Theory Bertrand
theory Bertrand
imports
Complex_Main
"HOL-Number_Theory.Number_Theory"
"HOL-Library.Discrete"
"HOL-Decision_Procs.Approximation_Bounds"
"HOL-Library.Code_Target_Numeral"
Pratt_Certificate.Pratt_Certificate
begin
subsection ‹Auxiliary facts›
lemma ln_2_le: "ln 2 ≤ 355 / (512 :: real)"
proof -
have "ln 2 ≤ real_of_float (ub_ln2 12)" by (rule ub_ln2)
also have "ub_ln2 12 = Float 5680 (- 13)" by code_simp
finally show ?thesis by simp
qed
lemma ln_2_ge: "ln 2 ≥ (5677 / 8192 :: real)"
proof -
have "ln 2 ≥ real_of_float (lb_ln2 12)" by (rule lb_ln2)
also have "lb_ln2 12 = Float 5677 (-13)" by code_simp
finally show ?thesis by simp
qed
lemma ln_2_ge': "ln (2 :: real) ≥ 2/3" and ln_2_le': "ln (2 :: real) ≤ 16/23"
using ln_2_le ln_2_ge by simp_all
lemma of_nat_ge_1_iff: "(of_nat x :: 'a :: linordered_semidom) ≥ 1 ⟷ x ≥ 1"
using of_nat_le_iff[of 1 x] by (subst (asm) of_nat_1)
lemma floor_conv_div_nat:
"of_int (floor (real m / real n)) = real (m div n)"
by (subst floor_divide_of_nat_eq) simp
lemma frac_conv_mod_nat:
"frac (real m / real n) = real (m mod n) / real n"
by (cases "n = 0")
(simp_all add: frac_def floor_conv_div_nat field_simps of_nat_mult
[symmetric] of_nat_add [symmetric] del: of_nat_mult of_nat_add)
lemma of_nat_prod_mset: "prod_mset (image_mset of_nat A) = of_nat (prod_mset A)"
by (induction A) simp_all
lemma prod_mset_pos: "(⋀x :: 'a :: linordered_semidom. x ∈# A ⟹ x > 0) ⟹ prod_mset A > 0"
by (induction A) simp_all
lemma ln_msetprod:
assumes "⋀x. x ∈#I ⟹ x > 0"
shows "(∑p::nat∈#I. ln p) = ln (∏p∈#I. p)"
using assms by (induction I) (simp_all add: of_nat_prod_mset ln_mult prod_mset_pos)
lemma ln_fact: "ln (fact n) = (∑d=1..n. ln d)"
by (induction n) (simp_all add: ln_mult)
lemma overpower_lemma:
fixes f g :: "real ⇒ real"
assumes "f a ≤ g a"
assumes "⋀x. a ≤ x ⟹ ((λx. g x - f x) has_real_derivative (d x)) (at x)"
assumes "⋀x. a ≤ x ⟹ d x ≥ 0"
assumes "a ≤ x"
shows "f x ≤ g x"
proof (cases "a < x")
case True
with assms have "∃z. z > a ∧ z < x ∧ g x - f x - (g a - f a) = (x - a) * d z"
by (intro MVT2) auto
then obtain z where z: "z > a" "z < x" "g x - f x - (g a - f a) = (x - a) * d z" by blast
hence "f x = g x + (f a - g a) + (a - x) * d z" by (simp add: algebra_simps)
also from assms have "f a - g a ≤ 0" by (simp add: algebra_simps)
also from assms z have "(a - x) * d z ≤ 0 * d z"
by (intro mult_right_mono) simp_all
finally show ?thesis by simp
qed (insert assms, auto)
subsection ‹Preliminary definitions›
definition primepow_even :: "nat ⇒ bool" where
"primepow_even q ⟷ (∃ p k. 1 ≤ k ∧ prime p ∧ q = p^(2*k))"
definition primepow_odd :: "nat ⇒ bool" where
"primepow_odd q ⟷ (∃ p k. 1 ≤ k ∧ prime p ∧ q = p^(2*k+1))"
abbreviation (input) isprimedivisor :: "nat ⇒ nat ⇒ bool" where
"isprimedivisor q p ≡ prime p ∧ p dvd q"
definition pre_mangoldt :: "nat ⇒ nat" where
"pre_mangoldt d = (if primepow d then aprimedivisor d else 1)"
definition mangoldt_even :: "nat ⇒ real" where
"mangoldt_even d = (if primepow_even d then ln (real (aprimedivisor d)) else 0)"
definition mangoldt_odd :: "nat ⇒ real" where
"mangoldt_odd d = (if primepow_odd d then ln (real (aprimedivisor d)) else 0)"
definition mangoldt_1 :: "nat ⇒ real" where
"mangoldt_1 d = (if prime d then ln d else 0)"
definition psi :: "nat ⇒ real" where
"psi n = (∑d=1..n. mangoldt d)"
definition psi_even :: "nat ⇒ real" where
"psi_even n = (∑d=1..n. mangoldt_even d)"
definition psi_odd :: "nat ⇒ real" where
"psi_odd n = (∑d=1..n. mangoldt_odd d)"
abbreviation (input) psi_even_2 :: "nat ⇒ real" where
"psi_even_2 n ≡ (∑d=2..n. mangoldt_even d)"
abbreviation (input) psi_odd_2 :: "nat ⇒ real" where
"psi_odd_2 n ≡ (∑d=2..n. mangoldt_odd d)"
definition theta :: "nat ⇒ real" where
"theta n = (∑p=1..n. if prime p then ln (real p) else 0)"
subsection ‹Properties of prime powers›
lemma primepow_even_imp_primepow:
assumes "primepow_even n"
shows "primepow n"
proof -
from assms obtain p k where "1 ≤ k" "prime p" "n = p ^ (2 * k)"
unfolding primepow_even_def by blast
moreover from ‹1 ≤ k› have "2 * k > 0"
by simp
ultimately show ?thesis unfolding primepow_def by blast
qed
lemma primepow_odd_imp_primepow:
assumes "primepow_odd n"
shows "primepow n"
proof -
from assms obtain p k where "1 ≤ k" "prime p" "n = p ^ (2 * k + 1)"
unfolding primepow_odd_def by blast
moreover from ‹1 ≤ k› have "Suc (2 * k) > 0"
by simp
ultimately show ?thesis unfolding primepow_def
by (auto simp del: power_Suc)
qed
lemma primepow_odd_altdef:
"primepow_odd n ⟷
primepow n ∧ odd (multiplicity (aprimedivisor n) n) ∧ multiplicity (aprimedivisor n) n > 1"
proof (intro iffI conjI; (elim conjE)?)
