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Mirrors > Home > MPE Home > Th. List > rpvmasum | Structured version Visualization version GIF version |
Description: The sum of the von Mangoldt function over those integers 𝑛≡𝐴 (mod 𝑁) is asymptotic to log𝑥 / ϕ(𝑥) + 𝑂(1). Equation 9.4.3 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 2-May-2016.) (Proof shortened by Mario Carneiro, 26-May-2016.) |
Ref | Expression |
---|---|
rpvmasum.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
rpvmasum.l | ⊢ 𝐿 = (ℤRHom‘𝑍) |
rpvmasum.a | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
rpvmasum.u | ⊢ 𝑈 = (Unit‘𝑍) |
rpvmasum.b | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
rpvmasum.t | ⊢ 𝑇 = (◡𝐿 “ {𝐴}) |
Ref | Expression |
---|---|
rpvmasum | ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − (log‘𝑥))) ∈ 𝑂(1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpvmasum.z | . . . . . . . . . . . . . 14 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
2 | rpvmasum.l | . . . . . . . . . . . . . 14 ⊢ 𝐿 = (ℤRHom‘𝑍) | |
3 | rpvmasum.a | . . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
4 | 3 | adantr 480 | . . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) → 𝑁 ∈ ℕ) |
5 | eqid 2610 | . . . . . . . . . . . . . 14 ⊢ (DChr‘𝑁) = (DChr‘𝑁) | |
6 | eqid 2610 | . . . . . . . . . . . . . 14 ⊢ (Base‘(DChr‘𝑁)) = (Base‘(DChr‘𝑁)) | |
7 | eqid 2610 | . . . . . . . . . . . . . 14 ⊢ (0g‘(DChr‘𝑁)) = (0g‘(DChr‘𝑁)) | |
8 | fveq2 6103 | . . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 = 𝑛 → (𝐿‘𝑚) = (𝐿‘𝑛)) | |
9 | 8 | fveq2d 6107 | . . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = 𝑛 → (𝑦‘(𝐿‘𝑚)) = (𝑦‘(𝐿‘𝑛))) |
10 | id 22 | . . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = 𝑛 → 𝑚 = 𝑛) | |
11 | 9, 10 | oveq12d 6567 | . . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑛 → ((𝑦‘(𝐿‘𝑚)) / 𝑚) = ((𝑦‘(𝐿‘𝑛)) / 𝑛)) |
12 | 11 | cbvsumv 14274 | . . . . . . . . . . . . . . . . 17 ⊢ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = Σ𝑛 ∈ ℕ ((𝑦‘(𝐿‘𝑛)) / 𝑛) |
13 | 12 | eqeq1i 2615 | . . . . . . . . . . . . . . . 16 ⊢ (Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0 ↔ Σ𝑛 ∈ ℕ ((𝑦‘(𝐿‘𝑛)) / 𝑛) = 0) |
14 | 13 | a1i 11 | . . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) → (Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0 ↔ Σ𝑛 ∈ ℕ ((𝑦‘(𝐿‘𝑛)) / 𝑛) = 0)) |
15 | 14 | rabbiia 3161 | . . . . . . . . . . . . . 14 ⊢ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} = {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑛 ∈ ℕ ((𝑦‘(𝐿‘𝑛)) / 𝑛) = 0} |
16 | simpr 476 | . . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) → 𝑓 ∈ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) | |
17 | 1, 2, 4, 5, 6, 7, 15, 16 | dchrisum0 25009 | . . . . . . . . . . . . 13 ⊢ ¬ (𝜑 ∧ 𝑓 ∈ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) |
18 | 17 | imnani 438 | . . . . . . . . . . . 12 ⊢ (𝜑 → ¬ 𝑓 ∈ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) |
19 | 18 | eq0rdv 3931 | . . . . . . . . . . 11 ⊢ (𝜑 → {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} = ∅) |
20 | 19 | fveq2d 6107 | . . . . . . . . . 10 ⊢ (𝜑 → (#‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) = (#‘∅)) |
21 | hash0 13019 | . . . . . . . . . 10 ⊢ (#‘∅) = 0 | |
22 | 20, 21 | syl6eq 2660 | . . . . . . . . 9 ⊢ (𝜑 → (#‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) = 0) |
23 | 22 | oveq2d 6565 | . . . . . . . 8 ⊢ (𝜑 → (1 − (#‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})) = (1 − 0)) |
24 | 1m0e1 11008 | . . . . . . . 8 ⊢ (1 − 0) = 1 | |
25 | 23, 24 | syl6eq 2660 | . . . . . . 7 ⊢ (𝜑 → (1 − (#‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})) = 1) |
26 | 25 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1 − (#‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})) = 1) |
