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Mirrors > Home > MPE Home > Th. List > dchrmusumlem | 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 ) |
dchrmusum.f | β’ πΉ = (π β β β¦ ((πβ(πΏβπ)) / π)) |
dchrmusum.c | β’ (π β πΆ β (0[,)+β)) |
dchrmusum.t | β’ (π β seq1( + , πΉ) β π) |
dchrmusum.2 | β’ (π β βπ¦ β (1[,)+β)(absβ((seq1( + , πΉ)β(ββπ¦)) β π)) β€ (πΆ / π¦)) |
Ref | Expression |
---|---|
dchrmusumlem | 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 | fzfid 13965 | . . . . . . 7 β’ ((π β§ π₯ β β+) β (1...(ββπ₯)) β Fin) | |
2 | dchrmusum.g | . . . . . . . . 9 β’ πΊ = (DChrβπ) | |
3 | rpvmasum.z | . . . . . . . . 9 β’ π = (β€/nβ€βπ) | |
4 | dchrmusum.d | . . . . . . . . 9 β’ π· = (BaseβπΊ) | |
5 | rpvmasum.l | . . . . . . . . 9 β’ πΏ = (β€RHomβπ) | |
6 | dchrmusum.b | . . . . . . . . . 10 β’ (π β π β π·) | |
7 | 6 | ad2antrr 724 | . . . . . . . . 9 β’ (((π β§ π₯ β β+) β§ π β (1...(ββπ₯))) β π β π·) |
8 | elfzelz 13528 | . . . . . . . . . 10 β’ (π β (1...(ββπ₯)) β π β β€) | |
9 | 8 | adantl 480 | . . . . . . . . 9 β’ (((π β§ π₯ β β+) β§ π β (1...(ββπ₯))) β π β β€) |
10 | 2, 3, 4, 5, 7, 9 | dchrzrhcl 27191 | . . . . . . . 8 β’ (((π β§ π₯ β β+) β§ π β (1...(ββπ₯))) β (πβ(πΏβπ)) β β) |
11 | elfznn 13557 | . . . . . . . . . . . . 13 β’ (π β (1...(ββπ₯)) β π β β) | |
12 | 11 | adantl 480 | . . . . . . . . . . . 12 β’ (((π β§ π₯ β β+) β§ π β (1...(ββπ₯))) β π β β) |
13 | mucl 27086 | . . . . . . . . . . . 12 β’ (π β β β (ΞΌβπ) β β€) | |
14 | 12, 13 | syl 17 | . . . . . . . . . . 11 β’ (((π β§ π₯ β β+) β§ π β (1...(ββπ₯))) β (ΞΌβπ) β β€) |
15 | 14 | zred 12691 | . . . . . . . . . 10 β’ (((π β§ π₯ β β+) β§ π β (1...(ββπ₯))) β (ΞΌβπ) β β) |
16 | 15, 12 | nndivred 12291 | . . . . . . . . 9 β’ (((π β§ π₯ β β+) β§ π β (1...(ββπ₯))) β ((ΞΌβπ) / π) β β) |
17 | 16 | recnd 11267 | . . . . . . . 8 β’ (((π β§ π₯ β β+) β§ π β (1...(ββπ₯))) β ((ΞΌβπ) / π) β β) |
18 | 10, 17 | mulcld 11259 | . . . . . . 7 β’ (((π β§ π₯ β β+) β§ π β (1...(ββπ₯))) β ((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) β β) |
19 | 1, 18 | fsumcl 15706 | . . . . . 6 β’ ((π β§ π₯ β β+) β Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) β β) |
20 | dchrmusum.t | . . . . . . . 8 β’ (π β seq1( + , πΉ) β π) | |
21 | climcl 15470 | . . . . . . . 8 β’ (seq1( + , πΉ) β π β π β β) | |
22 | 20, 21 | syl 17 | . . . . . . 7 β’ (π β π β β) |
23 | 22 | adantr 479 | . . . . . 6 β’ ((π β§ π₯ β β+) β π β β) |
24 | 19, 23 | mulcld 11259 | . . . . 5 β’ ((π β§ π₯ β β+) β (Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· π) β β) |
25 | rpvmasum.a | . . . . . . 7 β’ (π β π β β) | |
26 | dchrmusum.1 | . . . . . . 7 β’ 1 = (0gβπΊ) | |
27 | dchrmusum.n1 | . . . . . . 7 β’ (π β π β 1 ) | |
28 | dchrmusum.f | . . . . . . 7 β’ πΉ = (π β β β¦ ((πβ(πΏβπ)) / π)) | |
29 | dchrmusum.c | . . . . . . 7 β’ (π β πΆ β (0[,)+β)) | |
30 | dchrmusum.2 | . . . . . . 7 β’ (π β βπ¦ β (1[,)+β)(absβ((seq1( + , πΉ)β(ββπ¦)) β π)) β€ (πΆ / π¦)) | |
31 | 3, 5, 25, 2, 4, 26, 6, 27, 28, 29, 20, 30 | dchrisumn0 27467 | . . . . . 6 β’ (π β π β 0) |
32 | 31 | adantr 479 | . . . . 5 β’ ((π β§ π₯ β β+) β π β 0) |
33 | 24, 23, 32 | divrecd 12018 | . . . 4 β’ ((π β§ π₯ β β+) β ((Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· π) / π) = ((Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· π) Β· (1 / π))) |
34 | 19, 23, 32 | divcan4d 12021 | . . . 4 β’ ((π β§ π₯ β β+) β ((Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· π) / π) = Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π))) |
35 | 33, 34 | eqtr3d 2767 | . . 3 β’ ((π β§ π₯ β β+) β ((Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· π) Β· (1 / π)) = Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π))) |
