : ℕ → ℂ} {s : ℂ} (hf : LSeriesSummable f s)
    (hg : LSeriesSummable g s) :
    LSeriesSummable (f + g) s 
Arguments: {f g : ℕ → ℂ} {s : ℂ} (hf : LSeriesSummable f s) (hg : LSeriesSummable g s)
Goal: LSeriesSummable (f + g) s
-- LSeries_add
@[simp]
lemma LSeries_add {f g : ℕ → ℂ} {s : ℂ} (hf : LSeriesSummable f s) (hg : LSeriesSummable g s) :
    LSeries (f + g) s = LSeries f s + LSeries g s 
Arguments: {f g : ℕ → ℂ} {s : ℂ} (hf : LSeriesSummable f s) (hg : LSeriesSummable g s)
Goal: LSeries (f + g) s = LSeries f s + LSeries g s
-- LSeries.term_neg
lemma LSeries.term_neg (f : ℕ → ℂ) (s : ℂ) : term (-f) s = -term f s 
Arguments: (f : ℕ → ℂ) (s : ℂ)
Goal: LSeries.term (-f) s = -LSeries.term f s
-- LSeries.term_neg_apply
lemma LSeries.term_neg_apply (f : ℕ → ℂ) (s : ℂ) (n : ℕ) : term (-f) s n = -term f s n 
Arguments: (f : ℕ → ℂ) (s : ℂ) (n : ℕ)
Goal: LSeries.term (-f) s n = -LSeries.term f s n
-- LSeriesHasSum.neg
lemma LSeriesHasSum.neg {f : ℕ → ℂ} {s a : ℂ} (hf : LSeriesHasSum f s a) :
    LSeriesHasSum (-f) s (-a) 
Arguments: {f : ℕ → ℂ} {s a : ℂ} (hf : LSeriesHasSum f s a)
Goal: LSeriesHasSum (-f) s (-a)
-- LSeriesSummable.neg
lemma LSeriesSummable.neg {f : ℕ → ℂ} {s : ℂ} (hf : LSeriesSummable f s) :
    LSeriesSummable (-f) s 
Arguments: {f : ℕ → ℂ} {s : ℂ} (hf : LSeriesSummable f s)
Goal: LSeriesSummable (-f) s
-- LSeriesSummable.neg_iff
@[simp]
lemma LSeriesSummable.neg_iff {f : ℕ → ℂ} {s : ℂ} :
    LSeriesSummable (-f) s ↔ LSeriesSummable f s 
Arguments: {f : ℕ → ℂ} {s : ℂ}
Goal: LSeriesSummable (-f) s ↔ LSeriesSummable f s
-- LSeries_neg
@[simp]
lemma LSeries_neg (f : ℕ → ℂ) (s : ℂ) : LSeries (-f) s = -LSeries f s 
Arguments: (f : ℕ → ℂ) (s : ℂ)
Goal: LSeries (-f) s = -LSeries f s
-- LSeries.term_sub
lemma LSeries.term_sub (f g : ℕ → ℂ) (s : ℂ) : term (f - g) s = term f s - term g s 
Arguments: (f g : ℕ → ℂ) (s : ℂ)
Goal: LSeries.term (f - g) s = LSeries.term f s - LSeries.term g s
-- LSeries.term_sub_apply
lemma LSeries.term_sub_apply (f g : ℕ → ℂ) (s : ℂ) (n : ℕ) :
    term (f - g) s n = term f s n - term g s n 
Arguments: (f g : ℕ → ℂ) (s : ℂ) (n : ℕ)
Goal: LSeries.term (f - g) s n = LSeries.term f s n - LSeries.term g s n
-- LSeriesHasSum.sub
lemma LSeriesHasSum.sub {f g : ℕ → ℂ} {s a b : ℂ} (hf : LSeriesHasSum f s a)
    (hg : LSeriesHasSum g s b) :
    LSeriesHasSum (f - g) s (a - b) 
Arguments: {f g : ℕ → ℂ} {s a b : ℂ} (hf : LSeriesHasSum f s a) (hg : LSeriesHasSum g s b)
Goal: LSeriesHasSum (f - g) s (a - b)