Theory Prime_Number_Theorem
section ‹The Prime Number Theorem›
theory Prime_Number_Theorem
imports
Newman_Ingham_Tauberian
Prime_Counting_Functions
begin
unbundle prime_counting_notation
subsection ‹Constructing Newman's function›
text ‹
Starting from Mertens' first theorem, i.\,e.\ $\mathfrak M(x) = \ln x + O(1)$, we now
want to derive that $\mathfrak M(x) = \ln x + c + o(1)$. This result is considerably stronger
and it implies the Prime Number Theorem quite directly.
In order to do this, we define the Dirichlet series
\[f(s) = \sum_{n=1}^\infty \frac{\mathfrak{M}(n)}{n^s}\ .\]
We will prove that this series extends meromorphically to $\mathfrak{R}(s)\geq 1$ and
apply Ingham's theorem to it (after we subtracted its pole at $s = 1$).
›
definition fds_newman where
"fds_newman = fds (λn. complex_of_real (𝔐 n))"
lemma fds_nth_newman:
"fds_nth fds_newman n = of_real (𝔐 n)"
by (simp add: fds_newman_def fds_nth_fds)
lemma norm_fds_nth_newman:
"norm (fds_nth fds_newman n) = 𝔐 n"
unfolding fds_nth_newman norm_of_real
by (intro abs_of_nonneg sum_nonneg divide_nonneg_pos) (auto dest: prime_ge_1_nat)
text ‹
The Dirichlet series $f(s) + \zeta'(s)$ has the coefficients $\mathfrak{M}(n) - \ln n$,
so by Mertens' first theorem, $f(s) + \zeta'(s)$ has bounded coefficients.
›
lemma bounded_coeffs_newman_minus_deriv_zeta:
defines "f ≡ fds_newman + fds_deriv fds_zeta"
shows "Bseq (λn. fds_nth f n)"
proof -
have "(λn. 𝔐 (real n) - ln (real n)) ∈ O(λ_. 1)"
using mertens_bounded by (rule landau_o.big.compose) real_asymp
from natfun_bigo_1E[OF this, of 1]
obtain c where c: "c ≥ 1" "⋀n. ¦𝔐 (real n) - ln (real n)¦ ≤ c" by auto
show ?thesis
proof (intro BseqI[of c] allI)
fix n :: nat
show "norm (fds_nth f n) ≤ c"
proof (cases "n = 0")
case False
hence "fds_nth f n = of_real (𝔐 n - ln n)"
by (simp add: f_def fds_nth_newman fds_nth_deriv fds_nth_zeta scaleR_conv_of_real)
also from ‹n ≠ 0› have "norm … ≤ c"
using c(2)[of n] by (simp add: in_Reals_norm)
finally show ?thesis .
qed (insert c, auto)
qed (insert c, auto)
qed
text ‹
A Dirichlet series with bounded coefficients converges for all $s$ with
$\mathfrak{R}(s)>1$ and so does $\zeta'(s)$, so we can conclude that $f(s)$ does as well.
›
lemma abs_conv_abscissa_newman: "abs_conv_abscissa fds_newman ≤ 1"
and conv_abscissa_newman: "conv_abscissa fds_newman ≤ 1"
proof -
define f where "f = fds_newman + fds_deriv fds_zeta"
have "abs_conv_abscissa f ≤ 1"
using bounded_coeffs_newman_minus_deriv_zeta unfolding f_def
by (rule bounded_coeffs_imp_abs_conv_abscissa_le_1)
hence "abs_conv_abscissa (f - fds_deriv fds_zeta) ≤ 1"
by (intro abs_conv_abscissa_diff_leI) (auto simp: abs_conv_abscissa_deriv)
also have "f - fds_deriv fds_zeta = fds_newman" by (simp add: f_def)
finally show "abs_conv_abscissa fds_newman ≤ 1" .
from conv_le_abs_conv_abscissa and this show "conv_abscissa fds_newman ≤ 1"
by (rule order.trans)
qed
text ‹
We now change the order of summation to obtain an alternative form of $f(s)$ in terms of a
sum of Hurwitz $\zeta$ functions.
›
lemma eval_fds_newman_conv_infsetsum:
assumes s: "Re s > 1"
shows "eval_fds fds_newman s = (∑⇩ap | prime p. (ln (real p) / real p) * hurwitz_zeta p s)"
"(λp. ln (real p) / real p * hurwitz_zeta p s) abs_summable_on {p. prime p}"
proof -
from s have conv: "fds_abs_converges fds_newman s"
by (intro fds_abs_converges le_less_trans[OF abs_conv_abscissa_newman]) auto
define f where "f = (λn p. ln (real p) / real p / of_nat n powr s)"
have eq: "(∑⇩an∈{p..}. f n p) = ln (real p) / real p * hurwitz_zeta p s" if "prime p" for p
proof -
have "(∑⇩an∈{p..}. f n p) = (∑⇩ax∈{p..}. (ln (real p) / of_nat p) * (1 / of_nat x powr s))"
by (simp add: f_def)
also have "… = (ln (real p) / of_nat p) * (∑⇩ax∈{p..}. 1 / of_nat x powr s)"
using abs_summable_hurwitz_zeta[of s 0 p] that s
by (intro infsetsum_cmult_right) (auto dest: prime_gt_0_nat)
also have "(∑⇩ax∈{p..}. 1 / of_nat x powr s) = hurwitz_zeta p s"
using s that by (subst hurwitz_zeta_nat_conv_infsetsum(2))
(auto dest: prime_gt_0_nat simp: field_simps powr_minus)
finally show ?thesis .
