(* File: Prime_Harmonic.thy Author: Manuel Eberl <eberlm@in.tum.de> A lower bound for the partial sums of the prime harmonic series, and a proof of its divergence. (#81 on the list of 100 mathematical theorems) *) section ‹The Prime Harmonic Series› theory Prime_Harmonic imports "HOL-Analysis.Analysis" "HOL-Number_Theory.Number_Theory" Prime_Harmonic_Misc Squarefree_Nat begin subsection ‹Auxiliary equalities and inequalities› text ‹ First of all, we prove the following result about rearranging a product over a set into a sum over all subsets of that set. › lemma prime_harmonic_aux1: fixes A :: "'a :: field set" shows "finite A ⟹ (∏x∈A. 1 + 1 / x) = (∑x∈Pow A. 1 / ∏x)" proof (induction rule: finite_induct) fix a :: 'a and A :: "'a set" assume a: "a ∉ A" and fin: "finite A" assume IH: "(∏x∈A. 1 + 1 / x) = (∑x∈Pow A. 1 / ∏x)" from a and fin have "(∏x∈insert a A. 1 + 1 / x) = (1 + 1 / a) * (∏x∈A. 1 + 1 / x)" by simp also from fin have "… = (∑x∈Pow A. 1 / ∏x) + (∑x∈Pow A. 1 / (a * ∏x))" by (subst IH) (auto simp add: algebra_simps sum_divide_distrib) also from fin a have "(∑x∈Pow A. 1 / (a * ∏x)) = (∑x∈Pow A. 1 / ∏(insert a x))" by (intro sum.cong refl, subst prod.insert) (auto dest: finite_subset) also from a have "… = (∑x∈insert a ` Pow A. 1 / ∏x)" by (subst sum.reindex) (auto simp: inj_on_def) also from fin a have "(∑x∈Pow A. 1 / ∏x) + … = (∑x∈Pow A ∪ insert a ` Pow A. 1 / ∏x)" by (intro sum.union_disjoint [symmetric]) (simp, simp, blast) also have "Pow A ∪ insert a ` Pow A = Pow (insert a A)" by (simp only: Pow_insert) finally show " (∏x∈insert a A. 1 + 1 / x) = (∑x∈Pow (insert a A). 1 / ∏x)" . qed simp text ‹ Next, we prove a simple and reasonably accurate upper bound for the sum of the squares of any subset of the natural numbers, derived by simple telescoping. Our upper bound is approximately 1.67; the exact value is $\frac{\pi^2}{6} \approx 1.64$. (cf. Basel problem) › lemma prime_harmonic_aux2: assumes "finite (A :: nat set)" shows "(∑k∈A. 1 / (real k ^ 2)) ≤ 5/3" proof - define n where "n = max 2 (Max A)" have n: "n ≥ Max A" "n ≥ 2" by (auto simp: n_def) with assms have "A ⊆ {0..n}" by (auto intro: order.trans[OF Max_ge]) hence "(∑k∈A. 1 / (real k ^ 2)) ≤ (∑k=0..n. 1 / (real k ^ 2))" by (intro sum_mono2) auto also from n have "… = 1 + (∑k=Suc 1..n. 1 / (real k ^ 2))" by (simp add: sum.atLeast_Suc_atMost) also have "(∑k=Suc 1..n. 1 / (real k ^ 2)) ≤ (∑k=Suc 1..n. 1 / (real k ^ 2 - 1/4))" unfolding power2_eq_square by (intro sum_mono divide_left_mono mult_pos_pos) (linarith, simp_all add: field_simps less_1_mult) also have "… = (∑k=Suc 1..n. 1 / (real k - 1/2) - 1 / (real (Suc k) - 1/2))" by (intro sum.cong refl) (simp_all add: field_simps power2_eq_square) also from n have "… = 2 / 3 - 1 / (1 / 2 + real n)" by (subst sum_telescope') simp_all also have "1 + … ≤ 5/3" by simp finally show ?thesis by - simp qed subsection ‹Estimating the partial sums of the Prime Harmonic Series› text ‹ We are now ready to show our main result: the value of the partial prime harmonic sum over all primes no greater than $n$ is bounded from below by the $n$-th harmonic number $H_n$ minus some constant. In our case, this constant will be $\frac{5}{3}$. As mentioned before, using a proof of the Basel problem can improve this to $\frac{\pi^2}{6}$, but the improvement is very small and the proof of the Basel problem is a very complex one. The exact asymptotic behaviour of the partial sums is actually $\ln (\ln n) + M$, where $M$ is the Meissel--Mertens constant (approximately 