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Theorem dchrmusum 24930

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 )

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

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

Proof of Theorem dchrmusum Dummy variables 𝑦 𝑐 𝑡 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpvmasum.z	. . 3 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
2		rpvmasum.l	. . 3 ⊢ 𝐿 = (ℤRHom‘𝑍)
3		rpvmasum.a	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4		dchrmusum.g	. . 3 ⊢ 𝐺 = (DChr‘𝑁)
5		dchrmusum.d	. . 3 ⊢ 𝐷 = (Base‘𝐺)
6		dchrmusum.1	. . 3 ⊢ 1 = (0_g‘𝐺)
7		dchrmusum.b	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
8		dchrmusum.n1	. . 3 ⊢ (𝜑 → 𝑋 ≠ 1 )
9		eqid 2609	. . 3 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	dchrmusumlema 24899	. 2 ⊢ (𝜑 → ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))
11	3	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑁 ∈ ℕ)
12	7	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑋 ∈ 𝐷)
13	8	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑋 ≠ 1 )
14		simprl 789	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑐 ∈ (0[,)+∞))
15		simprrl 799	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡)
16		simprrr 800	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦))
17	1, 2, 11, 4, 5, 6, 12, 13, 9, 14, 15, 16	dchrmusumlem 24928	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛))) ∈ 𝑂(1))
18	17	rexlimdvaa 3013	. . 3 ⊢ (𝜑 → (∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛))) ∈ 𝑂(1)))
19	18	exlimdv 1847	. 2 ⊢ (𝜑 → (∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛))) ∈ 𝑂(1)))
20	10, 19	mpd 15	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 382 = wceq 1474 ∃wex 1694 ∈ wcel 1976 ≠ wne 2779 ∀wral 2895 ∃wrex 2896 class class class wbr 4577 ↦ cmpt 4637 ‘cfv 5790 (class class class)co 6527 0cc0 9792 1c1 9793 + caddc 9795 · cmul 9797 +∞cpnf 9927 ≤ cle 9931 − cmin 10117 / cdiv 10533 ℕcn 10867 ℝ⁺crp 11664 [,)cico 12004 ...cfz 12152 ⌊cfl 12408 seqcseq 12618 abscabs 13768 ⇝ cli 14009 𝑂(1)co1 14011 Σcsu 14210 Basecbs 15641 0_gc0g 15869 ℤRHomczrh 19612 ℤ/nℤczn 19615 μcmu 24538 DChrcdchr 24674

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1712 ax-4 1727 ax-5 1826 ax-6 1874 ax-7 1921 ax-8 1978 ax-9 1985 ax-10 2005 ax-11 2020 ax-12 2032 ax-13 2232 ax-ext 2589 ax-rep 4693 ax-sep 4703 ax-nul 4712 ax-pow 4764 ax-pr 4828 ax-un 6824 ax-inf2 8398 ax-cnex 9848 ax-resscn 9849 ax-1cn 9850 ax-icn 9851 ax-addcl 9852 ax-addrcl 9853 ax-mulcl 9854 ax-mulrcl 9855 ax-mulcom 9856 ax-addass 9857 ax-mulass 9858 ax-distr 9859 ax-i2m1 9860 ax-1ne0 9861 ax-1rid 9862 ax-rnegex 9863 ax-rrecex 9864 ax-cnre 9865 ax-pre-lttri 9866 ax-pre-lttrn 9867 ax-pre-ltadd 9868 ax-pre-mulgt0 9869 ax-pre-sup 9870 ax-addf 9871 ax-mulf 9872