qed
have norm_f: "norm (f n p) = ln p / p / n powr Re s" if "prime p" for n p :: nat
by (auto simp: f_def norm_divide norm_mult norm_powr_real_powr)
from conv have "(λn. norm (fds_nth fds_newman n / n powr s)) abs_summable_on UNIV"
by (intro abs_summable_on_normI) (simp add: fds_abs_converges_altdef')
also have "(λn. norm (fds_nth fds_newman n / n powr s)) =
(λn. ∑p | prime p ∧ p ≤ n. norm (f n p))"
by (auto simp: norm_divide norm_fds_nth_newman sum_divide_distrib primes_M_def
prime_sum_upto_def norm_mult norm_f norm_powr_real_powr intro!: sum.cong)
finally have summable1: "(λ(n,p). f n p) abs_summable_on (SIGMA n:UNIV. {p. prime p ∧ p ≤ n})"
using conv by (subst abs_summable_on_Sigma_iff) auto
also have "?this ⟷ (λ(p,n). f n p) abs_summable_on
(λ(n,p). (p,n)) ` (SIGMA n:UNIV. {p. prime p ∧ p ≤ n})"
by (subst abs_summable_on_reindex_iff [symmetric]) (auto simp: case_prod_unfold inj_on_def)
also have "(λ(n,p). (p,n)) ` (SIGMA n:UNIV. {p. prime p ∧ p ≤ n}) =
(SIGMA p:{p. prime p}. {p..})" by auto
finally have summable2: "(λ(p,n). f n p) abs_summable_on …" .
from abs_summable_on_Sigma_project1'[OF this]
have "(λp. ∑⇩an∈{p..}. f n p) abs_summable_on {p. prime p}" by auto
also have "?this ⟷ (λp. ln (real p) / real p * hurwitz_zeta p s) abs_summable_on {p. prime p}"
by (intro abs_summable_on_cong eq) auto
finally show … .
have "eval_fds fds_newman s =
(∑⇩an. ∑p | prime p ∧ p ≤ n. ln (real p) / real p / of_nat n powr s)"
using conv by (simp add: eval_fds_altdef fds_nth_newman sum_divide_distrib
primes_M_def prime_sum_upto_def)
also have "… = (∑⇩an. ∑⇩ap | prime p ∧ p ≤ n. f n p)"
unfolding f_def by (subst infsetsum_finite) auto
also have "… = (∑⇩a(n, p) ∈ (SIGMA n:UNIV. {p. prime p ∧ p ≤ n}). f n p)"
using summable1 by (subst infsetsum_Sigma) auto
also have "… = (∑⇩a(p, n) ∈ (λ(n,p). (p, n)) ` (SIGMA n:UNIV. {p. prime p ∧ p ≤ n}). f n p)"
by (subst infsetsum_reindex) (auto simp: case_prod_unfold inj_on_def)
also have "(λ(n,p). (p, n)) ` (SIGMA n:UNIV. {p. prime p ∧ p ≤ n}) =
(SIGMA p:{p. prime p}. {p..})" by auto
also have "(∑⇩a(p,n)∈…. f n p) = (∑⇩ap | prime p. ∑⇩an∈{p..}. f n p)"
using summable2 by (subst infsetsum_Sigma) auto
also have "(∑⇩ap | prime p. ∑⇩an∈{p..}. f n p) =
(∑⇩ap | prime p. ln (real p) / real p * hurwitz_zeta p s)"
by (intro infsetsum_cong eq) auto
finally show "eval_fds fds_newman s =
(∑⇩ap | prime p. (ln (real p) / real p) * hurwitz_zeta p s)" .
qed
text ‹
We now define a meromorphic continuation of $f(s)$ on $\mathfrak{R}(s) > \frac{1}{2}$.
To construct $f(s)$, we express it as
\[f(s) = \frac{1}{z-1}\left(\bar f(s) - \frac{\zeta'(s)}{\zeta(s)}\right)\ ,\]
where $\bar f(s)$ (which we shall call ‹pre_newman›) is a function that is analytic on
$\Re(s) > \frac{1}{2}$, which can be shown fairly easily using the Weierstra{\ss} M test.
$\zeta'(s)/\zeta(s)$ is meromorphic except for a single pole at $s = 1$ and one $k$-th order
pole for any $k$-th order zero of $\zeta$, but for the Prime Number Theorem, we are only
concerned with the area $\mathfrak{R}(s) \geq 1$, where $\zeta$ does not have any zeros.
Taken together, this means that $f(s)$ is analytic for $\mathfrak{R}(s)\geq 1$ except for a
double pole at $s = 1$, which we will take care of later.
›
context
fixes A :: "nat ⇒ complex ⇒ complex" and B :: "nat ⇒ complex ⇒ complex"
defines "A ≡ (λp s. (s - 1) * pre_zeta (real p) s -
of_nat p / (of_nat p powr s * (of_nat p powr s - 1)))"
defines "B ≡ (λp s. of_real (ln (real p)) / of_nat p * A p s)"
begin
definition pre_newman :: "complex ⇒ complex" where
"pre_newman s = (∑p. if prime p then B p s else 0)"
definition newman where "newman s = 1 / (s - 1) * (pre_newman s - deriv zeta s / zeta s)"
text ‹
The sum used in the definition of ‹pre_newman› converges uniformly on any disc within the
half-space with $\mathfrak{R}(s) > \frac{1}{2}$ by the Weierstra{\ss} M test.
