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Mirrors > Home > MPE Home > Th. List > dchrmusumlem | Structured version Visualization version GIF version |
Description: The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by 𝑛, is bounded. Equation 9.4.16 of [Shapiro], p. 379. (Contributed by Mario Carneiro, 12-May-2016.) |
Ref | Expression |
---|---|
rpvmasum.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
rpvmasum.l | ⊢ 𝐿 = (ℤRHom‘𝑍) |
rpvmasum.a | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
dchrmusum.g | ⊢ 𝐺 = (DChr‘𝑁) |
dchrmusum.d | ⊢ 𝐷 = (Base‘𝐺) |
dchrmusum.1 | ⊢ 1 = (0g‘𝐺) |
dchrmusum.b | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
dchrmusum.n1 | ⊢ (𝜑 → 𝑋 ≠ 1 ) |
dchrmusum.f | ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) |
dchrmusum.c | ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) |
dchrmusum.t | ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑇) |
dchrmusum.2 | ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦)) |
Ref | Expression |
---|---|
dchrmusumlem | ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛))) ∈ 𝑂(1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzfid 13329 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1...(⌊‘𝑥)) ∈ Fin) | |
2 | dchrmusum.g | . . . . . . . . 9 ⊢ 𝐺 = (DChr‘𝑁) | |
3 | rpvmasum.z | . . . . . . . . 9 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
4 | dchrmusum.d | . . . . . . . . 9 ⊢ 𝐷 = (Base‘𝐺) | |
5 | rpvmasum.l | . . . . . . . . 9 ⊢ 𝐿 = (ℤRHom‘𝑍) | |
6 | dchrmusum.b | . . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
7 | 6 | ad2antrr 722 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷) |
8 | elfzelz 12896 | . . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℤ) | |
9 | 8 | adantl 482 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℤ) |
10 | 2, 3, 4, 5, 7, 9 | dchrzrhcl 25748 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑛)) ∈ ℂ) |
11 | elfznn 12924 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ) | |
12 | 11 | adantl 482 | . . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ) |
13 | mucl 25645 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (μ‘𝑛) ∈ ℤ) | |
14 | 12, 13 | syl 17 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ) |
15 | 14 | zred 12075 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℝ) |
16 | 15, 12 | nndivred 11679 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℝ) |
17 | 16 | recnd 10657 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℂ) |
18 | 10, 17 | mulcld 10649 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) ∈ ℂ) |
19 | 1, 18 | fsumcl 15078 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) ∈ ℂ) |
20 | dchrmusum.t | . . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑇) | |
21 | climcl 14844 | . . . . . . . 8 ⊢ (seq1( + , 𝐹) ⇝ 𝑇 → 𝑇 ∈ ℂ) | |
22 | 20, 21 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ ℂ) |
23 | 22 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑇 ∈ ℂ) |
24 | 19, 23 | mulcld 10649 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇) ∈ ℂ) |
25 | rpvmasum.a | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
26 | dchrmusum.1 | . . . . . . 7 ⊢ 1 = (0g‘𝐺) | |
27 | dchrmusum.n1 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ 1 ) | |
28 | dchrmusum.f | . . . . . . 7 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) | |
29 | dchrmusum.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) | |
30 | dchrmusum.2 | . . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦)) | |
31 | 3, 5, 25, 2, 4, 26, 6, 27, 28, 29, 20, 30 | dchrisumn0 26024 | . . . . . 6 ⊢ (𝜑 → 𝑇 ≠ 0) |
32 | 31 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑇 ≠ 0) |
