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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 13956 | . . . . . . 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 725 | . . . . . . . . 9 β’ (((π β§ π₯ β β+) β§ π β (1...(ββπ₯))) β π β π·) |
8 | elfzelz 13519 | . . . . . . . . . 10 β’ (π β (1...(ββπ₯)) β π β β€) | |
9 | 8 | adantl 481 | . . . . . . . . 9 β’ (((π β§ π₯ β β+) β§ π β (1...(ββπ₯))) β π β β€) |
10 | 2, 3, 4, 5, 7, 9 | dchrzrhcl 27152 | . . . . . . . 8 β’ (((π β§ π₯ β β+) β§ π β (1...(ββπ₯))) β (πβ(πΏβπ)) β β) |
11 | elfznn 13548 | . . . . . . . . . . . . 13 β’ (π β (1...(ββπ₯)) β π β β) | |
12 | 11 | adantl 481 | . . . . . . . . . . . 12 β’ (((π β§ π₯ β β+) β§ π β (1...(ββπ₯))) β π β β) |
13 | mucl 27047 | . . . . . . . . . . . 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 12682 | . . . . . . . . . 10 β’ (((π β§ π₯ β β+) β§ π β (1...(ββπ₯))) β (ΞΌβπ) β β) |
16 | 15, 12 | nndivred 12282 | . . . . . . . . 9 β’ (((π β§ π₯ β β+) β§ π β (1...(ββπ₯))) β ((ΞΌβπ) / π) β β) |
17 | 16 | recnd 11258 | . . . . . . . 8 β’ (((π β§ π₯ β β+) β§ π β (1...(ββπ₯))) β ((ΞΌβπ) / π) β β) |
18 | 10, 17 | mulcld 11250 | . . . . . . 7 β’ (((π β§ π₯ β β+) β§ π β (1...(ββπ₯))) β ((πβ(πΏβπ)) Β· ((ΞΌβπ) / π)) β β) |
19 | 1, 18 | fsumcl 15697 | . . . . . 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 15461 | . . . . . . . 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 480 | . . . . . 6 β’ ((π β§ π₯ β β+) β π β β) |
24 | 19, 23 | mulcld 11250 | . . . . 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 27428 | . . . . . 6 β’ (π β π β 0) |
32 | 31 | adantr 480 | . . . . 5 β’ ((π β§ π₯ β β+) β π β 0) |
33 | 24, 23, 32 | divrecd 12009 | . . . 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 12012 | . . . 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 2769 | . . 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 5242 | . 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 11999 | . . . 4 β’ (π β (1 / π) β β) |
38 | 37 | adantr 480 | . . 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 27401 | . . 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 12999 | . . . 4 β’ β+ β β | |
41 | o1const 15582 | . . . 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 586 | . . 3 β’ (π β (π₯ β β+ β¦ (1 / π)) β π(1)) |
43 | 24, 38, 39, 42 | o1mul2 15587 | . 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 2829 | 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 395 = wceq 1534 β wcel 2099 β wne 2935 βwral 3056 β wss 3944 class class class wbr 5142 β¦ cmpt 5225 βcfv 6542 (class class class)co 7414 βcc 11122 βcr 11123 0cc0 11124 1c1 11125 + caddc 11127 Β· cmul 11129 +βcpnf 11261 β€ cle 11265 β cmin 11460 / cdiv 11887 βcn 12228 β€cz 12574 β+crp 12992 [,)cico 13344 ...cfz 13502 βcfl 13773 seqcseq 13984 abscabs 15199 β cli 15446 π(1)co1 15448 Ξ£csu 15650 Basecbs 17165 0gc0g 17406 β€RHomczrh 21405 β€/nβ€czn 21408 ΞΌcmu 27001 DChrcdchr 27139 |
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 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203 ax-mulf 11204 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-disj 5108 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-rpss 7720 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-tpos 8223 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-oadd 8482 df-omul 8483 df-er 8716 df-ec 8718 df-qs 8722 df-map 8836 df-pm 8837 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-fi 9420 df-sup 9451 df-inf 9452 df-oi 9519 df-dju 9910 df-card 9948 df-acn 9951 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-xnn0 12561 df-z 12575 df-dec 12694 df-uz 12839 df-q 12949 df-rp 12993 df-xneg 13110 df-xadd 13111 df-xmul 13112 df-ioo 13346 df-ioc 13347 df-ico 13348 df-icc 13349 df-fz 13503 df-fzo 13646 df-fl 13775 df-mod 13853 df-seq 13985 df-exp 14045 df-fac 14251 df-bc 14280 df-hash 14308 df-word 14483 df-concat 14539 df-s1 14564 df-shft 15032 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-limsup 15433 df-clim 15450 df-rlim 15451 df-o1 15452 df-lo1 15453 df-sum 15651 df-ef 16029 df-e 16030 df-sin 16031 df-cos 16032 df-tan 16033 df-pi 16034 df-dvds 16217 df-gcd 16455 df-prm 16628 df-numer 16692 df-denom 16693 df-phi 16720 df-pc 16791 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17389 df-topn 17390 df-0g 17408 df-gsum 17409 df-topgen 17410 df-pt 17411 df-prds 17414 df-xrs 17469 df-qtop 17474 df-imas 17475 df-qus 17476 df-xps 17477 df-mre 17551 df-mrc 17552 df-acs 17554 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-mhm 18725 df-submnd 18726 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19008 df-subg 19062 df-nsg 19063 df-eqg 19064 df-ghm 19152 df-gim 19197 df-ga 19225 df-cntz 19252 df-oppg 19281 df-od 19467 df-gex 19468 df-pgp 19469 df-lsm 19575 df-pj1 19576 df-cmn 19721 df-abl 19722 df-cyg 19817 df-dprd 19936 df-dpj 19937 df-mgp 20059 df-rng 20077 df-ur 20106 df-ring 20159 df-cring 20160 df-oppr 20255 df-dvdsr 20278 df-unit 20279 df-invr 20309 df-dvr 20322 df-rhm 20393 df-subrng 20465 df-subrg 20490 df-drng 20608 df-lmod 20727 df-lss 20798 df-lsp 20838 df-sra 21040 df-rgmod 21041 df-lidl 21086 df-rsp 21087 df-2idl 21126 df-psmet 21251 df-xmet 21252 df-met 21253 df-bl 21254 df-mopn 21255 df-fbas 21256 df-fg 21257 df-cnfld 21260 df-zring 21353 df-zrh 21409 df-zn 21412 df-top 22770 df-topon 22787 df-topsp 22809 df-bases 22823 df-cld 22897 df-ntr 22898 df-cls 22899 df-nei 22976 df-lp 23014 df-perf 23015 df-cn 23105 df-cnp 23106 df-haus 23193 df-cmp 23265 df-tx 23440 df-hmeo 23633 df-fil 23724 df-fm 23816 df-flim 23817 df-flf 23818 df-xms 24200 df-ms 24201 df-tms 24202 df-cncf 24772 df-0p 25573 df-limc 25769 df-dv 25770 df-ply 26096 df-idp 26097 df-coe 26098 df-dgr 26099 df-quot 26200 df-ulm 26287 df-log 26464 df-cxp 26465 df-atan 26773 df-em 26899 df-cht 27003 df-vma 27004 df-chp 27005 df-ppi 27006 df-mu 27007 df-dchr 27140 |
This theorem is referenced by: dchrmusum 27431 |
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