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Mirrors > Home > MPE Home > Th. List > dchrvmasumlem | 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 |
---|---|
dchrvmasumlem | ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) ∈ 𝑂(1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpvmasum.z | . . . . . . . 8 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
2 | rpvmasum.l | . . . . . . . 8 ⊢ 𝐿 = (ℤRHom‘𝑍) | |
3 | rpvmasum.a | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
4 | dchrmusum.g | . . . . . . . 8 ⊢ 𝐺 = (DChr‘𝑁) | |
5 | dchrmusum.d | . . . . . . . 8 ⊢ 𝐷 = (Base‘𝐺) | |
6 | dchrmusum.1 | . . . . . . . 8 ⊢ 1 = (0g‘𝐺) | |
7 | dchrmusum.b | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
8 | dchrmusum.n1 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ≠ 1 ) | |
9 | dchrmusum.f | . . . . . . . 8 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) | |
10 | dchrmusum.c | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) | |
11 | dchrmusum.t | . . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑇) | |
12 | dchrmusum.2 | . . . . . . . 8 ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦)) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | dchrisumn0 26089 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ≠ 0) |
14 | 13 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑇 ≠ 0) |
15 | ifnefalse 4477 | . . . . . 6 ⊢ (𝑇 ≠ 0 → if(𝑇 = 0, (log‘𝑥), 0) = 0) | |
16 | 14, 15 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → if(𝑇 = 0, (log‘𝑥), 0) = 0) |
17 | 16 | oveq2d 7164 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑇 = 0, (log‘𝑥), 0)) = (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + 0)) |
18 | fzfid 13333 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1...(⌊‘𝑥)) ∈ Fin) | |
19 | 7 | ad2antrr 724 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷) |
20 | elfzelz 12900 | . . . . . . . . 9 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℤ) | |
21 | 20 | adantl 484 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℤ) |
22 | 4, 1, 5, 2, 19, 21 | dchrzrhcl 25813 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑛)) ∈ ℂ) |
23 | elfznn 12928 | . . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ) | |
24 | 23 | adantl 484 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ) |
25 | vmacl 25687 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ → (Λ‘𝑛) ∈ ℝ) | |
26 | nndivre 11670 | . . . . . . . . . 10 ⊢ (((Λ‘𝑛) ∈ ℝ ∧ 𝑛 ∈ ℕ) → ((Λ‘𝑛) / 𝑛) ∈ ℝ) | |
27 | 25, 26 | mpancom 686 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → ((Λ‘𝑛) / 𝑛) ∈ ℝ) |
28 | 24, 27 | syl 17 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℝ) |
29 | 28 | recnd 10661 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((Λ‘𝑛) / 𝑛) ∈ ℂ) |
30 | 22, 29 | mulcld 10653 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ) |
31 | 18, 30 | fsumcl 15082 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) ∈ ℂ) |
32 | 31 | addid1d 10832 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + 0) = Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) |
33 | 17, 32 | eqtrd 2854 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑇 = 0, (log‘𝑥), 0)) = Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) |
34 | 33 | mpteq2dva 5152 | . 2 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑇 = 0, (log‘𝑥), 0))) = (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)))) |
35 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | dchrvmasumif 26071 | . 2 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑇 = 0, (log‘𝑥), 0))) ∈ 𝑂(1)) |
36 | 34, 35 | eqeltrrd 2912 | 1 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) ∈ 𝑂(1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ∀wral 3136 ifcif 4465 class class class wbr 5057 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7148 ℝcr 10528 0cc0 10529 1c1 10530 + caddc 10532 · cmul 10534 +∞cpnf 10664 ≤ cle 10668 − cmin 10862 / cdiv 11289 ℕcn 11630 ℤcz 11973 ℝ+crp 12381 [,)cico 12732 ...cfz 12884 ⌊cfl 13152 seqcseq 13361 abscabs 14585 ⇝ cli 14833 𝑂(1)co1 14835 Σcsu 15034 Basecbs 16475 0gc0g 16705 ℤRHomczrh 20639 ℤ/nℤczn 20642 logclog 25130 Λcvma 25661 DChrcdchr 25800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-inf2 9096 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-addf 10608 ax-mulf 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 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 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-rpss 7441 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-tpos 7884 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-omul 8099 df-er 8281 df-ec 8283 df-qs 8287 df-map 8400 df-pm 8401 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-fi 8867 df-sup 8898 df-inf 8899 df-oi 8966 df-dju 9322 df-card 9360 df-acn 9363 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-xnn0 11960 df-z 11974 df-dec 12091 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-ioo 12734 df-ioc 12735 df-ico 12736 df-icc 12737 df-fz 12885 df-fzo 13026 df-fl 13154 df-mod 13230 df-seq 13362 df-exp 13422 df-fac 13626 df-bc 13655 df-hash 13683 df-word 13854 df-concat 13915 df-s1 13942 df-shft 14418 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-limsup 14820 df-clim 14837 df-rlim 14838 df-o1 14839 df-lo1 14840 df-sum 15035 df-ef 15413 df-e 15414 df-sin 15415 df-cos 15416 df-tan 15417 df-pi 15418 df-dvds 15600 df-gcd 15836 df-prm 16008 df-numer 16067 df-denom 16068 df-phi 16095 df-pc 16166 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-qus 16774 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-mhm 17948 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-nsg 18269 df-eqg 18270 df-ghm 18348 df-gim 18391 df-ga 18412 df-cntz 18439 df-oppg 18466 df-od 18648 df-gex 18649 df-pgp 18650 df-lsm 18753 df-pj1 18754 df-cmn 18900 df-abl 18901 df-cyg 18989 df-dprd 19109 df-dpj 19110 df-mgp 19232 df-ur 19244 df-ring 19291 df-cring 19292 df-oppr 19365 df-dvdsr 19383 df-unit 19384 df-invr 19414 df-dvr 19425 df-rnghom 19459 df-drng 19496 df-subrg 19525 df-lmod 19628 df-lss 19696 df-lsp 19736 df-sra 19936 df-rgmod 19937 df-lidl 19938 df-rsp 19939 df-2idl 19997 df-psmet 20529 df-xmet 20530 df-met 20531 df-bl 20532 df-mopn 20533 df-fbas 20534 df-fg 20535 df-cnfld 20538 df-zring 20610 df-zrh 20643 df-zn 20646 df-top 21494 df-topon 21511 df-topsp 21533 df-bases 21546 df-cld 21619 df-ntr 21620 df-cls 21621 df-nei 21698 df-lp 21736 df-perf 21737 df-cn 21827 df-cnp 21828 df-haus 21915 df-cmp 21987 df-tx 22162 df-hmeo 22355 df-fil 22446 df-fm 22538 df-flim 22539 df-flf 22540 df-xms 22922 df-ms 22923 df-tms 22924 df-cncf 23478 df-0p 24263 df-limc 24456 df-dv 24457 df-ply 24770 df-idp 24771 df-coe 24772 df-dgr 24773 df-quot 24872 df-ulm 24957 df-log 25132 df-cxp 25133 df-atan 25437 df-em 25562 df-cht 25666 df-vma 25667 df-chp 25668 df-ppi 25669 df-mu 25670 df-dchr 25801 |
This theorem is referenced by: dchrvmasum 26093 |
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