33 | 24, 23, 32 | divrecd 11407 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇) / 𝑇) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇) · (1 / 𝑇))) |
34 | 19, 23, 32 | divcan4d 11410 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇) / 𝑇) = Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛))) |
35 | 33, 34 | eqtr3d 2855 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇) · (1 / 𝑇)) = Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛))) |
36 | 35 | mpteq2dva 5152 | . 2 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇) · (1 / 𝑇))) = (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)))) |
37 | 22, 31 | reccld 11397 | . . . 4 ⊢ (𝜑 → (1 / 𝑇) ∈ ℂ) |
38 | 37 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1 / 𝑇) ∈ ℂ) |
39 | 3, 5, 25, 2, 4, 26, 6, 27, 28, 29, 20, 30 | dchrmusum2 25997 | . . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇)) ∈ 𝑂(1)) |
40 | rpssre 12384 | . . . 4 ⊢ ℝ+ ⊆ ℝ | |
41 | o1const 14964 | . . . 4 ⊢ ((ℝ+ ⊆ ℝ ∧ (1 / 𝑇) ∈ ℂ) → (𝑥 ∈ ℝ+ ↦ (1 / 𝑇)) ∈ 𝑂(1)) | |
42 | 40, 37, 41 | sylancr 587 | . . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (1 / 𝑇)) ∈ 𝑂(1)) |
43 | 24, 38, 39, 42 | o1mul2 14969 | . 2 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇) · (1 / 𝑇))) ∈ 𝑂(1)) |
44 | 36, 43 | eqeltrrd 2911 | 1 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛))) ∈ 𝑂(1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ⊆ wss 3933 class class class wbr 5057 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℝcr 10524 0cc0 10525 1c1 10526 + caddc 10528 · cmul 10530 +∞cpnf 10660 ≤ cle 10664 − cmin 10858 / cdiv 11285 ℕcn 11626 ℤcz 11969 ℝ+crp 12377 [,)cico 12728 ...cfz 12880 ⌊cfl 13148 seqcseq 13357 abscabs 14581 ⇝ cli 14829 𝑂(1)co1 14831 Σcsu 15030 Basecbs 16471 0gc0g 16701 ℤRHomczrh 20575 ℤ/nℤczn 20578 μcmu 25599 DChrcdchr 25735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-disj 5023 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-rpss 7438 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-omul 8096 df-er 8278 df-ec 8280 df-qs 8284 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-dju 9318 df-card 9356 df-acn 9359 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-fac 13622 df-bc 13651 df-hash 13679 df-word 13850 df-concat 13911 df-s1 13938 df-shft 14414 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-o1 14835 df-lo1 14836 df-sum 15031 df-ef 15409 df-e 15410 df-sin 15411 df-cos 15412 df-tan 15413 df-pi 15414 df-dvds 15596 df-gcd 15832 df-prm 16004 df-numer 16063 df-denom 16064 df-phi 16091 df-pc 16162 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-qus 16770 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-nsg 18215 df-eqg 18216 df-ghm 18294 df-gim 18337 df-ga 18358 df-cntz 18385 df-oppg 18412 df-od 18585 df-gex 18586 df-pgp 18587 df-lsm 18690 df-pj1 18691 df-cmn 18837 df-abl 18838 df-cyg 18926 df-dprd 19046 df-dpj 19047 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-rnghom 19396 df-drng 19433 df-subrg 19462 df-lmod 19565 df-lss 19633 df-lsp 19673 df-sra 19873 df-rgmod 19874 df-lidl 19875 df-rsp 19876 df-2idl 19933 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-zring 20546 df-zrh 20579 df-zn 20582 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-lp 21672 df-perf 21673 df-cn 21763 df-cnp 21764 df-haus 21851 df-cmp 21923 df-tx 22098 df-hmeo 22291 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-xms 22857 df-ms 22858 df-tms 22859 df-cncf 23413 df-0p 24198 df-limc 24391 df-dv 24392 df-ply 24705 df-idp 24706 df-coe 24707 df-dgr 24708 df-quot 24807 df-ulm 24892 df-log 25067 df-cxp 25068 df-atan 25372 df-em 25497 df-cht 25601 df-vma 25602 df-chp 25603 df-ppi 25604 df-mu 25605 df-dchr 25736 |
This theorem is referenced by: dchrmusum 26027 |
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