assume "primepow_odd n"
then obtain p k where n: "k ≥ 1" "prime p" "n = p ^ (2 * k + 1)"
by (auto simp: primepow_odd_def)
thus "odd (multiplicity (aprimedivisor n) n)" "multiplicity (aprimedivisor n) n > 1"
by (simp_all add: aprimedivisor_primepow prime_elem_multiplicity_mult_distrib)
next
assume A: "primepow n" and B: "odd (multiplicity (aprimedivisor n) n)"
and C: "multiplicity (aprimedivisor n) n > 1"
from A obtain p k where n: "k ≥ 1" "prime p" "n = p ^ k"
by (auto simp: primepow_def Suc_le_eq)
with B C have "odd k" "k > 1"
by (simp_all add: aprimedivisor_primepow prime_elem_multiplicity_mult_distrib)
then obtain j where j: "k = 2 * j + 1" "j > 0" by (auto elim!: oddE)
with n show "primepow_odd n" by (auto simp: primepow_odd_def intro!: exI[of _ p, OF exI[of _ j]])
qed (auto dest: primepow_odd_imp_primepow)
lemma primepow_even_altdef:
"primepow_even n ⟷ primepow n ∧ even (multiplicity (aprimedivisor n) n)"
proof (intro iffI conjI; (elim conjE)?)
assume "primepow_even n"
then obtain p k where n: "k ≥ 1" "prime p" "n = p ^ (2 * k)"
by (auto simp: primepow_even_def)
thus "even (multiplicity (aprimedivisor n) n)"
by (simp_all add: aprimedivisor_primepow prime_elem_multiplicity_mult_distrib)
next
assume A: "primepow n" and B: "even (multiplicity (aprimedivisor n) n)"
from A obtain p k where n: "k ≥ 1" "prime p" "n = p ^ k"
by (auto simp: primepow_def Suc_le_eq)
with B have "even k"
by (simp_all add: aprimedivisor_primepow prime_elem_multiplicity_mult_distrib)
then obtain j where j: "k = 2 * j" by (auto elim!: evenE)
from j n have "j ≠ 0" by (intro notI) simp_all
with j n show "primepow_even n"
by (auto simp: primepow_even_def intro!: exI[of _ p, OF exI[of _ j]])
qed (auto dest: primepow_even_imp_primepow)
lemma primepow_odd_mult:
assumes "d > Suc 0"
shows "primepow_odd (aprimedivisor d * d) ⟷ primepow_even d"
using assms
by (auto simp: primepow_odd_altdef primepow_even_altdef primepow_mult_aprimedivisorI
aprimedivisor_primepow prime_aprimedivisor' aprimedivisor_dvd'
prime_elem_multiplicity_mult_distrib prime_elem_aprimedivisor_nat
dest!: primepow_multD)
lemma pre_mangoldt_primepow:
assumes "primepow n" "aprimedivisor n = p"
shows "pre_mangoldt n = p"
using assms by (simp add: pre_mangoldt_def)
lemma pre_mangoldt_notprimepow:
assumes "¬primepow n"
shows "pre_mangoldt n = 1"
using assms by (simp add: pre_mangoldt_def)
lemma primepow_cases:
"primepow d ⟷
( primepow_even d ∧ ¬ primepow_odd d ∧ ¬ prime d) ∨
(¬ primepow_even d ∧ primepow_odd d ∧ ¬ prime d) ∨
(¬ primepow_even d ∧ ¬ primepow_odd d ∧ prime d)"
by (auto simp: primepow_even_altdef primepow_odd_altdef multiplicity_aprimedivisor_Suc_0_iff
elim!: oddE intro!: Nat.gr0I)
subsection ‹Deriving a recurrence for the psi function›
lemma ln_fact_bounds:
assumes "n > 0"
shows "abs(ln (fact n) - n * ln n + n) ≤ 1 + ln n"
proof -
have "∀n∈{0<..}. ∃z>real n. z < real (n + 1) ∧ real (n + 1) * ln (real (n + 1)) -
real n * ln (real n) = (real (n + 1) - real n) * (ln z + 1)"
by (intro ballI MVT2) (auto intro!: derivative_eq_intros)
hence "∀n∈{0<..}. ∃z>real n. z < real (n + 1) ∧ real (n + 1) * ln (real (n + 1)) -
real n * ln (real n) = (ln z + 1)" by (simp add: algebra_simps)
from bchoice[OF this] obtain k :: "nat ⇒ real"
where lb: "real n < k n" and ub: "k n < real (n + 1)" and
mvt: "real (n+1) * ln (real (n+1)) - real n * ln (real n) = ln (k n) + 1"
if "n > 0" for n::nat by blast
have *: "(n + 1) * ln (n + 1) = (∑i=1..n. ln(k i) + 1)" for n::nat
proof (induction n)
case (Suc n)
have "(∑i = 1..n+1. ln (k i) + 1) = (∑i = 1..n. ln (k i) + 1) + ln (k (n+1)) + 1"
by simp
also from Suc.IH have "(∑i = 1..n. ln (k i) + 1) = real (n+1) * ln (real (n+1))" ..
also from mvt[of "n+1"] have "… = real (n+2) * ln (real (n+2)) - ln (k (n+1)) - 1"
by simp
finally show ?case
by simp
qed simp
have **: "abs((∑i=1..n+1. ln i) - ((n+1) * ln (n+1) - (n+1))) ≤ 1 + ln(n+1)" for n::nat
proof -
have "(∑i=1..n+1. ln i) ≤ (∑i=1..n. ln i) + ln (n+1)"
by simp
also have "(∑i=1..n. ln i) ≤ (∑i=1..n. ln (k i))"
by (intro sum_mono, subst ln_le_cancel_iff) (auto simp: Suc_le_eq dest: lb ub)
also have "… = (∑i=1..n. ln (k i) + 1) - n"
by (simp add: sum.distrib)
also from * have "… = (n+1) * ln (n+1) - n"
by simp
finally have a_minus_b: "(∑i=1..n+1. ln i) - ((n+1) * ln (n+1) - (n+1)) ≤ 1 + ln (n+1)"
by simp
from * have "(n+1) * ln (n+1) - n = (∑i=1..n. ln (k i) + 1) - n"
by simp
also have "… = (∑i=1..n. ln (k i))"
by (simp add: sum.distrib)
also have "… ≤ (∑i=1..n. ln (i+1))"
by (intro sum_mono, subst ln_le_cancel_iff) (auto simp: Suc_le_eq dest: lb ub)
also from sum.shift_bounds_cl_nat_ivl[of "ln" 1 1 n] have "… = (∑i=1+1..n+1. ln i)" ..