27 | 26 | oveq2d 6565 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥) · (1 − (#‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}))) = ((log‘𝑥) · 1)) |
28 | relogcl 24126 | . . . . . . . 8 ⊢ (𝑥 ∈ ℝ+ → (log‘𝑥) ∈ ℝ) | |
29 | 28 | adantl 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℝ) |
30 | 29 | recnd 9947 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (log‘𝑥) ∈ ℂ) |
31 | 30 | mulid1d 9936 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥) · 1) = (log‘𝑥)) |
32 | 27, 31 | eqtrd 2644 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((log‘𝑥) · (1 − (#‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}))) = (log‘𝑥)) |
33 | 32 | oveq2d 6565 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (#‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0})))) = (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − (log‘𝑥))) |
34 | 33 | mpteq2dva 4672 | . 2 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (#‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}))))) = (𝑥 ∈ ℝ+ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − (log‘𝑥)))) |
35 | eqid 2610 | . . 3 ⊢ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} = {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} | |
36 | rpvmasum.u | . . 3 ⊢ 𝑈 = (Unit‘𝑍) | |
37 | rpvmasum.b | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
38 | rpvmasum.t | . . 3 ⊢ 𝑇 = (◡𝐿 “ {𝐴}) | |
39 | 17 | pm2.21i 115 | . . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ {𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) → 𝐴 = (1r‘𝑍)) |
40 | 1, 2, 3, 5, 6, 7, 35, 36, 37, 38, 39 | rpvmasum2 25001 | . 2 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − ((log‘𝑥) · (1 − (#‘{𝑦 ∈ ((Base‘(DChr‘𝑁)) ∖ {(0g‘(DChr‘𝑁))}) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}))))) ∈ 𝑂(1)) |
41 | 34, 40 | eqeltrrd 2689 | 1 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (((ϕ‘𝑁) · Σ𝑛 ∈ ((1...(⌊‘𝑥)) ∩ 𝑇)((Λ‘𝑛) / 𝑛)) − (log‘𝑥))) ∈ 𝑂(1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 ∖ cdif 3537 ∩ cin 3539 ∅c0 3874 {csn 4125 ↦ cmpt 4643 ◡ccnv 5037 “ cima 5041 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 · cmul 9820 − cmin 10145 / cdiv 10563 ℕcn 10897 ℝ+crp 11708 ...cfz 12197 ⌊cfl 12453 #chash 12979 𝑂(1)co1 14065 Σcsu 14264 ϕcphi 15307 Basecbs 15695 0gc0g 15923 1rcur 18324 Unitcui 18462 ℤRHomczrh 19667 ℤ/nℤczn 19670 logclog 24105 Λcvma 24618 DChrcdchr 24757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-rpss 6835 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-xnn0 11241 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-o1 14069 df-lo1 14070 df-sum 14265 df-ef 14637 df-e 14638 df-sin 14639 df-cos 14640 df-pi 14642 df-dvds 14822 df-gcd 15055 df-prm 15224 df-numer 15281 df-denom 15282 df-phi 15309 df-pc 15380 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-qus 15992 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-nsg 17415 df-eqg 17416 df-ghm 17481 df-gim 17524 df-ga 17546 df-cntz 17573 df-oppg 17599 df-od 17771 df-gex 17772 df-pgp 17773 df-lsm 17874 df-pj1 17875 df-cmn 18018 df-abl 18019 df-cyg 18103 df-dprd 18217 df-dpj 18218 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-rnghom 18538 df-drng 18572 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-sra 18993 df-rgmod 18994 df-lidl 18995 df-rsp 18996 df-2idl 19053 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-zring 19638 df-zrh 19671 df-zn 19674 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-cmp 21000 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-0p 23243 df-limc 23436 df-dv 23437 df-ply 23748 df-idp 23749 df-coe 23750 df-dgr 23751 df-quot 23850 df-log 24107 df-cxp 24108 df-em 24519 df-cht 24623 df-vma 24624 df-chp 24625 df-ppi 24626 df-mu 24627 df-dchr 24758 |
This theorem is referenced by: rplogsum 25016 |
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