›
lemma uniform_limit_pre_newman:
assumes r: "r ≥ 0" "Re s - r > 1 / 2"
shows "uniform_limit (cball s r)
(λn s. ∑p<n. if prime p then B p s else 0) pre_newman at_top"
proof -
from r have Re: "Re z > 1 / 2" if "dist s z ≤ r" for z
using abs_Re_le_cmod[of "s - z"] r that
by (auto simp: dist_norm abs_if split: if_splits)
define x where "x = Re s - r"
from r Re have "x > 1 / 2" by (auto simp: x_def)
define C where "C = (norm s + r + 1) * (norm s + r) / x"
define M where "M = (λp::nat. ln p * (C / p powr (x + 1) + 1 / (p powr x * (p powr x - 1))))"
show ?thesis unfolding pre_newman_def
proof (intro Weierstrass_m_test_ev[OF eventually_mono[OF eventually_gt_at_top[of 1]]] ballI)
show "summable M"
proof (rule summable_comparison_test_bigo)
define ε where "ε = min (2 * x - 1) x / 2"
from ‹x > 1 / 2› have ε: "ε > 0" "1 + ε < 2 * x" "1 + ε < x + 1"
by (auto simp: ε_def min_def field_simps)
show "M ∈ O(λn. n powr (- 1 - ε))" unfolding M_def distrib_left
by (intro sum_in_bigo) (use ε in real_asymp)+
from ε show "summable (λn. norm (n powr (- 1 - ε)))"
by (simp add: summable_real_powr_iff)
qed
next
fix p :: nat and z assume p: "p > 1" and z: "z ∈ cball s r"
from z r Re[of z] have x: "Re z ≥ x" "x > 1 / 2" and "Re z > 1 / 2"
using abs_Re_le_cmod[of "s - z"] by (auto simp: x_def algebra_simps dist_norm)
have norm_z: "norm z ≤ norm s + r"
using z norm_triangle_ineq2[of z s] r by (auto simp: dist_norm norm_minus_commute)
from ‹p > 1› and x and r have "M p ≥ 0"
by (auto simp: C_def M_def intro!: mult_nonneg_nonneg add_nonneg_nonneg divide_nonneg_pos)
have bound: "norm ((z - 1) * pre_zeta p z) ≤
norm (z - 1) * (norm z / (Re z * p powr Re z))"
using pre_zeta_bound'[of z p] p ‹Re z > 1 / 2›
unfolding norm_mult by (intro mult_mono pre_zeta_bound) auto
have "norm (B p z) = ln p / p * norm (A p z)"
unfolding B_def using ‹p > 1› by (simp add: B_def norm_mult norm_divide)
also have "… ≤ ln p / p * (norm (z - 1) * norm z / Re z / p powr Re z +
p / (p powr Re z * (p powr Re z - 1)))"
unfolding A_def using ‹p > 1› and ‹Re z > 1 / 2› and bound
by (intro mult_left_mono order.trans[OF norm_triangle_ineq4 add_mono] mult_left_mono)
(auto simp: norm_divide norm_mult norm_powr_real_powr
intro!: divide_left_mono order.trans[OF _ norm_triangle_ineq2])
also have "… = ln p * (norm (z - 1) * norm z / Re z / p powr (Re z + 1) +
1 / (p powr Re z * (p powr Re z - 1)))"
using ‹p > 1› by (simp add: field_simps powr_add powr_minus)
also have "norm (z - 1) * norm z / Re z / p powr (Re z + 1) ≤ C / p powr (x + 1)"
unfolding C_def using r ‹Re z > 1 / 2› norm_z p x
by (intro mult_mono frac_le powr_mono order.trans[OF norm_triangle_ineq4]) auto
also have "1 / (p powr Re z * (p powr Re z - 1)) ≤
1 / (p powr x * (p powr x - 1))" using ‹p > 1› x
by (intro divide_left_mono mult_mono powr_mono diff_right_mono mult_pos_pos)
(auto simp: ge_one_powr_ge_zero)
finally have "norm (B p z) ≤ M p"
using ‹p > 1› by (simp add: mult_left_mono M_def)
with ‹M p ≥ 0› show "norm (if prime p then B p z else 0) ≤ M p" by simp
qed
qed
lemma sums_pre_newman: "Re s > 1 / 2 ⟹ (λp. if prime p then B p s else 0) sums pre_newman s"
using tendsto_uniform_limitI[OF uniform_limit_pre_newman[of 0 s]] by (auto simp: sums_def)
lemma analytic_pre_newman [THEN analytic_on_subset, analytic_intros]:
"pre_newman analytic_on {s. Re s > 1 / 2}"
proof -
have holo: "(λs::complex. if prime p then B p s else 0) holomorphic_on X"
if "X ⊆ {s. Re s > 1 / 2}" for X and p :: nat using that
by (cases "prime p")
(auto intro!: holomorphic_intros simp: B_def A_def dest!: prime_gt_1_nat)
have holo': "pre_newman holomorphic_on ball s r" if r: "r ≥ 0" "Re s - r > 1 / 2" for s r
proof -
from r have Re: "Re z > 1 / 2" if "dist s z ≤ r" for z
using abs_Re_le_cmod[of "s - z"] r that by (auto simp: dist_norm abs_if split: if_splits)
show ?thesis
by (rule holomorphic_uniform_limit[OF _ uniform_limit_pre_newman[of r s]])
(insert that Re, auto intro!: always_eventually holomorphic_on_imp_continuous_on
holomorphic_intros holo)
qed
show ?thesis unfolding analytic_on_def
proof safe
fix s assume "Re s > 1 / 2"
thus "∃r>0. pre_newman holomorphic_on ball s r"
by (intro exI[of _ "(Re s - 1 / 2) / 2"] conjI holo') (auto simp: field_simps)
qed
qed
lemma holomorphic_pre_newman [holomorphic_intros]:
"X ⊆ {s. Re s > 1 / 2} ⟹ pre_newman holomorphic_on X"
using analytic_pre_newman by (rule analytic_imp_holomorphic)
lemma eval_fds_newman:
assumes s: "Re s > 1"
shows "eval_fds fds_newman s = newman s"
proof -
have eq: "(ln (real p) / real p) * hurwitz_zeta p s =
1 / (s - 1) * (ln (real p) / (p powr s - 1) + B p s)"
if p: "prime p" for p
proof -
have "(ln (real p) / real p) * hurwitz_zeta p s =
ln (real p) / real p * (p powr (1 - s) / (s - 1) + pre_zeta p s)"
using s by (auto simp add: hurwitz_zeta_def)
also have "… = 1 / (s - 1) * (ln (real p) / (p powr s - 1) + B p s)"
using p s by (simp add: divide_simps powr_diff B_def)
(auto simp: A_def field_simps dest: prime_gt_1_nat)?