0.261). › theorem prime_harmonic_lower: assumes n: "n ≥ 2" shows "(∑p←primes_upto n. 1 / real p) ≥ ln (harm n) - ln (5/3)" proof - ― ‹the set of primes that we will allow in the squarefree part› define P where "P n = set (primes_upto n)" for n { fix n :: nat have "finite (P n)" by (simp add: P_def) } note [simp] = this ― ‹The function that combines the squarefree part and the square part› define f where "f = (λ(R, s :: nat). ∏R * s^2)" ― ‹@{term f} is injective if the squarefree part contains only primes and the square part is positive.› have inj: "inj_on f (Pow (P n)×{1..n})" proof (rule inj_onI, clarify, rule conjI) fix A1 A2 :: "nat set" and s1 s2 :: nat assume A: "A1 ⊆ P n" "A2 ⊆ P n" "s1 ∈ {1..n}" "s2 ∈ {1..n}" "f (A1, s1) = f (A2, s2)" have fin: "finite A1" "finite A2" by (rule A(1,2)[THEN finite_subset], simp)+ show "A1 = A2" "s1 = s2" by ((rule squarefree_decomposition_unique2'[of A1 s1 A2 s2], insert A fin, auto simp: f_def P_def set_primes_upto)[])+ qed ― ‹@{term f} hits every number between @{term "1::nat"} and @{term "n"}. It also hits a lot of other numbers, but we do not care about those, since we only need a lower bound.› have surj: "{1..n} ⊆ f ` (Pow (P n)×{1..n})" proof fix x assume x: "x ∈ {1..n}" have "x = f (squarefree_part x, square_part x)" by (simp add: f_def squarefree_decompose) moreover have "squarefree_part x ∈ Pow (P n)" using squarefree_part_subset[of x] x by (auto simp: P_def set_primes_upto intro: order.trans[OF squarefree_part_le[of _ x]]) moreover have "square_part x ∈ {1..n}" using x by (auto simp: Suc_le_eq intro: order.trans[OF square_part_le[of x]]) ultimately show "x ∈ f ` (Pow (P n)×{1..n})" by simp qed ― ‹We now show the main result by rearranging the sum over all primes to a product over all all squarefree parts times a sum over all square parts, and then applying some simple-minded approximation› have "harm n = (∑n=1..n. 1 / real n)" by (simp add: harm_def field_simps) also from surj have "… ≤ (∑n∈f ` (Pow (P n)×{1..n}). 1 / real n)" by (intro sum_mono2 finite_imageI finite_cartesian_product) simp_all also from inj have "… = (∑x∈Pow (P n)×{1..n}. 1 / real (f x))" by (subst sum.reindex) simp_all also have "… = (∑A∈Pow (P n). 1 / real (∏A)) * (∑k=1..n. 1 / (real k)^2)" unfolding f_def by (subst sum_product, subst sum.cartesian_product) (simp add: case_prod_beta) also have "… ≤ (∑A∈Pow (P n). 1 / real (∏A)) * (5/3)" by (intro mult_left_mono prime_harmonic_aux2 sum_nonneg) (auto simp: P_def intro!: prod_nonneg) also have "(∑A∈Pow (P n). 1 / real (∏A)) = (∑A∈((`) real) ` Pow (P n). 1 / ∏A)" by (subst sum.reindex) (auto simp: inj_on_def inj_image_eq_iff prod.reindex) also have "((`) real) ` Pow (P n) = Pow (real ` P n)" by (intro image_Pow_surj refl) also have "(∑A∈Pow (real ` P n). 1 / ∏A) = (∏x∈real ` P n. 1 + 1 / x)" by (intro prime_harmonic_aux1 [symmetric] finite_imageI) simp_all also have "… = (∏i∈P n. 1 + 1 / real i)" by (subst prod.reindex) (auto simp: inj_on_def) also have "… ≤ (∏i∈P n. exp (1 / real i))" by (intro prod_mono) auto also have "… = exp (∑i∈P n. 1 / real i)" by (simp add: exp_sum) finally have "ln (harm n) ≤ ln (… * (5/3))" using n by (subst ln_le_cancel_iff) simp_all hence "ln (harm n) - ln (5/3) ≤ (∑i∈P n. 1 / real i)" by (subst (asm) ln_mult) (simp_all add: algebra_simps) thus ?thesis unfolding P_def by (subst (asm) sum.distinct_set_conv_list) simp_all qed text ‹ We can use the inequality $\ln (n + 1) \le H_n$ to estimate the asymptotic growth of the partial prime harmonic series. Note