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Mirrors > Home > MPE Home > Th. List > dchrmusum | Structured version Visualization version GIF version |
Description: The sum of the MΓΆbius function multiplied by a non-principal Dirichlet character, divided by π, is bounded. Equation 9.4.16 of [Shapiro], p. 379. (Contributed by Mario Carneiro, 12-May-2016.) |
Ref | Expression |
---|---|
rpvmasum.z | β’ π = (β€/nβ€βπ) |
rpvmasum.l | β’ πΏ = (β€RHomβπ) |
rpvmasum.a | β’ (π β π β β) |
dchrmusum.g | β’ πΊ = (DChrβπ) |
dchrmusum.d | β’ π· = (BaseβπΊ) |
dchrmusum.1 | β’ 1 = (0gβπΊ) |
dchrmusum.b | β’ (π β π β π·) |
dchrmusum.n1 | β’ (π β π β 1 ) |
Ref | Expression |
---|---|
dchrmusum | β’ (π β (π₯ β β+ β¦ Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π))) β π(1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpvmasum.z | . . 3 β’ π = (β€/nβ€βπ) | |
2 | rpvmasum.l | . . 3 β’ πΏ = (β€RHomβπ) | |
3 | rpvmasum.a | . . 3 β’ (π β π β β) | |
4 | dchrmusum.g | . . 3 β’ πΊ = (DChrβπ) | |
5 | dchrmusum.d | . . 3 β’ π· = (BaseβπΊ) | |
6 | dchrmusum.1 | . . 3 β’ 1 = (0gβπΊ) | |
7 | dchrmusum.b | . . 3 β’ (π β π β π·) | |
8 | dchrmusum.n1 | . . 3 β’ (π β π β 1 ) | |
9 | eqid 2732 | . . 3 β’ (π β β β¦ ((πβ(πΏβπ)) / π)) = (π β β β¦ ((πβ(πΏβπ)) / π)) | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | dchrmusumlema 26993 | . 2 β’ (π β βπ‘βπ β (0[,)+β)(seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π))) β π‘ β§ βπ¦ β (1[,)+β)(absβ((seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π)))β(ββπ¦)) β π‘)) β€ (π / π¦))) |
11 | 3 | adantr 481 | . . . . 5 β’ ((π β§ (π β (0[,)+β) β§ (seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π))) β π‘ β§ βπ¦ β (1[,)+β)(absβ((seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π)))β(ββπ¦)) β π‘)) β€ (π / π¦)))) β π β β) |
12 | 7 | adantr 481 | . . . . 5 β’ ((π β§ (π β (0[,)+β) β§ (seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π))) β π‘ β§ βπ¦ β (1[,)+β)(absβ((seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π)))β(ββπ¦)) β π‘)) β€ (π / π¦)))) β π β π·) |
13 | 8 | adantr 481 | . . . . 5 β’ ((π β§ (π β (0[,)+β) β§ (seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π))) β π‘ β§ βπ¦ β (1[,)+β)(absβ((seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π)))β(ββπ¦)) β π‘)) β€ (π / π¦)))) β π β 1 ) |
14 | simprl 769 | . . . . 5 β’ ((π β§ (π β (0[,)+β) β§ (seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π))) β π‘ β§ βπ¦ β (1[,)+β)(absβ((seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π)))β(ββπ¦)) β π‘)) β€ (π / π¦)))) β π β (0[,)+β)) | |
15 | simprrl 779 | . . . . 5 β’ ((π β§ (π β (0[,)+β) β§ (seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π))) β π‘ β§ βπ¦ β (1[,)+β)(absβ((seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π)))β(ββπ¦)) β π‘)) β€ (π / π¦)))) β seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π))) β π‘) | |
16 | simprrr 780 | . . . . 5 β’ ((π β§ (π β (0[,)+β) β§ (seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π))) β π‘ β§ βπ¦ β (1[,)+β)(absβ((seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π)))β(ββπ¦)) β π‘)) β€ (π / π¦)))) β βπ¦ β (1[,)+β)(absβ((seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π)))β(ββπ¦)) β π‘)) β€ (π / π¦)) | |
17 | 1, 2, 11, 4, 5, 6, 12, 13, 9, 14, 15, 16 | dchrmusumlem 27022 | . . . 4 β’ ((π β§ (π β (0[,)+β) β§ (seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π))) β π‘ β§ βπ¦ β (1[,)+β)(absβ((seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π)))β(ββπ¦)) β π‘)) β€ (π / π¦)))) β (π₯ β β+ β¦ Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π))) β π(1)) |
18 | 17 | rexlimdvaa 3156 | . . 3 β’ (π β (βπ β (0[,)+β)(seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π))) β π‘ β§ βπ¦ β (1[,)+β)(absβ((seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π)))β(ββπ¦)) β π‘)) β€ (π / π¦)) β (π₯ β β+ β¦ Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π))) β π(1))) |