also have "… = (∑i=1..n+1. ln i)"
by (rule sum.mono_neutral_left) auto
finally have b_minus_a: "((n+1) * ln (n+1) - (n+1)) - (∑i=1..n+1. ln i) ≤ 1"
by simp
have "0 ≤ ln (n+1)"
by simp
with b_minus_a have "((n+1) * ln (n+1) - (n+1)) - (∑i=1..n+1. ln i) ≤ 1 + ln (n+1)"
by linarith
with a_minus_b show ?thesis
by linarith
qed
from ‹n > 0› have "n ≥ 1" by simp
thus ?thesis
proof (induction n rule: dec_induct)
case base
then show ?case by simp
next
case (step n)
from ln_fact[of "n+1"] **[of n] show ?case by simp
qed
qed
lemma ln_fact_diff_bounds:
"abs(ln (fact n) - 2 * ln (fact (n div 2)) - n * ln 2) ≤ 4 * ln (if n = 0 then 1 else n) + 3"
proof (cases "n div 2 = 0")
case True
hence "n ≤ 1" by simp
with ln_le_minus_one[of "2::real"] show ?thesis by (cases n) simp_all
next
case False
then have "n > 1" by simp
let ?a = "real n * ln 2"
let ?b = "4 * ln (real n) + 3"
let ?l1 = "ln (fact (n div 2))"
let ?a1 = "real (n div 2) * ln (real (n div 2)) - real (n div 2)"
let ?b1 = "1 + ln (real (n div 2))"
let ?l2 = "ln (fact n)"
let ?a2 = "real n * ln (real n) - real n"
let ?b2 = "1 + ln (real n)"
have abs_a: "abs(?a - (?a2 - 2 * ?a1)) ≤ ?b - 2 * ?b1 - ?b2"
proof (cases "even n")
case True
then have "real (2 * (n div 2)) = real n"
by simp
then have n_div_2: "real (n div 2) = real n / 2"
by simp
from ‹n > 1› have *: "abs(?a - (?a2 - 2 * ?a1)) = 0"
by (simp add: n_div_2 ln_div algebra_simps)
from ‹even n› and ‹n > 1› have "0 ≤ ln (real n) - ln (real (n div 2))"
by (auto elim: evenE)
also have "2 * … ≤ 3 * ln (real n) - 2 * ln (real (n div 2))"
using ‹n > 1› by (auto intro!: ln_ge_zero)
also have "… = ?b - 2 * ?b1 - ?b2" by simp
finally show ?thesis using * by simp
next
case False
then have "real (2 * (n div 2)) = real (n - 1)"
by simp
with ‹n > 1› have n_div_2: "real (n div 2) = (real n - 1) / 2"
by simp
from ‹odd n› ‹n div 2 ≠ 0› have "n ≥ 3"
by presburger
have "?a - (?a2 - 2 * ?a1) = real n * ln 2 - real n * ln (real n) + real n +
2 * real (n div 2) * ln (real (n div 2)) - 2* real (n div 2)"
by (simp add: algebra_simps)
also from n_div_2 have "2 * real (n div 2) = real n - 1"
by simp
also have "real n * ln 2 - real n * ln (real n) + real n +
(real n - 1) * ln (real (n div 2)) - (real n - 1)
= real n * (ln (real n - 1) - ln (real n)) - ln (real (n div 2)) + 1"
using ‹n > 1› by (simp add: algebra_simps n_div_2 ln_div)
finally have lhs: "abs(?a - (?a2 - 2 * ?a1)) =
abs(real n * (ln (real n - 1) - ln (real n)) - ln (real (n div 2)) + 1)"
by simp
from ‹n > 1› have "real n * (ln (real n - 1) - ln (real n)) ≤ 0"
by (simp add: algebra_simps mult_left_mono)
moreover from ‹n > 1› have "ln (real (n div 2)) ≤ ln (real n)" by simp
moreover {
have "exp 1 ≤ (3::real)" by (rule exp_le)
also from ‹n ≥ 3› have "… ≤ exp (ln (real n))" by simp
finally have "ln (real n) ≥ 1" by simp
}
ultimately have ub: "real n * (ln (real n - 1) - ln (real n)) - ln(real (n div 2)) + 1 ≤
3 * ln (real n) - 2 * ln(real (n div 2))" by simp
have mon: "real n' * (ln (real n') - ln (real n' - 1)) ≤
real n * (ln (real n) - ln (real n - 1))"
if "n ≥ 3" "n' ≥ n" for n n'::nat
proof (rule DERIV_nonpos_imp_nonincreasing[where f = "λx. x * (ln x - ln (x - 1))"])
fix t assume t: "real n ≤ t" "t ≤ real n'"
with that have "1 / (t - 1) ≥ ln (1 + 1/(t - 1))"
by (intro ln_add_one_self_le_self) simp_all
also from t that have "ln (1 + 1/(t - 1)) = ln t- ln (t - 1)"
by (simp add: ln_div [symmetric] field_simps)
finally have "ln t - ln (t - 1) ≤ 1 / (t - 1)" .
with that t
show "∃y. ((λx. x * (ln x - ln (x - 1))) has_field_derivative y) (at t) ∧ y ≤ 0"
by (intro exI[of _ "1 / (1 - t) + ln t - ln (t - 1)"])
(force intro!: derivative_eq_intros simp: field_simps)+
qed (use that in simp_all)
from ‹n > 1› have "ln 2 = ln (real n) - ln (real n / 2)"
by (simp add: ln_div)
also from ‹n > 1› have "… ≤ ln (real n) - ln (real (n div 2))"
by simp
finally have *: "3*ln 2 + ln(real (n div 2)) ≤ 3* ln(real n) - 2* ln(real (n div 2))"
by simp
have "- real n * (ln (real n - 1) - ln (real n)) + ln(real (n div 2)) - 1 =
real n * (ln (real n) - ln (real n - 1)) - 1 + ln(real (n div 2))"
by (simp add: algebra_simps)
also have "real n * (ln (real n) - ln (real n - 1)) ≤ 3 * (ln 3 - ln (3 - 1))"
using mon[OF _ ‹n ≥ 3›] by simp
also {
have "Some (Float 3 (-1)) = ub_ln 1 3" by code_simp
from ub_ln(1)[OF this] have "ln 3 ≤ (1.6 :: real)" by simp
also have "1.6 - 1 / 3 ≤ 2 * (2/3 :: real)" by simp
also have "2/3 ≤ ln (2 :: real)" by (rule ln_2_ge')
finally have "ln 3 - 1 / 3 ≤ 2 * ln (2 :: real)" by simp
}
hence "3 * (ln 3 - ln (3 - 1)) - 1 ≤ 3 * ln (2 :: real)" by simp
also note *
finally have "- real n * (ln (real n - 1) - ln (real n)) + ln(real (n div 2)) - 1 ≤
3 * ln (real n) - 2 * ln (real (n div 2))" by simp
hence lhs': "abs(real n * (ln (real n - 1) - ln (real n)) - ln(real (n div 2)) + 1) ≤
3 * ln (real n) - 2 * ln (real (n div 2))"
using ub by simp
have rhs: "?b - 2 * ?b1 - ?b2 = 3* ln (real n) - 2 * ln (real (n div 2))"
by simp
from ‹n > 1› have "ln (real (n div 2)) ≤ 3* ln (real n) - 2* ln (real (n div 2))"
by simp
with rhs lhs lhs' show ?thesis
by simp
qed
then have minus_a: "-?a ≤ ?b - 2 * ?b1 - ?b2 - (?a2 - 2 * ?a1)"
by simp
from abs_a have a: "?a ≤ ?b - 2 * ?b1 - ?b2 + ?a2 - 2 * ?a1"
by (simp)
from ln_fact_bounds[of "n div 2"] False have abs_l1: "abs(?l1 - ?a1) ≤ ?b1"
by (simp add: algebra_simps)
then have minus_l1: "?a1 - ?l1 ≤ ?b1"
by linarith
from abs_l1 have l1: "?l1 - ?a1 ≤ ?b1"
by linarith
from ln_fact_bounds[of n] False have abs_l2: "abs(?l2 - ?a2) ≤ ?b2"
by (simp add: algebra_simps)
then have l2: "?l2 - ?a2 ≤ ?b2"
by simp
from abs_l2 have minus_l2: "?a2 - ?l2 ≤ ?b2"
by simp
from minus_a minus_l1 l2 have "?l2 - 2 * ?l1 - ?a ≤ ?b"
by simp
moreover from a l1 minus_l2 have "- ?l2 + 2 * ?l1 + ?a ≤ ?b"
by simp
ultimately have "abs((?l2 - 2*?l1) - ?a) ≤ ?b"
by simp
then show ?thesis
by simp
qed
lemma ln_primefact:
assumes "n ≠ (0::nat)"
shows "ln n = (∑d=1..n. if primepow d ∧ d dvd n then ln (aprimedivisor d) else 0)"
(is "?lhs = ?rhs")
proof -
have "?rhs = (∑d∈{x ∈ {1..n}. primepow x ∧ x dvd n}. ln (real (aprimedivisor d)))"
unfolding primepow_factors_def by (subst sum.inter_filter [symmetric]) simp_all
also have "{x ∈ {1..n}. primepow x ∧ x dvd n} = primepow_factors n"
using assms by (auto simp: primepow_factors_def dest: dvd_imp_le primepow_gt_Suc_0)
finally have *: "(∑d∈primepow_factors n. ln (real (aprimedivisor d))) = ?rhs" ..
