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Theorem dchrmusumlem 27449

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.)

**Hypotheses**
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 = (0_g‘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))

**Assertion**
Ref	Expression
dchrmusumlem	⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛))) ∈ 𝑂(1))

Distinct variable groups: 𝑥,𝑛,𝑦, 1 𝐶,𝑛,𝑥,𝑦 𝑛,𝐹,𝑥,𝑦 𝑥,𝑎,𝑦 𝑛,𝑁,𝑥,𝑦 𝜑,𝑛,𝑥 𝑇,𝑛,𝑥,𝑦 𝑛,𝑍,𝑥,𝑦 𝐷,𝑛,𝑥,𝑦 𝑛,𝑎,𝐿,𝑥,𝑦 𝑋,𝑎,𝑛,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑦,𝑎) 𝐶(𝑎) 𝐷(𝑎) 𝑇(𝑎) 1 (𝑎) 𝐹(𝑎) 𝐺(𝑥,𝑦,𝑛,𝑎) 𝑁(𝑎) 𝑍(𝑎)

**Proof of Theorem dchrmusumlem**
Step	Hyp	Ref	Expression
1		fzfid 13965	. . . . . . 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 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
8		elfzelz 13528	. . . . . . . . . 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 27172	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑛)) ∈ ℂ)
11		elfznn 13557	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
12	11	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
13		mucl 27067	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (μ‘𝑛) ∈ ℤ)
14	12, 13	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
15	14	zred 12691	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℝ)
16	15, 12	nndivred 12291	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℝ)
17	16	recnd 11267	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℂ)
18	10, 17	mulcld 11259	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) ∈ ℂ)
19	1, 18	fsumcl 15706	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) ∈ ℂ)
20		dchrmusum.t	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑇)
21		climcl 15470	. . . . . . . 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 11259	. . . . 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 = (0_g‘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 27448	. . . . . 6 ⊢ (𝜑 → 𝑇 ≠ 0)
32	31	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑇 ≠ 0)
33	24, 23, 32	divrecd 12018	. . . 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 12021	. . . 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 2770	. . 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 5243	. 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 12008	. . . 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 27421	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇)) ∈ 𝑂(1))
40		rpssre 13008	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
41		o1const 15591	. . . 4 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (1 / 𝑇) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑇)) ∈ 𝑂(1))
42	40, 37, 41	sylancr 586	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑇)) ∈ 𝑂(1))
43	24, 38, 39, 42	o1mul2 15596	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇) · (1 / 𝑇))) ∈ 𝑂(1))
44	36, 43	eqeltrrd 2830	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 1534 ∈ wcel 2099 ≠ wne 2936 ∀wral 3057 ⊆ wss 3945 class class class wbr 5143 ↦ cmpt 5226 ‘cfv 6543 (class class class)co 7415 ℂcc 11131 ℝcr 11132 0cc0 11133 1c1 11134 + caddc 11136 · cmul 11138 +∞cpnf 11270 ≤ cle 11274 − cmin 11469 / cdiv 11896 ℕcn 12237 ℤcz 12583 ℝ⁺crp 13001 [,)cico 13353 ...cfz 13511 ⌊cfl 13782 seqcseq 13993 abscabs 15208 ⇝ cli 15455 𝑂(1)co1 15457 Σcsu 15659 Basecbs 17174 0_gc0g 17415 ℤRHomczrh 21419 ℤ/nℤczn 21422 μcmu 27021 DChrcdchr 27159

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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 ax-addf 11212 ax-mulf 11213

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-disj 5109 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7680 df-rpss 7723 df-om 7866 df-1st 7988 df-2nd 7989 df-supp 8161 df-tpos 8226 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-2o 8482 df-oadd 8485 df-omul 8486 df-er 8719 df-ec 8721 df-qs 8725 df-map 8841 df-pm 8842 df-ixp 8911 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-fsupp 9381 df-fi 9429 df-sup 9460 df-inf 9461 df-oi 9528 df-dju 9919 df-card 9957 df-acn 9960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-xnn0 12570 df-z 12584 df-dec 12703 df-uz 12848 df-q 12958 df-rp 13002 df-xneg 13119 df-xadd 13120 df-xmul 13121 df-ioo 13355 df-ioc 13356 df-ico 13357 df-icc 13358 df-fz 13512 df-fzo 13655 df-fl 13784 df-mod 13862 df-seq 13994 df-exp 14054 df-fac 14260 df-bc 14289 df-hash 14317 df-word 14492 df-concat 14548 df-s1 14573 df-shft 15041 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-limsup 15442 df-clim 15459 df-rlim 15460 df-o1 15461 df-lo1 15462 df-sum 15660 df-ef 16038 df-e 16039 df-sin 16040 df-cos 16041 df-tan 16042 df-pi 16043 df-dvds 16226 df-gcd 16464 df-prm 16637 df-numer 16701 df-denom 16702 df-phi 16729 df-pc 16800 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17398 df-topn 17399 df-0g 17417 df-gsum 17418 df-topgen 17419 df-pt 17420 df-prds 17423 df-xrs 17478 df-qtop 17483 df-imas 17484 df-qus 17485 df-xps 17486 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-mhm 18734 df-submnd 18735 df-grp 18887 df-minusg 18888 df-sbg 18889 df-mulg 19018 df-subg 19072 df-nsg 19073 df-eqg 19074 df-ghm 19162 df-gim 19207 df-ga 19235 df-cntz 19262 df-oppg 19291 df-od 19477 df-gex 19478 df-pgp 19479 df-lsm 19585 df-pj1 19586 df-cmn 19731 df-abl 19732 df-cyg 19827 df-dprd 19946 df-dpj 19947 df-mgp 20069 df-rng 20087 df-ur 20116 df-ring 20169 df-cring 20170 df-oppr 20267 df-dvdsr 20290 df-unit 20291 df-invr 20321 df-dvr 20334 df-rhm 20405 df-subrng 20477 df-subrg 20502 df-drng 20620 df-lmod 20739 df-lss 20810 df-lsp 20850 df-sra 21052 df-rgmod 21053 df-lidl 21098 df-rsp 21099 df-2idl 21138 df-psmet 21265 df-xmet 21266 df-met 21267 df-bl 21268 df-mopn 21269 df-fbas 21270 df-fg 21271 df-cnfld 21274 df-zring 21367 df-zrh 21423 df-zn 21426 df-top 22790 df-topon 22807 df-topsp 22829 df-bases 22843 df-cld 22917 df-ntr 22918 df-cls 22919 df-nei 22996 df-lp 23034 df-perf 23035 df-cn 23125 df-cnp 23126 df-haus 23213 df-cmp 23285 df-tx 23460 df-hmeo 23653 df-fil 23744 df-fm 23836 df-flim 23837 df-flf 23838 df-xms 24220 df-ms 24221 df-tms 24222 df-cncf 24792 df-0p 25593 df-limc 25789 df-dv 25790 df-ply 26116 df-idp 26117 df-coe 26118 df-dgr 26119 df-quot 26220 df-ulm 26307 df-log 26484 df-cxp 26485 df-atan 26793 df-em 26919 df-cht 27023 df-vma 27024 df-chp 27025 df-ppi 27026 df-mu 27027 df-dchr 27160

This theorem is referenced by: dchrmusum 27451

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