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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 2731 | . . 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 27233 | . 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 480 | . . . . 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 480 | . . . . 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 480 | . . . . 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 768 | . . . . 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 778 | . . . . 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 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 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 27262 | . . . 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 3155 | . . 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 1935 | . 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 395 = wceq 1540 βwex 1780 β wcel 2105 β wne 2939 βwral 3060 βwrex 3069 class class class wbr 5148 β¦ cmpt 5231 βcfv 6543 (class class class)co 7412 0cc0 11114 1c1 11115 + caddc 11117 Β· cmul 11119 +βcpnf 11250 β€ cle 11254 β cmin 11449 / cdiv 11876 βcn 12217 β+crp 12979 [,)cico 13331 ...cfz 13489 βcfl 13760 seqcseq 13971 abscabs 15186 β cli 15433 π(1)co1 15435 Ξ£csu 15637 Basecbs 17149 0gc0g 17390 β€RHomczrh 21269 β€/nβ€czn 21272 ΞΌcmu 26836 DChrcdchr 26972 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 ax-mulf 11194 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 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 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-rpss 7717 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-tpos 8215 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-oadd 8474 df-omul 8475 df-er 8707 df-ec 8709 df-qs 8713 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9509 df-dju 9900 df-card 9938 df-acn 9941 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-xnn0 12550 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ioc 13334 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-mod 13840 df-seq 13972 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-word 14470 df-concat 14526 df-s1 14551 df-shft 15019 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-limsup 15420 df-clim 15437 df-rlim 15438 df-o1 15439 df-lo1 15440 df-sum 15638 df-ef 16016 df-e 16017 df-sin 16018 df-cos 16019 df-tan 16020 df-pi 16021 df-dvds 16203 df-gcd 16441 df-prm 16614 df-numer 16676 df-denom 16677 df-phi 16704 df-pc 16775 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-qus 17460 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-mulg 18988 df-subg 19040 df-nsg 19041 df-eqg 19042 df-ghm 19129 df-gim 19174 df-ga 19196 df-cntz 19223 df-oppg 19252 df-od 19438 df-gex 19439 df-pgp 19440 df-lsm 19546 df-pj1 19547 df-cmn 19692 df-abl 19693 df-cyg 19788 df-dprd 19907 df-dpj 19908 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-rhm 20364 df-subrng 20435 df-subrg 20460 df-drng 20503 df-lmod 20617 df-lss 20688 df-lsp 20728 df-sra 20931 df-rgmod 20932 df-lidl 20933 df-rsp 20934 df-2idl 21007 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-fbas 21142 df-fg 21143 df-cnfld 21146 df-zring 21219 df-zrh 21273 df-zn 21276 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-nei 22823 df-lp 22861 df-perf 22862 df-cn 22952 df-cnp 22953 df-haus 23040 df-cmp 23112 df-tx 23287 df-hmeo 23480 df-fil 23571 df-fm 23663 df-flim 23664 df-flf 23665 df-xms 24047 df-ms 24048 df-tms 24049 df-cncf 24619 df-0p 25420 df-limc 25616 df-dv 25617 df-ply 25938 df-idp 25939 df-coe 25940 df-dgr 25941 df-quot 26041 df-ulm 26126 df-log 26302 df-cxp 26303 df-atan 26609 df-em 26734 df-cht 26838 df-vma 26839 df-chp 26840 df-ppi 26841 df-mu 26842 df-dchr 26973 |
This theorem is referenced by: (None) |
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