finally show ?thesis .
qed
have "(λp. (ln (real p) / real p) * hurwitz_zeta p s) abs_summable_on {p. prime p}"
using s by (intro eval_fds_newman_conv_infsetsum)
hence "(λp. 1 / (s - 1) * (ln (real p) / (p powr s - 1) + B p s))
abs_summable_on {p. prime p}"
by (subst (asm) abs_summable_on_cong[OF eq refl]) auto
hence summable:
"(λp. ln (real p) / (p powr s - 1) + B p s) abs_summable_on {p. prime p}"
using s by (subst (asm) abs_summable_on_cmult_right_iff) auto
from s have [simp]: "s ≠ 1" by auto
have "eval_fds fds_newman s =
(∑⇩ap | prime p. (ln (real p) / real p) * hurwitz_zeta p s)"
using s by (rule eval_fds_newman_conv_infsetsum)
also have "… = (∑⇩ap | prime p. 1 / (s - 1) * (ln (real p) / (p powr s - 1) + B p s))"
by (intro infsetsum_cong eq) auto
also have "… = 1 / (s - 1) * (∑⇩ap | prime p. ln (real p) / (p powr s - 1) + B p s)"
(is "_ = _ * ?S") by (rule infsetsum_cmult_right[OF summable])
also have "?S = (∑p. if prime p then
ln (real p) / (p powr s - 1) + B p s else 0)"
by (subst infsetsum_nat[OF summable]) auto
also have "… = (∑p. (if prime p then ln (real p) / (p powr s - 1) else 0) +
(if prime p then B p s else 0))"
by (intro suminf_cong) auto
also have "… = pre_newman s - deriv zeta s / zeta s"
using sums_pre_newman[of s] sums_logderiv_zeta[of s] s
by (subst suminf_add [symmetric]) (auto simp: sums_iff)
finally show ?thesis by (simp add: newman_def)
qed
end
text ‹
Next, we shall attempt to get rid of the pole by subtracting suitable multiples of $\zeta(s)$
and $\zeta'(s)$. To this end, we shall first prove the following alternative definition of
$\zeta'(s)$:
›
lemma deriv_zeta_eq':
assumes "0 < Re s" "s ≠ 1"
shows "deriv zeta s = deriv (λz. pre_zeta 1 z * (z - 1)) s / (s - 1) -
(pre_zeta 1 s * (s - 1) + 1) / (s - 1)⇧2"
(is "_ = ?rhs")
proof (rule DERIV_imp_deriv)
have [derivative_intros]: "(pre_zeta 1 has_field_derivative deriv (pre_zeta 1) s) (at s)"
by (intro holomorphic_derivI[of _ UNIV] holomorphic_intros) auto
have *: "deriv (λz. pre_zeta 1 z * (z - 1)) s = deriv (pre_zeta 1) s * (s - 1) + pre_zeta 1 s"
by (subst deriv_mult)
(auto intro!: holomorphic_on_imp_differentiable_at[of _ UNIV] holomorphic_intros)
hence "((λs. pre_zeta 1 s + 1 / (s - 1)) has_field_derivative
deriv (pre_zeta 1) s - 1 / ((s - 1) * (s - 1))) (at s)"
using assms by (auto intro!: derivative_eq_intros)
also have "deriv (pre_zeta 1) s - 1 / ((s - 1) * (s - 1)) = ?rhs"
using * assms by (simp add: divide_simps power2_eq_square, simp add: field_simps)
also have "((λs. pre_zeta 1 s + 1 / (s - 1)) has_field_derivative ?rhs) (at s) ⟷
(zeta has_field_derivative ?rhs) (at s)"
using assms
by (intro has_field_derivative_cong_ev eventually_mono[OF t1_space_nhds[of _ 1]])
(auto simp: zeta_def hurwitz_zeta_def)
finally show … .
qed
text ‹
From this, it follows that $(s - 1) \zeta'(s) - \zeta'(s) / \zeta(s)$ is analytic
for $\mathfrak{R}(s) \geq 1$:
›
lemma analytic_zeta_derivdiff:
obtains a where
"(λz. if z = 1 then a else (z - 1) * deriv zeta z - deriv zeta z / zeta z)
analytic_on {s. Re s ≥ 1}"
proof
have neq: "pre_zeta 1 z * (z - 1) + 1 ≠ 0" if "Re z ≥ 1" for z
using zeta_Re_ge_1_nonzero[of z] that
by (cases "z = 1") (auto simp: zeta_def hurwitz_zeta_def divide_simps)
let ?g = "λz. (1 - inverse (pre_zeta 1 z * (z - 1) + 1)) * ((z - 1) *
deriv ((λu. pre_zeta 1 u * (u - 1))) z - (pre_zeta 1 z * (z - 1) + 1))"
show "(λz. if z = 1 then deriv ?g 1 else (z - 1) * deriv zeta z - deriv zeta z / zeta z)
analytic_on {s. Re s ≥ 1}" (is "?f analytic_on _")
proof (rule pole_theorem_analytic_0)
show "?g analytic_on {s. 1 ≤ Re s}" using neq
by (auto intro!: analytic_intros)
next
show "∃d>0. ∀w∈ball z d - {1}. ?g w = (w - 1) * ?f w"
if z: "z ∈ {s. 1 ≤ Re s}" for z
proof -
have *: "isCont (λz. pre_zeta 1 z * (z - 1) + 1) z"
by (auto intro!: continuous_intros)
obtain e where "e > 0" and e: "⋀y. dist z y < e ⟹ pre_zeta (Suc 0) y * (y-1) + 1 ≠ 0"
using continuous_at_avoid [OF * neq[of z]] z by auto
show ?thesis
proof (intro exI ballI conjI)
fix w
assume w: "w ∈ ball z (min e 1) - {1}"
then have "Re w > 0"
using complex_Re_le_cmod [of "z-w"] z by (simp add: dist_norm)
with w show "?g w = (w - 1) * (if w = 1 then deriv ?g 1 else
(w - 1) * deriv zeta w - deriv zeta w / zeta w)"
by (subst (1 2) deriv_zeta_eq',
simp_all add: zeta_def hurwitz_zeta_def divide_simps e power2_eq_square)
(simp_all add: algebra_simps)?