from in_prime_factors_imp_prime prime_gt_0_nat
have pf_pos: "⋀p. p∈#prime_factorization n ⟹ p > 0"
by blast
from ln_msetprod[of "prime_factorization n", OF pf_pos] assms
have "ln n = (∑p∈#prime_factorization n. ln p)"
by (simp add: of_nat_prod_mset)
also from * sum_prime_factorization_conv_sum_primepow_factors[of n ln, OF assms(1)]
have "… = ?rhs" by simp
finally show ?thesis .
qed
context
begin
private lemma divisors:
fixes x d::nat
assumes "x ∈ {1..n}"
assumes "d dvd x"
shows "∃k∈{1..n div d}. x = d * k"
proof -
from assms have "x ≤ n"
by simp
then have ub: "x div d ≤ n div d"
by (simp add: div_le_mono ‹x ≤ n›)
from assms have "1 ≤ x div d" by (auto elim!: dvdE)
with ub have "x div d ∈ {1..n div d}"
by simp
with ‹d dvd x› show ?thesis by (auto intro!: bexI[of _ "x div d"])
qed
lemma ln_fact_conv_mangoldt: "ln (fact n) = (∑d=1..n. mangoldt d * floor (n / d))"
proof -
have *: "(∑da=1..n. if primepow da ∧ da dvd d then ln (aprimedivisor da) else 0) =
(∑(da::nat)=1..d. if primepow da ∧ da dvd d then ln (aprimedivisor da) else 0)"
if d: "d ∈ {1..n}" for d
by (rule sum.mono_neutral_right, insert d) (auto dest: dvd_imp_le)
have "(∑d=1..n. ∑da=1..d. if primepow da ∧
da dvd d then ln (aprimedivisor da) else 0) =
(∑d=1..n. ∑da=1..n. if primepow da ∧
da dvd d then ln (aprimedivisor da) else 0)"
by (rule sum.cong) (insert *, simp_all)
also have "… = (∑da=1..n. ∑d=1..n. if primepow da ∧
da dvd d then ln (aprimedivisor da) else 0)"
by (rule sum.swap)
also have "… = sum (λd. mangoldt d * floor (n/d)) {1..n}"
proof (rule sum.cong)
fix d assume d: "d ∈ {1..n}"
have "(∑da = 1..n. if primepow d ∧ d dvd da then ln (real (aprimedivisor d)) else 0) =
(∑da = 1..n. if d dvd da then mangoldt d else 0)"
by (intro sum.cong) (simp_all add: mangoldt_def)
also have "… = mangoldt d * real (card {x. x ∈ {1..n} ∧ d dvd x})"
by (subst sum.inter_filter [symmetric]) (simp_all add: algebra_simps)
also {
have "{x. x ∈ {1..n} ∧ d dvd x} = {x. ∃k ∈{1..n div d}. x=k*d}"
proof safe
fix x assume "x ∈ {1..n}" "d dvd x"
thus "∃k∈{1..n div d}. x = k * d" using divisors[of x n d] by auto
next
fix x k assume k: "k ∈ {1..n div d}"
from k have "k * d ≤ n div d * d" by (intro mult_right_mono) simp_all
also have "n div d * d ≤ n div d * d + n mod d" by (rule le_add1)
also have "… = n" by simp
finally have "k * d ≤ n" .
thus "k * d ∈ {1..n}" using d k by auto
qed auto
also have "… = (λk. k*d) ` {1..n div d}"
by fast
also have "card … = card {1..n div d}"
by (rule card_image) (simp add: inj_on_def)
also have "… = n div d"
by simp
also have "... = ⌊n / d⌋"
by (simp add: floor_divide_of_nat_eq)
finally have "real (card {x. x ∈ {1..n} ∧ d dvd x}) = real_of_int ⌊n / d⌋"
by force
}
finally show "(∑da = 1..n. if primepow d ∧ d dvd da then ln (real (aprimedivisor d)) else 0) =
mangoldt d * real_of_int ⌊real n / real d⌋" .
qed simp_all
finally have "(∑d=1..n. ∑da=1..d. if primepow da ∧
da dvd d then ln (aprimedivisor da) else 0) =
sum (λd. mangoldt d * floor (n/d)) {1..n}" .
with ln_primefact have "(∑d=1..n. ln d) =
(∑d=1..n. mangoldt d * floor (n/d))"
by simp
with ln_fact show ?thesis
by simp
qed
end
context
begin
private lemma div_2_mult_2_bds:
fixes n d :: nat
assumes "d > 0"
shows "0 ≤ ⌊n / d⌋ - 2 * ⌊(n div 2) / d⌋" "⌊n / d⌋ - 2 * ⌊(n div 2) / d⌋ ≤ 1"
proof -
have "⌊2::real⌋ * ⌊(n div 2) / d⌋ ≤ ⌊2 * ((n div 2) / d)⌋"
by (rule le_mult_floor) simp_all
also from assms have "… ≤ ⌊n / d⌋" by (intro floor_mono) (simp_all add: field_simps)
finally show "0 ≤ ⌊n / d⌋ - 2 * ⌊(n div 2) / d⌋" by (simp add: algebra_simps)
next
have "real (n div d) ≤ real (2 * ((n div 2) div d) + 1)"
by (subst div_mult2_eq [symmetric], simp only: mult.commute, subst div_mult2_eq) simp
thus "⌊n / d⌋ - 2 * ⌊(n div 2) / d⌋ ≤ 1"
unfolding of_nat_add of_nat_mult floor_conv_div_nat [symmetric] by simp_all
qed
private lemma n_div_d_eq_1: "d ∈ {n div 2 + 1..n} ⟹ ⌊real n / real d⌋ = 1"
by (cases "n = d") (auto simp: field_simps intro: floor_eq)
lemma psi_bounds_ln_fact:
shows "ln (fact n) - 2 * ln (fact (n div 2)) ≤ psi n"
"psi n - psi (n div 2) ≤ ln (fact n) - 2 * ln (fact (n div 2))"
proof -
fix n::nat
let ?k = "n div 2" and ?d = "n mod 2"
have *: "⌊?k / d⌋ = 0" if "d > ?k" for d
proof -
from that div_less have "0 = ?k div d" by simp
also have "… = ⌊?k / d⌋" by (rule floor_divide_of_nat_eq [symmetric])
finally show "⌊?k / d⌋ = 0" by simp
qed
have sum_eq: "(∑d=1..2*?k+?d. mangoldt d * ⌊?k / d⌋) = (∑d=1..?k. mangoldt d * ⌊?k / d⌋)"
by (intro sum.mono_neutral_right) (auto simp: *)
from ln_fact_conv_mangoldt have "ln (fact n) = (∑d=1..n. mangoldt d * ⌊n / d⌋)" .
