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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 27003 | . 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 27032 | . . . 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 7411 0cc0 11112 1c1 11113 + caddc 11115 Β· cmul 11117 +βcpnf 11247 β€ cle 11251 β cmin 11446 / cdiv 11873 βcn 12214 β+crp 12976 [,)cico 13328 ...cfz 13486 βcfl 13757 seqcseq 13968 abscabs 15183 β cli 15430 π(1)co1 15432 Ξ£csu 15634 Basecbs 17146 0gc0g 17387 β€RHomczrh 21055 β€/nβ€czn 21058 ΞΌcmu 26606 DChrcdchr 26742 |
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 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192 |
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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-rpss 7715 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-omul 8473 df-er 8705 df-ec 8707 df-qs 8711 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-dju 9898 df-card 9936 df-acn 9939 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-xnn0 12547 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-ioo 13330 df-ioc 13331 df-ico 13332 df-icc 13333 df-fz 13487 df-fzo 13630 df-fl 13759 df-mod 13837 df-seq 13969 df-exp 14030 df-fac 14236 df-bc 14265 df-hash 14293 df-word 14467 df-concat 14523 df-s1 14548 df-shft 15016 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-limsup 15417 df-clim 15434 df-rlim 15435 df-o1 15436 df-lo1 15437 df-sum 15635 df-ef 16013 df-e 16014 df-sin 16015 df-cos 16016 df-tan 16017 df-pi 16018 df-dvds 16200 df-gcd 16438 df-prm 16611 df-numer 16673 df-denom 16674 df-phi 16701 df-pc 16772 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-hom 17223 df-cco 17224 df-rest 17370 df-topn 17371 df-0g 17389 df-gsum 17390 df-topgen 17391 df-pt 17392 df-prds 17395 df-xrs 17450 df-qtop 17455 df-imas 17456 df-qus 17457 df-xps 17458 df-mre 17532 df-mrc 17533 df-acs 17535 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-mhm 18673 df-submnd 18674 df-grp 18824 df-minusg 18825 df-sbg 18826 df-mulg 18953 df-subg 19005 df-nsg 19006 df-eqg 19007 df-ghm 19092 df-gim 19135 df-ga 19156 df-cntz 19183 df-oppg 19212 df-od 19398 df-gex 19399 df-pgp 19400 df-lsm 19506 df-pj1 19507 df-cmn 19652 df-abl 19653 df-cyg 19748 df-dprd 19867 df-dpj 19868 df-mgp 19990 df-ur 20007 df-ring 20060 df-cring 20061 df-oppr 20154 df-dvdsr 20175 df-unit 20176 df-invr 20206 df-dvr 20219 df-rnghom 20255 df-subrg 20321 df-drng 20363 df-lmod 20477 df-lss 20548 df-lsp 20588 df-sra 20791 df-rgmod 20792 df-lidl 20793 df-rsp 20794 df-2idl 20863 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-fbas 20947 df-fg 20948 df-cnfld 20951 df-zring 21024 df-zrh 21059 df-zn 21062 df-top 22403 df-topon 22420 df-topsp 22442 df-bases 22456 df-cld 22530 df-ntr 22531 df-cls 22532 df-nei 22609 df-lp 22647 df-perf 22648 df-cn 22738 df-cnp 22739 df-haus 22826 df-cmp 22898 df-tx 23073 df-hmeo 23266 df-fil 23357 df-fm 23449 df-flim 23450 df-flf 23451 df-xms 23833 df-ms 23834 df-tms 23835 df-cncf 24401 df-0p 25194 df-limc 25390 df-dv 25391 df-ply 25709 df-idp 25710 df-coe 25711 df-dgr 25712 df-quot 25811 df-ulm 25896 df-log 26072 df-cxp 26073 df-atan 26379 df-em 26504 df-cht 26608 df-vma 26609 df-chp 26610 df-ppi 26611 df-mu 26612 df-dchr 26743 |
This theorem is referenced by: (None) |
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