19 | 18 | exlimdv 1936 | . 2 β’ (π β (βπ‘βπ β (0[,)+β)(seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π))) β π‘ β§ βπ¦ β (1[,)+β)(absβ((seq1( + , (π β β β¦ ((πβ(πΏβπ)) / π)))β(ββπ¦)) β π‘)) β€ (π / π¦)) β (π₯ β β+ β¦ Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π))) β π(1))) |
20 | 10, 19 | mpd 15 | 1 β’ (π β (π₯ β β+ β¦ Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π))) β π(1)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 βwex 1781 β wcel 2106 β wne 2940 βwral 3061 βwrex 3070 class class class wbr 5148 β¦ cmpt 5231 βcfv 6543 (class class class)co 7408 0cc0 11109 1c1 11110 + caddc 11112 Β· cmul 11114 +βcpnf 11244 β€ cle 11248 β cmin 11443 / cdiv 11870 βcn 12211 β+crp 12973 [,)cico 13325 ...cfz 13483 βcfl 13754 seqcseq 13965 abscabs 15180 β cli 15427 π(1)co1 15429 Ξ£csu 15631 Basecbs 17143 0gc0g 17384 β€RHomczrh 21048 β€/nβ€czn 21051 ΞΌcmu 26596 DChrcdchr 26732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 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 7364 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7669 df-rpss 7712 df-om 7855 df-1st 7974 df-2nd 7975 df-supp 8146 df-tpos 8210 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-rdg 8409 df-1o 8465 df-2o 8466 df-oadd 8469 df-omul 8470 df-er 8702 df-ec 8704 df-qs 8708 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-acn 9936 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-xnn0 12544 df-z 12558 df-dec 12677 df-uz 12822 df-q 12932 df-rp 12974 df-xneg 13091 df-xadd 13092 df-xmul 13093 df-ioo 13327 df-ioc 13328 df-ico 13329 df-icc 13330 df-fz 13484 df-fzo 13627 df-fl 13756 df-mod 13834 df-seq 13966 df-exp 14027 df-fac 14233 df-bc 14262 df-hash 14290 df-word 14464 df-concat 14520 df-s1 14545 df-shft 15013 df-cj 15045 df-re 15046 df-im 15047 df-sqrt 15181 df-abs 15182 df-limsup 15414 df-clim 15431 df-rlim 15432 df-o1 15433 df-lo1 15434 df-sum 15632 df-ef 16010 df-e 16011 df-sin 16012 df-cos 16013 df-tan 16014 df-pi 16015 df-dvds 16197 df-gcd 16435 df-prm 16608 df-numer 16670 df-denom 16671 df-phi 16698 df-pc 16769 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-ress 17173 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17367 df-topn 17368 df-0g 17386 df-gsum 17387 df-topgen 17388 df-pt 17389 df-prds 17392 df-xrs 17447 df-qtop 17452 df-imas 17453 df-qus 17454 df-xps 17455 df-mre 17529 df-mrc 17530 df-acs 17532 df-mgm 18560 df-sgrp 18609 df-mnd 18625 df-mhm 18670 df-submnd 18671 df-grp 18821 df-minusg 18822 df-sbg 18823 df-mulg 18950 df-subg 19002 df-nsg 19003 df-eqg 19004 df-ghm 19089 df-gim 19132 df-ga 19153 df-cntz 19180 df-oppg 19209 df-od 19395 df-gex 19396 df-pgp 19397 df-lsm 19503 df-pj1 19504 df-cmn 19649 df-abl 19650 df-cyg 19745 df-dprd 19864 df-dpj 19865 df-mgp 19987 df-ur 20004 df-ring 20057 df-cring 20058 df-oppr 20149 df-dvdsr 20170 df-unit 20171 df-invr 20201 df-dvr 20214 df-rnghom 20250 df-subrg 20316 df-drng 20358 df-lmod 20472 df-lss 20542 df-lsp 20582 df-sra 20784 df-rgmod 20785 df-lidl 20786 df-rsp 20787 df-2idl 20856 df-psmet 20935 df-xmet 20936 df-met 20937 df-bl 20938 df-mopn 20939 df-fbas 20940 df-fg 20941 df-cnfld 20944 df-zring 21017 df-zrh 21052 df-zn 21055 df-top 22395 df-topon 22412 df-topsp 22434 df-bases 22448 df-cld 22522 df-ntr 22523 df-cls 22524 df-nei 22601 df-lp 22639 df-perf 22640 df-cn 22730 df-cnp 22731 df-haus 22818 df-cmp 22890 df-tx 23065 df-hmeo 23258 df-fil 23349 df-fm 23441 df-flim 23442 df-flf 23443 df-xms 23825 df-ms 23826 df-tms 23827 df-cncf 24393 df-0p 25186 df-limc 25382 df-dv 25383 df-ply 25701 df-idp 25702 df-coe 25703 df-dgr 25704 df-quot 25803 df-ulm 25888 df-log 26064 df-cxp 26065 df-atan 26369 df-em 26494 df-cht 26598 df-vma 26599 df-chp 26600 df-ppi 26601 df-mu 26602 df-dchr 26733 |
This theorem is referenced by: (None) |
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