qed (use ‹e > 0› in auto)
qed
qed auto
qed
text ‹
Finally, $f(s) + \zeta'(s) + c\zeta(s)$ is analytic.
›
lemma analytic_newman_variant:
obtains c a where
"(λz. if z = 1 then a else newman z + deriv zeta z + c * zeta z) analytic_on {s. Re s ≥ 1}"
proof -
obtain c where
c: "(λz. if z = 1 then c else (z - 1) * deriv zeta z - deriv zeta z / zeta z)
analytic_on {s. Re s ≥ 1}"
using analytic_zeta_derivdiff by blast
let ?g = "λz. pre_newman z +
(if z = 1 then c
else (z - 1) * deriv zeta z -
deriv zeta z / zeta z) - (c + pre_newman 1) * (pre_zeta 1 z * (z - 1) + 1)"
have "(λz. if z = 1 then deriv ?g 1 else newman z + deriv zeta z + (-(c + pre_newman 1)) * zeta z)
analytic_on {s. Re s ≥ 1}" (is "?f analytic_on _")
proof (rule pole_theorem_analytic_0)
show "?g analytic_on {s. 1 ≤ Re s}"
by (intro c analytic_intros) auto
next
show "∃d>0. ∀w∈ball z d - {1}. ?g w = (w - 1) * ?f w"
if "z ∈ {s. 1 ≤ Re s}" for z using that
by (intro exI[of _ 1], simp_all add: newman_def divide_simps zeta_def hurwitz_zeta_def)
(auto simp: field_simps)?
qed auto
with that show ?thesis by blast
qed
subsection ‹The asymptotic expansion of ‹𝔐››
text ‹
Our next goal is to show the key result that $\mathfrak{M}(x) = \ln n + c + o(1)$.
As a first step, we invoke Ingham's Tauberian theorem on the function we have
just defined and obtain that the sum
\[\sum\limits_{n=1}^\infty \frac{\mathfrak{M}(n) - \ln n + c}{n}\]
exists.
›
lemma mertens_summable:
obtains c :: real where "summable (λn. (𝔐 n - ln n + c) / n)"
proof -
from analytic_newman_variant obtain c a where
analytic: "(λz. if z = 1 then a else newman z + deriv zeta z + c * zeta z)
analytic_on {s. Re s ≥ 1}" .
define f where "f = (λz. if z = 1 then a else newman z + deriv zeta z + c * zeta z)"
have analytic: "f analytic_on {s. Re s ≥ 1}" using analytic by (simp add: f_def)
define F where "F = fds_newman + fds_deriv fds_zeta + fds_const c * fds_zeta"
note le = conv_abscissa_add_leI conv_abscissa_deriv_le conv_abscissa_newman conv_abscissa_mult_const_left
note intros = le le[THEN le_less_trans] le[THEN order.trans] fds_converges
have eval_F: "eval_fds F s = f s" if s: "Re s > 1" for s
proof -
have "eval_fds F s = eval_fds (fds_newman + fds_deriv fds_zeta) s +
eval_fds (fds_const c * fds_zeta) s"
unfolding F_def using s by (subst eval_fds_add) (auto intro!: intros)
also have "… = f s" using s unfolding f_def
by (subst eval_fds_add)
(auto intro!: intros simp: eval_fds_newman eval_fds_deriv_zeta eval_fds_mult eval_fds_zeta)
finally show ?thesis .
qed
have conv: "fds_converges F s" if "Re s ≥ 1" for s
proof (rule Newman_Ingham_1)
have "(λn. 𝔐 (real n) - ln (real n)) ∈ O(λ_. 1)"
using mertens_bounded by (rule landau_o.big.compose) real_asymp
from natfun_bigo_1E[OF this, of 1]
obtain c' where c': "c' ≥ 1" "⋀n. ¦𝔐 (real n) - ln (real n)¦ ≤ c'" by auto
have "Bseq (fds_nth F)"
proof (intro BseqI allI)
fix n :: nat
show "norm (fds_nth F n) ≤ (c' + norm c)" unfolding F_def using c'
by (auto simp: fds_nth_zeta fds_nth_deriv fds_nth_newman scaleR_conv_of_real in_Reals_norm
intro!: order.trans[OF norm_triangle_ineq] add_mono)
qed (insert c', auto intro: add_pos_nonneg)
thus "fds_nth F ∈ O(λ_. 1)" by (simp add: natfun_bigo_iff_Bseq)
next
show "f analytic_on {s. Re s ≥ 1}" by fact
next
show "eval_fds F s = f s" if "Re s > 1" for s using that by (rule eval_F)
qed (insert that, auto simp: F_def intro!: intros)
from conv[of 1] have "summable (λn. fds_nth F n / of_nat n)"
unfolding fds_converges_def by auto
also have "?this ⟷ summable (λn. (𝔐 n - Ln n + c) / n)"
by (intro summable_cong eventually_mono[OF eventually_gt_at_top[of 0]])
(auto simp: F_def fds_nth_newman fds_nth_deriv fds_nth_zeta scaleR_conv_of_real
intro!: sum.cong dest: prime_gt_0_nat)
finally have "summable (λn. (𝔐 n - Re (Ln (of_nat n)) + Re c) / n)"
by (auto dest: summable_Re)
also have "?this ⟷ summable (λn. (𝔐 n - ln n + Re c) / n)"
by (intro summable_cong eventually_mono[OF eventually_gt_at_top[of 0]]) (auto intro!: sum.cong)
finally show ?thesis using that[of "Re c"] by blast
qed
text ‹
Next, we prove a lemma given by Newman stating that if the sum $\sum a_n / n$ exists and
$a_n + \ln n$ is nondecreasing, then $a_n$ must tend to 0. Unfortunately, the proof is
rather tedious, but so is the paper version by Newman.