also have "… = (∑d=1..n. mangoldt d * ⌊(2 * (n div 2) + n mod 2) / d⌋)"
by simp
also have "… ≤ (∑d=1..n. mangoldt d * (2 * ⌊?k / d⌋ + 1))"
using div_2_mult_2_bds(2)[of _ n]
by (intro sum_mono mult_left_mono, subst of_int_le_iff)
(auto simp: algebra_simps mangoldt_nonneg)
also have "… = 2 * (∑d=1..n. mangoldt d * ⌊(n div 2) / d⌋) + (∑d=1..n. mangoldt d)"
by (simp add: algebra_simps sum.distrib sum_distrib_left)
also have "… = 2 * (∑d=1..2*?k+?d. mangoldt d * ⌊(n div 2) / d⌋) + (∑d=1..n. mangoldt d)"
by presburger
also from sum_eq have "… = 2 * (∑d=1..?k. mangoldt d * ⌊(n div 2) / d⌋) + (∑d=1..n. mangoldt d)"
by presburger
also from ln_fact_conv_mangoldt psi_def have "… = 2 * ln (fact ?k) + psi n"
by presburger
finally show "ln (fact n) - 2 * ln (fact (n div 2)) ≤ psi n"
by simp
next
fix n::nat
let ?k = "n div 2" and ?d = "n mod 2"
from psi_def have "psi n - psi ?k = (∑d=1..2*?k+?d. mangoldt d) - (∑d=1..?k. mangoldt d)"
by presburger
also have "… = sum mangoldt ({1..2 * (n div 2) + n mod 2} - {1..n div 2})"
by (subst sum_diff) simp_all
also have "… = (∑d∈({1..2 * (n div 2) + n mod 2} - {1..n div 2}).
(if d ≤ ?k then 0 else mangoldt d))"
by (intro sum.cong) simp_all
also have "… = (∑d=1..2*?k+?d. (if d ≤ ?k then 0 else mangoldt d))"
by (intro sum.mono_neutral_left) auto
also have "… = (∑d=1..n. (if d ≤ ?k then 0 else mangoldt d))"
by presburger
also have "… = (∑d=1..n. (if d ≤ ?k then mangoldt d * 0 else mangoldt d))"
by (intro sum.cong) simp_all
also from div_2_mult_2_bds(1) have "… ≤ (∑d=1..n. (if d ≤ ?k then mangoldt d * (⌊n/d⌋ - 2 * ⌊?k/d⌋) else mangoldt d))"
by (intro sum_mono)
(auto simp: algebra_simps mangoldt_nonneg intro!: mult_left_mono simp del: of_int_mult)
also from n_div_d_eq_1 have "… = (∑d=1..n. (if d ≤ ?k then mangoldt d * (⌊n/d⌋ - 2 * ⌊?k/d⌋) else mangoldt d * ⌊n/d⌋))"
by (intro sum.cong refl) auto
also have "… = (∑d=1..n. mangoldt d * real_of_int (⌊real n / real d⌋) -
(if d ≤ ?k then 2 * mangoldt d * real_of_int ⌊real ?k / real d⌋ else 0))"
by (intro sum.cong refl) (auto simp: algebra_simps)
also have "… = (∑d=1..n. mangoldt d * real_of_int (⌊real n / real d⌋)) -
(∑d=1..n. (if d ≤ ?k then 2 * mangoldt d * real_of_int ⌊real ?k / real d⌋ else 0))"
by (rule sum_subtractf)
also have "(∑d=1..n. (if d ≤ ?k then 2 * mangoldt d * real_of_int ⌊real ?k / real d⌋ else 0)) =
(∑d=1..?k. (if d ≤ ?k then 2 * mangoldt d * real_of_int ⌊real ?k / real d⌋ else 0))"
by (intro sum.mono_neutral_right) auto
also have "… = (∑d=1..?k. 2 * mangoldt d * real_of_int ⌊real ?k / real d⌋)"
by (intro sum.cong) simp_all
also have "… = 2 * (∑d=1..?k. mangoldt d * real_of_int ⌊real ?k / real d⌋)"
by (simp add: sum_distrib_left mult_ac)
also have "(∑d = 1..n. mangoldt d * real_of_int ⌊real n / real d⌋) - … =
ln (fact n) - 2 * ln (fact (n div 2))"
by (simp add: ln_fact_conv_mangoldt)
finally show "psi n - psi (n div 2) ≤ ln (fact n) - 2 * ln (fact (n div 2))" .
qed
end
lemma psi_bounds_induct:
"real n * ln 2 - (4 * ln (real (if n = 0 then 1 else n)) + 3) ≤ psi n"
"psi n - psi (n div 2) ≤ real n * ln 2 + (4 * ln (real (if n = 0 then 1 else n)) + 3)"
proof -
from le_imp_neg_le[OF ln_fact_diff_bounds]
have "n * ln 2 - (4 * ln (if n = 0 then 1 else n) + 3)
≤ n * ln 2 - abs(ln (fact n) - 2 * ln (fact (n div 2)) - n * ln 2)"
by simp
also have "… ≤ ln (fact n) - 2 * ln (fact (n div 2))"
by simp
also from psi_bounds_ln_fact (1) have "… ≤ psi n"
by simp
finally show "real n * ln 2 - (4 * ln (real (if n = 0 then 1 else n)) + 3) ≤ psi n" .
next
from psi_bounds_ln_fact (2) have "psi n - psi (n div 2) ≤ ln (fact n) - 2 * ln (fact (n div 2))" .
also have "… ≤ n * ln 2 + abs(ln (fact n) - 2 * ln (fact (n div 2)) - n * ln 2)"
by simp
also from ln_fact_diff_bounds [of n]
have "abs(ln (fact n) - 2 * ln (fact (n div 2)) - n * ln 2)
≤ (4 * ln (real (if n = 0 then 1 else n)) + 3)" by simp
finally show "psi n - psi (n div 2) ≤ real n * ln 2 + (4 * ln (real (if n = 0 then 1 else n)) + 3)"
by simp
qed
subsection ‹Bounding the psi function›
text ‹
In this section, we will first prove the relatively tight estimate
@{prop "psi n ≤ 3 / 2 + ln 2 * n"} for @{term "n ≤ 128"} and then use the
recurrence we have just derived to extend it to @{prop "psi n ≤ 551 / 256"} for
@{term "n ≤ 1024"}, at which point applying the recurrence can be used to prove
the same bound for arbitrarily big numbers.