36 | 35 | mpteq2dva 5244 | . 2 β’ (π β (π₯ β β+ β¦ ((Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· π) Β· (1 / π))) = (π₯ β β+ β¦ Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)))) |
37 | 22, 31 | reccld 12008 | . . . 4 β’ (π β (1 / π) β β) |
38 | 37 | adantr 479 | . . 3 β’ ((π β§ π₯ β β+) β (1 / π) β β) |
39 | 3, 5, 25, 2, 4, 26, 6, 27, 28, 29, 20, 30 | dchrmusum2 27440 | . . 3 β’ (π β (π₯ β β+ β¦ (Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· π)) β π(1)) |
40 | rpssre 13008 | . . . 4 β’ β+ β β | |
41 | o1const 15591 | . . . 4 β’ ((β+ β β β§ (1 / π) β β) β (π₯ β β+ β¦ (1 / π)) β π(1)) | |
42 | 40, 37, 41 | sylancr 585 | . . 3 β’ (π β (π₯ β β+ β¦ (1 / π)) β π(1)) |
43 | 24, 38, 39, 42 | o1mul2 15596 | . 2 β’ (π β (π₯ β β+ β¦ ((Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) Β· π) Β· (1 / π))) β π(1)) |
44 | 36, 43 | eqeltrrd 2826 | 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 394 = wceq 1533 β wcel 2098 β wne 2930 βwral 3051 β wss 3941 class class class wbr 5144 β¦ cmpt 5227 βcfv 6543 (class class class)co 7413 βcc 11131 βcr 11132 0cc0 11133 1c1 11134 + caddc 11136 Β· cmul 11138 +βcpnf 11270 β€ cle 11274 β cmin 11469 / cdiv 11896 βcn 12237 β€cz 12583 β+crp 13001 [,)cico 13353 ...cfz 13511 βcfl 13782 seqcseq 13993 abscabs 15208 β cli 15455 π(1)co1 15457 Ξ£csu 15659 Basecbs 17174 0gc0g 17415 β€RHomczrh 21424 β€/nβ€czn 21427 ΞΌcmu 27040 DChrcdchr 27178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 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 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 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 5110 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 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 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 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 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-rpss 7723 df-om 7866 df-1st 7987 df-2nd 7988 df-supp 8159 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-omul 8485 df-er 8718 df-ec 8720 df-qs 8724 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 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 18892 df-minusg 18893 df-sbg 18894 df-mulg 19023 df-subg 19077 df-nsg 19078 df-eqg 19079 df-ghm 19167 df-gim 19212 df-ga 19240 df-cntz 19267 df-oppg 19296 df-od 19482 df-gex 19483 df-pgp 19484 df-lsm 19590 df-pj1 19591 df-cmn 19736 df-abl 19737 df-cyg 19832 df-dprd 19951 df-dpj 19952 df-mgp 20074 df-rng 20092 df-ur 20121 df-ring 20174 df-cring 20175 df-oppr 20272 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-dvr 20339 df-rhm 20410 df-subrng 20482 df-subrg 20507 df-drng 20625 df-lmod 20744 df-lss 20815 df-lsp 20855 df-sra 21057 df-rgmod 21058 df-lidl 21103 df-rsp 21104 df-2idl 21143 df-psmet 21270 df-xmet 21271 df-met 21272 df-bl 21273 df-mopn 21274 df-fbas 21275 df-fg 21276 df-cnfld 21279 df-zring 21372 df-zrh 21428 df-zn 21431 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-cld 22936 df-ntr 22937 df-cls 22938 df-nei 23015 df-lp 23053 df-perf 23054 df-cn 23144 df-cnp 23145 df-haus 23232 df-cmp 23304 df-tx 23479 df-hmeo 23672 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-xms 24239 df-ms 24240 df-tms 24241 df-cncf 24811 df-0p 25612 df-limc 25808 df-dv 25809 df-ply 26135 df-idp 26136 df-coe 26137 df-dgr 26138 df-quot 26239 df-ulm 26326 df-log 26503 df-cxp 26504 df-atan 26812 df-em 26938 df-cht 27042 df-vma 27043 df-chp 27044 df-ppi 27045 df-mu 27046 df-dchr 27179 |
This theorem is referenced by: dchrmusum 27470 |
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