›
lemma sum_goestozero_lemma:
fixes d::real
assumes d: "¦∑i = M..N. a i / i¦ < d" and le: "⋀n. a n + ln n ≤ a (Suc n) + ln (Suc n)"
and "0 < M" "M < N"
shows "a M ≤ d * N / (real N - real M) + (real N - real M) / M ∧
-a N ≤ d * N / (real N - real M) + (real N - real M) / M"
proof -
have "0 ≤ d"
using assms by linarith+
then have "0 ≤ d * N / (N - M + 1)" by simp
then have le_dN: "⟦0 ≤ x ⟹ x ≤ d * N / (N - M + 1)⟧ ⟹ x ≤ d * N / (N - M + 1)" for x::real
by linarith
have le_a_ln: "a m + ln m ≤ a n + ln n" if "n ≥ m" for n m
by (rule transitive_stepwise_le) (use le that in auto)
have *: "x ≤ b ∧ y ≤ b" if "a ≤ b" "x ≤ a" "y ≤ a" for a b x y::real
using that by linarith
show ?thesis
proof (rule *)
show "d * N / (N - M) + ln (N / M) ≤ d * N / (real N - real M) + (real N - real M) / M"
using ‹0 < M› ‹M < N› ln_le_minus_one [of "N / M"]
by (simp add: of_nat_diff) (simp add: divide_simps)
next
have "a M - ln (N / M) ≤ (d * N) / (N - M + 1)"
proof (rule le_dN)
assume 0: "0 ≤ a M - ln (N / M)"
have "(Suc N - M) * (a M - ln (N / M)) / N = (∑i = M..N. (a M - ln (N / M)) / N)"
by simp
also have "… ≤ (∑i = M..N. a i / i)"
proof (rule sum_mono)
fix i
assume i: "i ∈ {M..N}"
with ‹0 < M› have "0 < i" by auto
have "(a M - ln (N / M)) / N ≤ (a M - ln (N / M)) / i"
using 0 using i ‹0 < M› by (simp add: frac_le_eq divide_simps mult_left_mono)
also have "a M + ln (real M) ≤ a i + ln (real N)"
by (rule order.trans[OF le_a_ln[of M i]]) (use i assms in auto)
hence "(a M - ln (N / M)) / i ≤ a i / real i"
using assms i by (intro divide_right_mono) (auto simp: ln_div field_simps)
finally show "(a M - ln (N / M)) / real N ≤ a i / real i" .
qed
finally have "((Suc N) - M) * (a M - ln (N / M)) / N ≤ ¦∑i = M..N. a i / i¦"
by simp
also have "… ≤ d" using d by simp
finally have "((Suc N) - M) * (a M - ln (N / M)) / N ≤ d" .
then show ?thesis
using ‹M < N› by (simp add: of_nat_diff field_simps)
qed
also have "… ≤ d * N / (N - M)"
using assms(1,4) by (simp add: field_simps)
finally show "a M ≤ d * N / (N - M) + ln (N / M)" by simp
next
have "- a N - ln (N / M) ≤ (d * N) / (N - M + 1)"
proof (rule le_dN)
assume 0: "0 ≤ - a N - ln (N / M)"
have "(∑i = M..N. a i / i) ≤ (∑i = M..N. (a N + ln (N / M)) / N)"
proof (rule sum_mono)
fix i
assume i: "i ∈ {M..N}"
with ‹0 < M› have "0 < i" by auto
have "a i + ln (real M) ≤ a N + ln (real N)"
by (rule order.trans[OF _ le_a_ln[of i N]]) (use i assms in auto)
hence "a i / i ≤ (a N + ln (N / M)) / i"
using assms(3,4) by (intro divide_right_mono) (auto simp: field_simps ln_div)
also have "… ≤ (a N + ln (N / M)) / N"
using i ‹i > 0› 0 by (intro divide_left_mono_neg) auto
finally show "a i / i ≤ (a N + ln (N / M)) / N" .
qed
also have "… = ((Suc N) - M) * (a N + ln (N / M)) / N"
by simp
finally have "(∑i = M..N. a i / i) ≤ (real (Suc N) - real M) * (a N + ln (N / M)) / N"
using ‹M < N› by (simp add: of_nat_diff)
then have "-((real (Suc N) - real M) * (a N + ln (N / M)) / N) ≤ ¦∑i = M..N. a i / i¦"
by linarith
also have "… ≤ d" using d by simp
finally have "- ((real (Suc N) - real M) * (a N + ln (N / M)) / N) ≤ d" .