First of all, we will prove the bound for @{term "n <= 128"} using reflection and
approximation.
›
context
begin
private lemma Ball_insertD:
assumes "∀x∈insert y A. P x"
shows "P y" "∀x∈A. P x"
using assms by auto
private lemma meta_eq_TrueE: "PROP A ≡ Trueprop True ⟹ PROP A"
by simp
private lemma pre_mangoldt_pos: "pre_mangoldt n > 0"
unfolding pre_mangoldt_def by (auto simp: primepow_gt_Suc_0)
private lemma psi_conv_pre_mangoldt: "psi n = ln (real (prod pre_mangoldt {1..n}))"
by (auto simp: psi_def mangoldt_def pre_mangoldt_def ln_prod primepow_gt_Suc_0 intro!: sum.cong)
private lemma eval_psi_aux1: "psi 0 = ln (real (numeral Num.One))"
by (simp add: psi_def)
private lemma eval_psi_aux2:
assumes "psi m = ln (real (numeral x))" "pre_mangoldt n = y" "m + 1 = n" "numeral x * y = z"
shows "psi n = ln (real z)"
proof -
from assms(2) [symmetric] have [simp]: "y > 0" by (simp add: pre_mangoldt_pos)
have "psi n = psi (Suc m)" by (simp add: assms(3) [symmetric])
also have "… = ln (real y * (∏x = Suc 0..m. real (pre_mangoldt x)))"
using assms(2,3) [symmetric] by (simp add: psi_conv_pre_mangoldt prod.nat_ivl_Suc' mult_ac)
also have "… = ln (real y) + psi m"
by (subst ln_mult) (simp_all add: pre_mangoldt_pos prod_pos psi_conv_pre_mangoldt)
also have "psi m = ln (real (numeral x))" by fact
also have "ln (real y) + … = ln (real (numeral x * y))" by (simp add: ln_mult)
finally show ?thesis by (simp add: assms(4) [symmetric])
qed
private lemma Ball_atLeast0AtMost_doubleton:
assumes "psi 0 ≤ 3 / 2 * ln 2 * real 0"
assumes "psi 1 ≤ 3 / 2 * ln 2 * real 1"
shows "(∀x∈{0..1}. psi x ≤ 3 / 2 * ln 2 * real x)"
using assms unfolding One_nat_def atLeast0_atMost_Suc ball_simps by auto
private lemma Ball_atLeast0AtMost_insert:
assumes "(∀x∈{0..m}. psi x ≤ 3 / 2 * ln 2 * real x)"
assumes "psi (numeral n) ≤ 3 / 2 * ln 2 * real (numeral n)" "m = pred_numeral n"
shows "(∀x∈{0..numeral n}. psi x ≤ 3 / 2 * ln 2 * real x)"
using assms
by (subst numeral_eq_Suc[of n], subst atLeast0_atMost_Suc,
subst ball_simps, simp only: numeral_eq_Suc [symmetric])
private lemma eval_psi_ineq_aux:
assumes "psi n = x" "x ≤ 3 / 2 * ln 2 * n"
shows "psi n ≤ 3 / 2 * ln 2 * n"
using assms by simp_all
private lemma eval_psi_ineq_aux2:
assumes "numeral m ^ 2 ≤ (2::nat) ^ (3 * n)"
shows "ln (real (numeral m)) ≤ 3 / 2 * ln 2 * real n"
proof -
have "ln (real (numeral m)) ≤ 3 / 2 * ln 2 * real n ⟷
2 * log 2 (real (numeral m)) ≤ 3 * real n"
by (simp add: field_simps log_def)
also have "2 * log 2 (real (numeral m)) = log 2 (real (numeral m ^ 2))"
by (subst of_nat_power, subst log_nat_power) simp_all
also have "… ≤ 3 * real n ⟷ real ((numeral m) ^ 2) ≤ 2 powr real (3 * n)"
by (subst Transcendental.log_le_iff) simp_all
also have "2 powr (3 * n) = real (2 ^ (3 * n))"
by (simp add: powr_realpow [symmetric])
also have "real ((numeral m) ^ 2) ≤ … ⟷ numeral m ^ 2 ≤ (2::nat) ^ (3 * n)"
by (rule of_nat_le_iff)
finally show ?thesis using assms by blast
qed
private lemma eval_psi_ineq_aux_mono:
assumes "psi n = x" "psi m = x" "psi n ≤ 3 / 2 * ln 2 * n" "n ≤ m"
shows "psi m ≤ 3 / 2 * ln 2 * m"
proof -
from assms have "psi m = psi n" by simp
also have "… ≤ 3 / 2 * ln 2 * n" by fact
also from ‹n ≤ m› have "… ≤ 3 / 2 * ln 2 * m" by simp
finally show ?thesis .
qed
lemma not_primepow_1_nat: "¬primepow (1 :: nat)" by auto
ML_file ‹bertrand.ML›
local_setup ‹fn lthy =>
let
fun tac ctxt =
let
val psi_cache = Bertrand.prove_psi ctxt 129
fun prove_psi_ineqs ctxt =
let
fun tac goal_ctxt =
HEADGOAL (resolve_tac goal_ctxt @{thms eval_psi_ineq_aux2} THEN'
Simplifier.simp_tac goal_ctxt)
fun prove_by_approx n thm =
let
val thm = thm RS @{thm eval_psi_ineq_aux}
val [prem] = Thm.prems_of thm
val prem = Goal.prove ctxt [] [] prem (tac o #context)
in
prem RS thm
end
fun prove_by_mono last_thm last_thm' thm =
let
val thm = @{thm eval_psi_ineq_aux_mono} OF [last_thm, thm, last_thm']
val [prem] = Thm.prems_of thm
val prem =
Goal.prove ctxt [] [] prem (fn {context = goal_ctxt, ...} =>
HEADGOAL (Simplifier.simp_tac goal_ctxt))
in
prem RS thm
end
fun go _ acc [] = acc
| go last acc ((n, x, thm) :: xs) =
let
val thm' =
case last of
NONE => prove_by_approx n thm
| SOME (last_x, last_thm, last_thm') =>
if last_x = x then
prove_by_mono last_thm last_thm' thm
else
prove_by_approx n thm
in
go (SOME (x, thm, thm')) (thm' :: acc) xs
end
in
rev o go NONE []
end
val psi_ineqs = prove_psi_ineqs ctxt psi_cache
fun prove_ball ctxt (thm1 :: thm2 :: thms) =
let
val thm = @{thm Ball_atLeast0AtMost_doubleton} OF [thm1, thm2]
fun solve_prem thm =
let
val thm' =
Goal.prove ctxt [] [] (Thm.cprem_of thm 1 |> Thm.term_of)
(fn {context = goal_ctxt, ...} =>
HEADGOAL (Simplifier.simp_tac goal_ctxt))
in
thm' RS thm
end
fun go thm thm' = (@{thm Ball_atLeast0AtMost_insert} OF [thm', thm]) |> solve_prem
in
fold go thms thm
end
| prove_ball _ _ = raise Match
in
HEADGOAL (resolve_tac ctxt [prove_ball ctxt psi_ineqs])
end
val thm = Goal.prove lthy [] [] @{prop "∀n∈{0..128}. psi n ≤ 3 / 2 * ln 2 * n"} (tac o #context)
in
Local_Theory.note ((@{binding psi_ubound_log_128}, []), [thm]) lthy |> snd
end
›
end
context
begin
private lemma psi_ubound_aux:
defines "f ≡ λx::real. (4 * ln x + 3) / (ln 2 * x)"
assumes "x ≥ 2" "x ≤ y"
shows "f x ≥ f y"
using assms(3)
proof (rule DERIV_nonpos_imp_nonincreasing, goal_cases)
case (1 t)
define f' where "f' = (λx. (1 - 4 * ln x) / x^2 / ln 2 :: real)"
from 1 assms(2) have "(f has_real_derivative f' t) (at t)" unfolding f_def f'_def
by (auto intro!: derivative_eq_intros simp: field_simps power2_eq_square)
moreover {
from ln_2_ge have "1/4 ≤ ln (2::real)" by simp
also from assms(2) 1 have "… ≤ ln t" by simp
finally have "ln t ≥ 1/4" .
