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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 14011 | . . . . . . 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 726 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷) |
8 | elfzelz 13561 | . . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℤ) | |
9 | 8 | adantl 481 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℤ) |
10 | 2, 3, 4, 5, 7, 9 | dchrzrhcl 27304 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑛)) ∈ ℂ) |
11 | elfznn 13590 | . . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ) | |
12 | 11 | adantl 481 | . . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ) |
13 | mucl 27199 | . . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (μ‘𝑛) ∈ ℤ) | |
14 | 12, 13 | syl 17 | . . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ) |
15 | 14 | zred 12720 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℝ) |
16 | 15, 12 | nndivred 12318 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℝ) |
17 | 16 | recnd 11287 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℂ) |
18 | 10, 17 | mulcld 11279 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) ∈ ℂ) |
19 | 1, 18 | fsumcl 15766 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) ∈ ℂ) |
20 | dchrmusum.t | . . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑇) | |
21 | climcl 15532 | . . . . . . . 8 ⊢ (seq1( + , 𝐹) ⇝ 𝑇 → 𝑇 ∈ ℂ) | |
22 | 20, 21 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ ℂ) |
23 | 22 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑇 ∈ ℂ) |
24 | 19, 23 | mulcld 11279 | . . . . 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 27580 | . . . . . 6 ⊢ (𝜑 → 𝑇 ≠ 0) |
32 | 31 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑇 ≠ 0) |
33 | 24, 23, 32 | divrecd 12044 | . . . 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 12047 | . . . 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 2777 | . . 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 5248 | . 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 12034 | . . . 4 ⊢ (𝜑 → (1 / 𝑇) ∈ ℂ) |
38 | 37 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (1 / 𝑇) ∈ ℂ) |
39 | 3, 5, 25, 2, 4, 26, 6, 27, 28, 29, 20, 30 | dchrmusum2 27553 | . . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇)) ∈ 𝑂(1)) |
40 | rpssre 13040 | . . . 4 ⊢ ℝ+ ⊆ ℝ | |
41 | o1const 15653 | . . . 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 15658 | . 2 ⊢ (𝜑 → (𝑥 ∈ ℝ+ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇) · (1 / 𝑇))) ∈ 𝑂(1)) |
44 | 36, 43 | eqeltrrd 2840 | 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 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ⊆ wss 3963 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 +∞cpnf 11290 ≤ cle 11294 − cmin 11490 / cdiv 11918 ℕcn 12264 ℤcz 12611 ℝ+crp 13032 [,)cico 13386 ...cfz 13544 ⌊cfl 13827 seqcseq 14039 abscabs 15270 ⇝ cli 15517 𝑂(1)co1 15519 Σcsu 15719 Basecbs 17245 0gc0g 17486 ℤRHomczrh 21528 ℤ/nℤczn 21531 μcmu 27153 DChrcdchr 27291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-rpss 7742 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-o1 15523 df-lo1 15524 df-sum 15720 df-ef 16100 df-e 16101 df-sin 16102 df-cos 16103 df-tan 16104 df-pi 16105 df-dvds 16288 df-gcd 16529 df-prm 16706 df-numer 16769 df-denom 16770 df-phi 16800 df-pc 16871 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-qus 17556 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-nsg 19155 df-eqg 19156 df-ghm 19244 df-gim 19290 df-ga 19321 df-cntz 19348 df-oppg 19377 df-od 19561 df-gex 19562 df-pgp 19563 df-lsm 19669 df-pj1 19670 df-cmn 19815 df-abl 19816 df-cyg 19911 df-dprd 20030 df-dpj 20031 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-2idl 21278 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-zring 21476 df-zrh 21532 df-zn 21535 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-cmp 23411 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-0p 25719 df-limc 25916 df-dv 25917 df-ply 26242 df-idp 26243 df-coe 26244 df-dgr 26245 df-quot 26348 df-ulm 26435 df-log 26613 df-cxp 26614 df-atan 26925 df-em 27051 df-cht 27155 df-vma 27156 df-chp 27157 df-ppi 27158 df-mu 27159 df-dchr 27292 |
This theorem is referenced by: dchrmusum 27583 |
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