that $H_n \sim \ln n + \gamma$ where $\gamma$ is the Euler--Mascheroni constant (approximately 0.577), so we lose some accuracy here. › corollary prime_harmonic_lower': assumes n: "n ≥ 2" shows "(∑p←primes_upto n. 1 / real p) ≥ ln (ln (n + 1)) - ln (5/3)" proof - from assms ln_le_harm[of n] have "ln (ln (real n + 1)) ≤ ln (harm n)" by simp also from assms have "… - ln (5/3) ≤ (∑p←primes_upto n. 1 / real p)" by (rule prime_harmonic_lower) finally show ?thesis by - simp qed (* TODO: Not needed in Isabelle 2016 *) lemma Bseq_eventually_mono: assumes "eventually (λn. norm (f n) ≤ norm (g n)) sequentially" "Bseq g" shows "Bseq f" proof - from assms(1) obtain N where N: "⋀n. n ≥ N ⟹ norm (f n) ≤ norm (g n)" by (auto simp: eventually_at_top_linorder) from assms(2) obtain K where K: "⋀n. norm (g n) ≤ K" by (blast elim!: BseqE) { fix n :: nat have "norm (f n) ≤ max K (Max {norm (f n) |n. n < N})" apply (cases "n < N") apply (rule max.coboundedI2, rule Max.coboundedI, auto) [] apply (rule max.coboundedI1, force intro: order.trans[OF N K]) done } thus ?thesis by (blast intro: BseqI') qed lemma Bseq_add: assumes "Bseq (f :: nat ⇒ 'a :: real_normed_vector)" shows "Bseq (λx. f x + c)" proof - from assms obtain K where K: "⋀x. norm (f x) ≤ K" unfolding Bseq_def by blast { fix x :: nat have "norm (f x + c) ≤ norm (f x) + norm c" by (rule norm_triangle_ineq) also have "norm (f x) ≤ K" by (rule K) finally have "norm (f x + c) ≤ K + norm c" by simp } thus ?thesis by (rule BseqI') qed lemma convergent_imp_Bseq: "convergent f ⟹ Bseq f" by (simp add: Cauchy_Bseq convergent_Cauchy) (* END TODO *) text ‹ We now use our last estimate to show that the prime harmonic series diverges. This is obvious, since it is bounded from below by $\ln (\ln (n + 1))$ minus some constant, which obviously tends to infinite. Directly using the divergence of the harmonic series would also be possible and shorten this proof a bit.. › corollary prime_harmonic_series_unbounded: "¬Bseq (λn. ∑p←primes_upto n. 1 / p)" (is "¬Bseq ?f") proof assume "Bseq ?f" hence "Bseq (λn. ?f n + ln (5/3))" by (rule Bseq_add) have "Bseq (λn. ln (ln (n + 1)))" proof (rule Bseq_eventually_mono) from eventually_ge_at_top[of "2::nat"] show "eventually (λn. norm (ln (ln (n + 1))) ≤ norm (?f n + ln (5/3))) sequentially" proof eventually_elim fix n :: nat assume n: "n ≥ 2" hence "norm (ln (ln (real n + 1))) = ln (ln (real n + 1))" using ln_ln_nonneg[of "real n + 1"] by simp also have "… ≤ ?f n + ln (5/3)" using prime_harmonic_lower'[OF n] by (simp add: algebra_simps) also have "?f n + ln (5/3) ≥ 0" by (intro add_nonneg_nonneg sum_list_nonneg) simp_all hence "?f n + ln (5/3) = norm (?f n + ln (5/3))" by simp finally show "norm (ln (ln (n + 1))) ≤ norm (?f n + ln (5/3))" by (simp add: add_ac) qed qed fact then obtain k where k: "k > 0" "⋀n. norm (ln (ln (real (n::nat) + 1))) ≤ k" by (auto elim!: BseqE simp: add_ac) define N where "N = nat ⌈exp (exp k)⌉" have N_pos: "N > 0" unfolding N_def by simp have "real N + 1 > exp (exp k)" unfolding N_def by linarith hence "ln (real N + 1) > ln (exp (exp k))" by (subst ln_less_cancel_iff) simp_all with N_pos have "ln (ln (real N + 1)) > ln (exp k)" by (subst ln_less_cancel_iff) simp_all hence "k < ln (ln (real N + 1))" by simp also have "… ≤ norm (ln (ln (real N + 1)))" by simp finally show False using k(2)[of N] by simp qed corollary prime_harmonic_series_diverges: "¬convergent (λn. ∑p←primes_upto n. 1 / p)" using prime_harmonic_series_unbounded convergent_imp_Bseq by blast end