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Mirrors > Home > MPE Home > Th. List > dchrvmasum | Structured version Visualization version GIF version |
Description: The sum of the von Mangoldt function multiplied by a non-principal Dirichlet character, divided by π, is bounded. Equation 9.4.8 of [Shapiro], p. 376. (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 |
---|---|
dchrvmasum | β’ (π β (π₯ β β+ β¦ Ξ£π β (1...(ββπ₯))((πβ(πΏβπ)) Β· ((Ξβπ) / 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 2725 | . . 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 27442 | . 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 479 | . . . . 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 479 | . . . . 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 479 | . . . . 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 | dchrvmasumlem 27472 | . . . 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· ((Ξβπ) / π))) β π(1)) |
18 | 17 | rexlimdvaa 3146 | . . 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· ((Ξβπ) / π))) β π(1))) |
19 | 18 | exlimdv 1928 | . 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· ((Ξβπ) / 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· ((Ξβπ) / π))) β π(1)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 βwex 1773 β wcel 2098 β wne 2930 βwral 3051 βwrex 3060 class class class wbr 5143 β¦ cmpt 5226 βcfv 6542 (class class class)co 7415 0cc0 11136 1c1 11137 + caddc 11139 Β· cmul 11141 +βcpnf 11273 β€ cle 11277 β cmin 11472 / cdiv 11899 βcn 12240 β+crp 13004 [,)cico 13356 ...cfz 13514 βcfl 13785 seqcseq 13996 abscabs 15211 β cli 15458 π(1)co1 15460 Ξ£csu 15662 Basecbs 17177 0gc0g 17418 β€RHomczrh 21427 β€/nβ€czn 21430 Ξcvma 27040 DChrcdchr 27181 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 ax-addf 11215 ax-mulf 11216 |
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 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 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 5109 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7681 df-rpss 7725 df-om 7868 df-1st 7989 df-2nd 7990 df-supp 8162 df-tpos 8228 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-2o 8484 df-oadd 8487 df-omul 8488 df-er 8721 df-ec 8723 df-qs 8727 df-map 8843 df-pm 8844 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-fsupp 9384 df-fi 9432 df-sup 9463 df-inf 9464 df-oi 9531 df-dju 9922 df-card 9960 df-acn 9963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-xnn0 12573 df-z 12587 df-dec 12706 df-uz 12851 df-q 12961 df-rp 13005 df-xneg 13122 df-xadd 13123 df-xmul 13124 df-ioo 13358 df-ioc 13359 df-ico 13360 df-icc 13361 df-fz 13515 df-fzo 13658 df-fl 13787 df-mod 13865 df-seq 13997 df-exp 14057 df-fac 14263 df-bc 14292 df-hash 14320 df-word 14495 df-concat 14551 df-s1 14576 df-shft 15044 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-limsup 15445 df-clim 15462 df-rlim 15463 df-o1 15464 df-lo1 15465 df-sum 15663 df-ef 16041 df-e 16042 df-sin 16043 df-cos 16044 df-tan 16045 df-pi 16046 df-dvds 16229 df-gcd 16467 df-prm 16640 df-numer 16704 df-denom 16705 df-phi 16732 df-pc 16803 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-hom 17254 df-cco 17255 df-rest 17401 df-topn 17402 df-0g 17420 df-gsum 17421 df-topgen 17422 df-pt 17423 df-prds 17426 df-xrs 17481 df-qtop 17486 df-imas 17487 df-qus 17488 df-xps 17489 df-mre 17563 df-mrc 17564 df-acs 17566 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-submnd 18738 df-grp 18895 df-minusg 18896 df-sbg 18897 df-mulg 19026 df-subg 19080 df-nsg 19081 df-eqg 19082 df-ghm 19170 df-gim 19215 df-ga 19243 df-cntz 19270 df-oppg 19299 df-od 19485 df-gex 19486 df-pgp 19487 df-lsm 19593 df-pj1 19594 df-cmn 19739 df-abl 19740 df-cyg 19835 df-dprd 19954 df-dpj 19955 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-cring 20178 df-oppr 20275 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-dvr 20342 df-rhm 20413 df-subrng 20485 df-subrg 20510 df-drng 20628 df-lmod 20747 df-lss 20818 df-lsp 20858 df-sra 21060 df-rgmod 21061 df-lidl 21106 df-rsp 21107 df-2idl 21146 df-psmet 21273 df-xmet 21274 df-met 21275 df-bl 21276 df-mopn 21277 df-fbas 21278 df-fg 21279 df-cnfld 21282 df-zring 21375 df-zrh 21431 df-zn 21434 df-top 22812 df-topon 22829 df-topsp 22851 df-bases 22865 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-lp 23056 df-perf 23057 df-cn 23147 df-cnp 23148 df-haus 23235 df-cmp 23307 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-xms 24242 df-ms 24243 df-tms 24244 df-cncf 24814 df-0p 25615 df-limc 25811 df-dv 25812 df-ply 26138 df-idp 26139 df-coe 26140 df-dgr 26141 df-quot 26242 df-ulm 26329 df-log 26506 df-cxp 26507 df-atan 26815 df-em 26941 df-cht 27045 df-vma 27046 df-chp 27047 df-ppi 27048 df-mu 27049 df-dchr 27182 |
This theorem is referenced by: (None) |
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