}
with 1 assms(2) have "f' t ≤ 0" by (simp add: f'_def field_simps)
ultimately show ?case by (intro exI[of _ "f' t"]) simp_all
qed
text ‹
These next rules are used in combination with @{thm psi_bounds_induct} and
@{thm psi_ubound_log_128} to extend the upper bound for @{term "psi"} from values no greater
than 128 to values no greater than 1024. The constant factor of the upper bound changes every
time, but once we have reached 1024, the recurrence is self-sustaining in the sense that we do
not have to adjust the constant factor anymore in order to double the range.
›
lemma psi_ubound_log_double_cases':
assumes "⋀n. n ≤ m ⟹ psi n ≤ c * ln 2 * real n" "n ≤ m'" "m' = 2*m"
"c ≤ c'" "c ≥ 0" "m ≥ 1" "c' ≥ 1 + c/2 + (4 * ln (m+1) + 3) / (ln 2 * (m+1))"
shows "psi n ≤ c' * ln 2 * real n"
proof (cases "n > m")
case False
hence "psi n ≤ c * ln 2 * real n" by (intro assms) simp_all
also have "c ≤ c'" by fact
finally show ?thesis by - (simp_all add: mult_right_mono)
next
case True
hence n: "n ≥ m+1" by simp
from psi_bounds_induct(2)[of n] True
have "psi n ≤ real n * ln 2 + 4 * ln (real n) + 3 + psi (n div 2)" by simp
also from assms have "psi (n div 2) ≤ c * ln 2 * real (n div 2)"
by (intro assms) simp_all
also have "real (n div 2) ≤ real n / 2" by simp
also have "c * ln 2 * … = c / 2 * ln 2 * real n" by simp
also have "real n * ln 2 + 4 * ln (real n) + 3 + … =
(1 + c/2) * ln 2 * real n + (4 * ln (real n) + 3)" by (simp add: field_simps)
also {
have "(4 * ln (real n) + 3) / (ln 2 * (real n)) ≤ (4 * ln (m+1) + 3) / (ln 2 * (m+1))"
using n assms by (intro psi_ubound_aux) simp_all
also from assms have "(4 * ln (m+1) + 3) / (ln 2 * (m+1)) ≤ c' - 1 - c/2"
by (simp add: algebra_simps)
finally have "4 * ln (real n) + 3 ≤ (c' - 1 - c/2) * ln 2 * real n"
using n by (simp add: field_simps)
}
also have "(1 + c / 2) * ln 2 * real n + (c' - 1 - c / 2) * ln 2 * real n = c' * ln 2 * real n"
by (simp add: field_simps)
finally show ?thesis using ‹c ≥ 0› by (simp_all add: mult_left_mono)
qed
end
lemma psi_ubound_log_double_cases:
assumes "∀n≤m. psi n ≤ c * ln 2 * real n"
"c' ≥ 1 + c/2 + (4 * ln (m+1) + 3) / (ln 2 * (m+1))"
"m' = 2*m" "c ≤ c'" "c ≥ 0" "m ≥ 1"
shows "∀n≤m'. psi n ≤ c' * ln 2 * real n"
using assms(1) by (intro allI impI assms psi_ubound_log_double_cases'[of m c _ m' c']) auto
lemma psi_ubound_log_1024:
"∀n≤1024. psi n ≤ 551 / 256 * ln 2 * real n"
proof -
from psi_ubound_log_128 have "∀n≤128. psi n ≤ 3 / 2 * ln 2 * real n" by simp
hence "∀n≤256. psi n ≤ 1025 / 512 * ln 2 * real n"
proof (rule psi_ubound_log_double_cases, goal_cases)
case 1
have "Some (Float 624 (- 7)) = ub_ln 9 129" by code_simp
from ub_ln(1)[OF this] and ln_2_ge show ?case by (simp add: field_simps)
qed simp_all
hence "∀n≤512. psi n ≤ 549 / 256 * ln 2 * real n"
proof (rule psi_ubound_log_double_cases, goal_cases)
case 1
have "Some (Float 180 (- 5)) = ub_ln 7 257" by code_simp
from ub_ln(1)[OF this] and ln_2_ge show ?case by (simp add: field_simps)
qed simp_all
thus "∀n≤1024. psi n ≤ 551 / 256 * ln 2 * real n"
proof (rule psi_ubound_log_double_cases, goal_cases)
case 1
have "Some (Float 203 (- 5)) = ub_ln 7 513" by code_simp
from ub_ln(1)[OF this] and ln_2_ge show ?case by (simp add: field_simps)
qed simp_all
qed
lemma psi_bounds_sustained_induct:
assumes "4 * ln (1 + 2 ^ j) + 3 ≤ d * ln 2 * (1 + 2^j)"
assumes "4 / (1 + 2^j) ≤ d * ln 2"
assumes "0 ≤ c"
assumes "c / 2 + d + 1 ≤ c"
assumes "j ≤ k"
assumes "⋀n. n ≤ 2^k ⟹ psi n ≤ c * ln 2 * n"
assumes "n ≤ 2^(Suc k)"
shows "psi n ≤ c * ln 2 * n"
proof (cases "n ≤ 2^k")
case True
with assms(6) show ?thesis .
next
case False
from psi_bounds_induct(2)
have "psi n - psi (n div 2) ≤ real n * ln 2 + (4 * ln (real (if n = 0 then 1 else n)) + 3)" .
also from False have "(if n = 0 then 1 else n) = n"
by simp
finally have "psi n ≤ real n * ln 2 + (4 * ln (real n) + 3) + psi (n div 2)"
by simp
also from assms(6,7) have "psi (n div 2) ≤ c * ln 2 * (n div 2)"
by simp
also have "real (n div 2) ≤ real n / 2"
by simp
also have "real n * ln 2 + (4 * ln (real n) + 3) + c * ln 2 * (n / 2) ≤ c * ln 2 * real n"
proof (rule overpower_lemma[of
"λx. x * ln 2 + (4 * ln x + 3) + c * ln 2 * (x / 2)" "1+2^j"
"λx. c * ln 2 * x" "λx. c * ln 2 - ln 2 - 4 / x - c / 2 * ln 2"
"real n"])
from assms(1) have "4 * ln (1 + 2^j) + 3 ≤ d * ln 2 * (1 + 2^j)" .