This theorem depends on definitions: df-bi 195 df-or 383 df-an 384 df-3or 1031 df-3an 1032 df-tru 1477 df-fal 1480 df-ex 1695 df-nf 1700 df-sb 1867 df-eu 2461 df-mo 2462 df-clab 2596 df-cleq 2602 df-clel 2605 df-nfc 2739 df-ne 2781 df-nel 2782 df-ral 2900 df-rex 2901 df-reu 2902 df-rmo 2903 df-rab 2904 df-v 3174 df-sbc 3402 df-csb 3499 df-dif 3542 df-un 3544 df-in 3546 df-ss 3553 df-pss 3555 df-nul 3874 df-if 4036 df-pw 4109 df-sn 4125 df-pr 4127 df-tp 4129 df-op 4131 df-uni 4367 df-int 4405 df-iun 4451 df-iin 4452 df-disj 4548 df-br 4578 df-opab 4638 df-mpt 4639 df-tr 4675 df-eprel 4939 df-id 4943 df-po 4949 df-so 4950 df-fr 4987 df-se 4988 df-we 4989 df-xp 5034 df-rel 5035 df-cnv 5036 df-co 5037 df-dm 5038 df-rn 5039 df-res 5040 df-ima 5041 df-pred 5583 df-ord 5629 df-on 5630 df-lim 5631 df-suc 5632 df-iota 5754 df-fun 5792 df-fn 5793 df-f 5794 df-f1 5795 df-fo 5796 df-f1o 5797 df-fv 5798 df-isom 5799 df-riota 6489 df-ov 6530 df-oprab 6531 df-mpt2 6532 df-of 6772 df-rpss 6812 df-om 6935 df-1st 7036 df-2nd 7037 df-supp 7160 df-tpos 7216 df-wrecs 7271 df-recs 7332 df-rdg 7370 df-1o 7424 df-2o 7425 df-oadd 7428 df-omul 7429 df-er 7606 df-ec 7608 df-qs 7612 df-map 7723 df-pm 7724 df-ixp 7772 df-en 7819 df-dom 7820 df-sdom 7821 df-fin 7822 df-fsupp 8136 df-fi 8177 df-sup 8208 df-inf 8209 df-oi 8275 df-card 8625 df-acn 8628 df-cda 8850 df-pnf 9932 df-mnf 9933 df-xr 9934 df-ltxr 9935 df-le 9936 df-sub 10119 df-neg 10120 df-div 10534 df-nn 10868 df-2 10926 df-3 10927 df-4 10928 df-5 10929 df-6 10930 df-7 10931 df-8 10932 df-9 10933 df-n0 11140 df-z 11211 df-dec 11326 df-uz 11520 df-q 11621 df-rp 11665 df-xneg 11778 df-xadd 11779 df-xmul 11780 df-ioo 12006 df-ioc 12007 df-ico 12008 df-icc 12009 df-fz 12153 df-fzo 12290 df-fl 12410 df-mod 12486 df-seq 12619 df-exp 12678 df-fac 12878 df-bc 12907 df-hash 12935 df-word 13100 df-concat 13102 df-s1 13103 df-shft 13601 df-cj 13633 df-re 13634 df-im 13635 df-sqrt 13769 df-abs 13770 df-limsup 13996 df-clim 14013 df-rlim 14014 df-o1 14015 df-lo1 14016 df-sum 14211 df-ef 14583 df-e 14584 df-sin 14585 df-cos 14586 df-pi 14588 df-dvds 14768 df-gcd 15001 df-prm 15170 df-numer 15227 df-denom 15228 df-phi 15255 df-pc 15326 df-struct 15643 df-ndx 15644 df-slot 15645 df-base 15646 df-sets 15647 df-ress 15648 df-plusg 15727 df-mulr 15728 df-starv 15729 df-sca 15730 df-vsca 15731 df-ip 15732 df-tset 15733 df-ple 15734 df-ds 15737 df-unif 15738 df-hom 15739 df-cco 15740 df-rest 15852 df-topn 15853 df-0g 15871 df-gsum 15872 df-topgen 15873 df-pt 15874 df-prds 15877 df-xrs 15931 df-qtop 15936 df-imas 15937 df-qus 15938 df-xps 15939 df-mre 16015 df-mrc 16016 df-acs 16018 df-mgm 17011 df-sgrp 17053 df-mnd 17064 df-mhm 17104 df-submnd 17105 df-grp 17194 df-minusg 17195 df-sbg 17196 df-mulg 17310 df-subg 17360 df-nsg 17361 df-eqg 17362 df-ghm 17427 df-gim 17470 df-ga 17492 df-cntz 17519 df-oppg 17545 df-od 17717 df-gex 17718 df-pgp 17719 df-lsm 17820 df-pj1 17821 df-cmn 17964 df-abl 17965 df-cyg 18049 df-dprd 18163 df-dpj 18164 df-mgp 18259 df-ur 18271 df-ring 18318 df-cring 18319 df-oppr 18392 df-dvdsr 18410 df-unit 18411 df-invr 18441 df-dvr 18452 df-rnghom 18484 df-drng 18518 df-subrg 18547 df-lmod 18634 df-lss 18700 df-lsp 18739 df-sra 18939 df-rgmod 18940 df-lidl 18941 df-rsp 18942 df-2idl 18999 df-psmet 19505 df-xmet 19506 df-met 19507 df-bl 19508 df-mopn 19509 df-fbas 19510 df-fg 19511 df-cnfld 19514 df-zring 19584 df-zrh 19616 df-zn 19619 df-top 20463 df-bases 20464 df-topon 20465 df-topsp 20466 df-cld 20575 df-ntr 20576 df-cls 20577 df-nei 20654 df-lp 20692 df-perf 20693 df-cn 20783 df-cnp 20784 df-haus 20871 df-cmp 20942 df-tx 21117 df-hmeo 21310 df-fil 21402 df-fm 21494 df-flim 21495 df-flf 21496 df-xms 21876 df-ms 21877 df-tms 21878 df-cncf 22420 df-0p 23160 df-limc 23353 df-dv 23354 df-ply 23665 df-idp 23666 df-coe 23667 df-dgr 23668 df-quot 23767 df-log 24024 df-cxp 24025 df-em 24436 df-cht 24540 df-vma 24541 df-chp 24542 df-ppi 24543 df-mu 24544 df-dchr 24675

This theorem is referenced by: (None)

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