then show ?thesis
using ‹M < N› by (simp add: of_nat_diff field_simps)
qed
also have "… ≤ d * N / real (N - M)"
using ‹0 < M› ‹M < N› ‹0 ≤ d› by (simp add: field_simps)
finally show "-a N ≤ d * N / real (N - M) + ln (N / M)" by simp
qed
qed
proposition sum_goestozero_theorem:
assumes summ: "summable (λi. a i / i)"
and le: "⋀n. a n + ln n ≤ a (Suc n) + ln (Suc n)"
shows "a ⇢ 0"
proof (clarsimp simp: lim_sequentially)
fix r::real
assume "r > 0"
have *: "∃n0. ∀n≥n0. ¦a n¦ < ε" if ε: "0 < ε" "ε < 1" for ε
proof -
have "0 < (ε / 8)⇧2" using ‹0 < ε› by simp
then obtain N0 where N0: "⋀m n. m ≥ N0 ⟹ norm (∑k=m..n. (λi. a i / i) k) < (ε / 8)⇧2"
by (metis summable_partial_sum_bound summ)
obtain N1 where "real N1 > 4 / ε"
using reals_Archimedean2[of "4 / ε"] ε by auto
hence "N1 ≠ 0" and N1: "1 / real N1 < ε / 4" using ε
by (auto simp: divide_simps mult_ac intro: Nat.gr0I)
have "¦a n¦ < ε" if n: "n ≥ 2 * N0 + N1 + 7" for n
proof -
define k where "k = ⌊n * ε/4⌋"
have "n * ε / 4 > 1" and "n * ε / 4 ≤ n / 4" and "n / 4 < n"
using less_le_trans[OF N1, of "n / N1 * ε / 4"] ‹N1 ≠ 0› ε n by (auto simp: field_simps)
hence k: "k > 0" "4 * k ≤ n" "nat k < n" "(n * ε / 4) - 1 < k" "k ≤ (n * ε / 4)"
unfolding k_def by linarith+
have "-a n < ε"
proof -
have "N0 ≤ n - nat k"
using n k by linarith
then have *: "¦∑k = n - nat k .. n. a k / k¦ < (ε / 8)⇧2"
using N0 [of "n - nat k" n] by simp
have "-a n ≤ (ε / 8)⇧2 * n / ⌊n * ε / 4⌋ + ⌊n * ε / 4⌋ / (n - k)"
using sum_goestozero_lemma [OF * le, THEN conjunct2] k by (simp add: of_nat_diff k_def)
also have "…< ε"
proof -
have "ε / 16 * n / k < 2"
using k by (auto simp: field_simps)
then have "ε * (ε / 16 * n / k) < ε * 2"
using ε mult_less_cancel_left_pos by blast
then have "(ε / 8)⇧2 * n / k < ε / 2"
by (simp add: field_simps power2_eq_square)
moreover have "k / (n - k) < ε / 2"
proof -
have "(ε + 2) * k < 4 * k" using k ε by simp
also have "… ≤ ε * real n" using k by (auto simp: field_simps)
finally show ?thesis using k by (auto simp: field_simps)
qed
ultimately show ?thesis unfolding k_def by linarith
qed
finally show ?thesis .
qed
moreover have "a n < ε"
proof -
have "N0 ≤ n" using n k by linarith
then have *: "¦∑k = n .. n + nat k. a k / k¦ < (ε/8)⇧2"
using N0 [of n "n + nat k"] by simp
have "a n ≤ (ε/8)⇧2 * (n + nat k) / k + k / n"
using sum_goestozero_lemma [OF * le, THEN conjunct1] k by (simp add: of_nat_diff)
also have "…< ε"
proof -
have "4 ≤ 28 * real_of_int k" using k by linarith
then have "ε/16 * n / k < 2" using k by (auto simp: field_simps)
have "ε * (real n + k) < 32 * k"
proof -
have "ε * n / 4 < k + 1" by (simp add: mult.commute k_def)
then have "ε * n < 4 * k + 4" by (simp add: divide_simps)
also have "… ≤ 8 * k" using k by auto
finally have 1: "ε * real n < 8 * k" .
have 2: "ε * k < k" using k ε by simp
show ?thesis using k add_strict_mono [OF 1 2] by (simp add: algebra_simps)
qed
then have "(ε / 8)⇧2 * real (n + nat k) / k < ε / 2"
using ε k by (simp add: divide_simps mult_less_0_iff power2_eq_square)
moreover have "k / n < ε / 2"
using k ε by (auto simp: k_def field_simps)
ultimately show ?thesis by linarith
qed
finally show ?thesis .
qed
ultimately show ?thesis by force
qed
then show ?thesis by blast
qed
show "∃n0. ∀n≥n0. ¦a n¦ < r"
using * [of "min r (1/5)"] ‹0 < r› by force
qed
text ‹
This leads us to the main intermediate result:
›
lemma Mertens_convergent: "convergent (λn::nat. 𝔐 n - ln n)"
proof -
obtain c where c: "summable (λn. (𝔐 n - ln n + c) / n)"
by (blast intro: mertens_summable)
then obtain l where l: "(λn. (𝔐 n - ln n + c) / n) sums l"
by (auto simp: summable_def)
have *: "(λn. 𝔐 n - ln n + c) ⇢ 0"
by (rule sum_goestozero_theorem[OF c]) auto
hence "(λn. 𝔐 n - ln n) ⇢ -c"
by (simp add: tendsto_iff dist_norm)
thus ?thesis by (rule convergentI)
qed
corollary 𝔐_minus_ln_limit:
obtains c where "((λx::real. 𝔐 x - ln x) ⤏ c) at_top"
proof -
from Mertens_convergent obtain c where "(λn. 𝔐 n - ln n) ⇢ c"
by (auto simp: convergent_def)
hence 1: "((λx::real. 𝔐 (nat ⌊x⌋) - ln (nat ⌊x⌋)) ⤏ c) at_top"
by (rule filterlim_compose) real_asymp
have 2: "((λx::real. ln (nat ⌊x⌋) - ln x) ⤏ 0) at_top"
by real_asymp
have 3: "((λx. 𝔐 x - ln x) ⤏ c) at_top"
using tendsto_add[OF 1 2] by simp
with that show ?thesis by blast
qed
subsection ‹The asymptotics of the prime-counting functions›
text ‹
We will now use the above result to prove the asymptotics of the prime-counting functions
$\vartheta(x) \sim x$, $\psi(x) \sim x$, and $\pi(x) \sim x / \ln x$. The last of these is
typically called the Prime Number Theorem, but since these functions can be expressed in terms
of one another quite easily, knowing the asymptotics of any of them immediately gives the
asymptotics of the other ones.