also from assms(4) have "d ≤ c - c/2 - 1"
by simp
also have "(…) * ln 2 * (1 + 2 ^ j) = c * ln 2 * (1 + 2 ^ j) - c / 2 * ln 2 * (1 + 2 ^ j)
- (1 + 2 ^ j) * ln 2"
by (simp add: left_diff_distrib)
finally have "4 * ln (1 + 2^j) + 3 ≤ c * ln 2 * (1 + 2 ^ j) - c / 2 * ln 2 * (1 + 2 ^ j)
- (1 + 2 ^ j) * ln 2"
by (simp add: add_pos_pos)
then show "(1 + 2 ^ j) * ln 2 + (4 * ln (1 + 2 ^ j) + 3)
+ c * ln 2 * ((1 + 2 ^ j) / 2) ≤ c * ln 2 * (1 + 2 ^ j)"
by simp
next
fix x::real
assume x: "1 + 2^j ≤ x"
moreover have "1 + 2 ^ j > (0::real)" by (simp add: add_pos_pos)
ultimately have x_pos: "x > 0" by linarith
show "((λx. c * ln 2 * x - (x * ln 2 + (4 * ln x + 3) + c * ln 2 * (x / 2)))
has_real_derivative c * ln 2 - ln 2 - 4 / x - c / 2 * ln 2) (at x)"
by (rule derivative_eq_intros refl | simp add: ‹0 < x›)+
from ‹0 < x› ‹0 < 1 + 2^j› have "0 < x * (1 + 2^j)"
by (rule mult_pos_pos)
have "4 / x ≤ 4 / (1 + 2^j)"
by (intro divide_left_mono mult_pos_pos add_pos_pos x x_pos) simp_all
also from assms(2) have "4 / (1 + 2^j) ≤ d * ln 2" .
also from assms(4) have "d ≤ c - c/2 - 1" by simp
also have "… * ln 2 = c * ln 2 - c/2 * ln 2 - ln 2" by (simp add: algebra_simps)
finally show "0 ≤ c * ln 2 - ln 2 - 4 / x - c / 2 * ln 2" by simp
next
have "1 + 2^j = real (1 + 2^j)" by simp
also from assms(5) have "… ≤ real (1 + 2^k)" by simp
also from False have "2^k ≤ n - 1" by simp
finally show "1 + 2^j ≤ real n" using False by simp
qed
finally show ?thesis using assms by - (simp_all add: mult_left_mono)
qed
lemma psi_bounds_sustained:
assumes "⋀n. n ≤ 2^k ⟹ psi n ≤ c * ln 2 * n"
assumes "4 * ln (1 + 2^k) + 3 ≤ (c/2 - 1) * ln 2 * (1 + 2^k)"
assumes "4 / (1 + 2^k) ≤ (c/2 - 1) * ln 2"
assumes "c ≥ 0"
shows "psi n ≤ c * ln 2 * n"
proof -
have "psi n ≤ c * ln 2 * n" if "n ≤ 2^j" for j n
using that
proof (induction j arbitrary: n)
case 0
with assms(4) 0 show ?case unfolding psi_def mangoldt_def by (cases n) auto
next
case (Suc j)
show ?case
proof (cases "k ≤ j")
case True
from assms(4) have c_div_2: "c/2 + (c/2 - 1) + 1 ≤ c"
by simp
from psi_bounds_sustained_induct[of k "c/2 -1" c j,
OF assms(2) assms(3) assms(4) c_div_2 True Suc.IH Suc.prems]
show ?thesis by simp
next
case False
then have j_lt_k: "Suc j ≤ k" by simp
from Suc.prems have "n ≤ 2 ^ Suc j" .
also have "(2::nat) ^ Suc j ≤ 2 ^ k"
using power_increasing[of "Suc j" k "2::nat", OF j_lt_k]
by simp
finally show ?thesis using assms(1) by simp
qed
qed
from less_exp this [of n n] show ?thesis by simp
qed
lemma psi_ubound_log: "psi n ≤ 551 / 256 * ln 2 * n"
proof (rule psi_bounds_sustained)
show "0 ≤ 551 / (256 :: real)" by simp
next
fix n :: nat assume "n ≤ 2 ^ 10"
with psi_ubound_log_1024 show "psi n ≤ 551 / 256 * ln 2 * real n" by auto
next
have "4 / (1 + 2 ^ 10) ≤ (551 / 256 / 2 - 1) * (2/3 :: real)"
by simp
also have "… ≤ (551 / 256 / 2 - 1) * ln 2"
by (intro mult_left_mono ln_2_ge') simp_all
finally show "4 / (1 + 2 ^ 10) ≤ (551 / 256 / 2 - 1) * ln (2 :: real)" .
next
have "Some (Float 16 (-1)) = ub_ln 3 1025" by code_simp
from ub_ln(1)[OF this] and ln_2_ge
have "2048 * ln 1025 + 1536 ≤ 39975 * (ln 2::real)" by simp
thus "4 * ln (1 + 2 ^ 10) + 3 ≤ (551 / 256 / 2 - 1) * ln 2 * (1 + 2 ^ 10 :: real)"
by simp
qed
lemma psi_ubound_3_2: "psi n ≤ 3/2 * n"
proof -
have "(551 / 256) * ln 2 ≤ (551 / 256) * (16/23 :: real)"
by (intro mult_left_mono ln_2_le') auto
also have "… ≤ 3 / 2" by simp
finally have "551 / 256 * ln 2 ≤ 3/(2::real)" .
with of_nat_0_le_iff mult_right_mono have "551 / 256 * ln 2 * n ≤ 3/2 * n"
by blast
with psi_ubound_log[of "n"] show ?thesis
by linarith
qed
subsection ‹Doubling psi and theta›
lemma psi_residues_compare_2:
"psi_odd_2 n ≤ psi_even_2 n"
proof -
have "psi_odd_2 n = (∑d∈{d. d ∈ {2..n} ∧ primepow_odd d}. mangoldt_odd d)"
unfolding mangoldt_odd_def by (rule sum.mono_neutral_right) auto
also have "… = (∑d∈{d. d ∈ {2..n} ∧ primepow_odd d}. ln (real (aprimedivisor d)))"
by (intro sum.cong refl) (simp add: mangoldt_odd_def)
also have "… ≤ (∑d∈{d. d ∈ {2..n} ∧ primepow_even d}. ln (real (aprimedivisor d)))"
proof (rule sum_le_included [where i = "λy. y * aprimedivisor y"]; clarify?)
fix d :: nat assume "d ∈ {2..n}" "primepow_odd d"
note d = this
then obtain p k where d': "k ≥ 1" "prime p" "d = p ^ (2*k+1)"
by (auto simp: primepow_odd_def)
from d' have "p ^ (2 * k) ≤ p ^ (2</