In this sense, all of the above are equivalent formulations of the Prime Number Theorem.
The one we shall tackle first, due to its strong connection to the $\mathfrak{M}$ function, is
$\vartheta(x) \sim x$.
We know that $\mathfrak{M}(x)$ has the asymptotic expansion
$\mathfrak{M}(x) = \ln x + c + o(1)$. We also know that
\[\vartheta(x) = x\mathfrak{M}(x) - \int\nolimits_2^x \mathfrak{M}(t) \,\mathrm{d}t\ .\]
Substituting in the above asymptotic equation, we obtain:
\begin{align*}
\vartheta(x) &= x\ln x + cx + o(x) - \int\nolimits_2^x \ln t + c + o(1) \,\mathrm{d}t\\
&= x\ln x + cx + o(x) - (x\ln x - x + cx + o(x))\\
&= x + o(x)
\end{align*}
In conclusion, $\vartheta(x) \sim x$.
›
theorem θ_asymptotics: "θ ∼[at_top] (λx. x)"
proof -
from 𝔐_minus_ln_limit obtain c where c: "((λx. 𝔐 x - ln x) ⤏ c) at_top"
by auto
define r where "r = (λx. 𝔐 x - ln x - c)"
have 𝔐_expand: "𝔐 = (λx. ln x + c + r x)"
by (simp add: r_def)
have r: "r ∈ o(λ_. 1)" unfolding r_def
using tendsto_add[OF c tendsto_const[of "-c"]] by (intro smalloI_tendsto) auto
define r' where "r' = (λx. integral {2..x} r)"
have integrable_r: "r integrable_on {x..y}"
if "2 ≤ x" for x y :: real using that unfolding r_def
by (intro integrable_diff integrable_primes_M)
(auto intro!: integrable_continuous_real continuous_intros)
hence integral: "(r has_integral r' x) {2..x}" if "x ≥ 2" for x
by (auto simp: has_integral_iff r'_def)
have r': "r' ∈ o(λx. x)" using integrable_r unfolding r'_def
by (intro integral_smallo[OF r]) (auto simp: filterlim_ident)
define C where "C = 2 * (c + ln 2 - 1)"
have "θ ∼[at_top] (λx. x + (r x * x + C - r' x))"
proof (intro asymp_equiv_refl_ev eventually_mono[OF eventually_gt_at_top])
fix x :: real assume x: "x > 2"
have "(𝔐 has_integral ((x * ln x - x + c * x) - (2 * ln 2 - 2 + c * 2) + r' x)) {2..x}"
unfolding 𝔐_expand using x
by (intro has_integral_add[OF fundamental_theorem_of_calculus integral])
(auto simp flip: has_real_derivative_iff_has_vector_derivative
intro!: derivative_eq_intros continuous_intros)
from has_integral_unique[OF θ_conv_𝔐_integral this]
show "θ x = x + (r x * x + C - r' x)" using x
by (simp add: field_simps 𝔐_expand C_def)
qed
also have "(λx. r x * x + C - r' x) ∈ o(λx. x)"
proof (intro sum_in_smallo r)
show "(λ_. C) ∈ o(λx. x)" by real_asymp
qed (insert landau_o.small_big_mult[OF r, of "λx. x"] r', simp_all)
hence "(λx. x + (r x * x + C - r' x)) ∼[at_top] (λx. x)"
by (subst asymp_equiv_add_right) auto
finally show ?thesis by auto
qed
text ‹
The various other forms of the Prime Number Theorem follow as simple corollaries.
›
corollary ψ_asymptotics: "ψ ∼[at_top] (λx. x)"
using θ_asymptotics PNT4_imp_PNT5 by simp
corollary prime_number_theorem: "π ∼[at_top] (λx. x / ln x)"
using θ_asymptotics PNT4_imp_PNT1 by simp
corollary ln_π_asymptotics: "(λx. ln (π x)) ∼[at_top] ln"
using prime_number_theorem PNT1_imp_PNT1' by simp
corollary π_ln_π_asymptotics: "(λx. π x * ln (π x)) ∼[at_top] (λx. x)"
using prime_number_theorem PNT1_imp_PNT2 by simp
corollary nth_prime_asymptotics: "(λn. real (nth_prime n)) ∼[at_top] (λn. real n * ln (real n))"
using π_ln_π_asymptotics PNT2_imp_PNT3 by simp
text ‹
The following versions use a little less notation.
›
corollary prime_number_theorem': "((λx. π x / (x / ln x)) ⤏ 1) at_top"
using prime_number_theorem
by (rule asymp_equivD_strong[OF _ eventually_mono[OF eventually_gt_at_top[of 1]]]) auto
corollary prime_number_theorem'':
"(λx. card {p. prime p ∧ real p ≤ x}) ∼[at_top] (λx. x / ln x)"
proof -
have "π = (λx. card {p. prime p ∧ real p ≤ x})"
by (intro ext) (simp add: π_def prime_sum_upto_def)
with prime_number_theorem show ?thesis by simp
qed
corollary prime_number_theorem''':
"(λn. card {p. prime p ∧ p ≤ n}) ∼[at_top] (λn. real n / ln (real n))"
proof -
have "(λn. card {p. prime p ∧ real p ≤ real n}) ∼[at_top] (λn. real n / ln (real n))"
using prime_number_theorem''
by (rule asymp_equiv_compose') (simp add: filterlim_real_sequentially)
thus ?thesis by simp
qed
unbundle no_prime_counting_notation
end