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Mirrors > Home > MPE Home > Th. List > rpvmasum | Structured version Visualization version GIF version |
Description: The sum of the von Mangoldt function over those integers πβ‘π΄ (mod π) is asymptotic to logπ₯ / Ο(π₯) + π(1). Equation 9.4.3 of [Shapiro], p. 375. (Contributed by Mario Carneiro, 2-May-2016.) (Proof shortened by Mario Carneiro, 26-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) |
rpvmasum.u | β’ π = (Unitβπ) |
rpvmasum.b | β’ (π β π΄ β π) |
rpvmasum.t | β’ π = (β‘πΏ β {π΄}) |
Ref | Expression |
---|---|
rpvmasum | β’ (π β (π₯ β β+ β¦ (((Οβπ) Β· Ξ£π β ((1...(ββπ₯)) β© π)((Ξβπ) / π)) β (logβπ₯))) β π(1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpvmasum.z | . . . . . . . . . . . . . 14 β’ π = (β€/nβ€βπ) | |
2 | rpvmasum.l | . . . . . . . . . . . . . 14 β’ πΏ = (β€RHomβπ) | |
3 | rpvmasum.a | . . . . . . . . . . . . . . 15 β’ (π β π β β) | |
4 | 3 | adantr 480 | . . . . . . . . . . . . . 14 β’ ((π β§ π β {π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0}) β π β β) |
5 | eqid 2726 | . . . . . . . . . . . . . 14 β’ (DChrβπ) = (DChrβπ) | |
6 | eqid 2726 | . . . . . . . . . . . . . 14 β’ (Baseβ(DChrβπ)) = (Baseβ(DChrβπ)) | |
7 | eqid 2726 | . . . . . . . . . . . . . 14 β’ (0gβ(DChrβπ)) = (0gβ(DChrβπ)) | |
8 | 2fveq3 6890 | . . . . . . . . . . . . . . . . . 18 β’ (π = π β (π¦β(πΏβπ)) = (π¦β(πΏβπ))) | |
9 | id 22 | . . . . . . . . . . . . . . . . . 18 β’ (π = π β π = π) | |
10 | 8, 9 | oveq12d 7423 | . . . . . . . . . . . . . . . . 17 β’ (π = π β ((π¦β(πΏβπ)) / π) = ((π¦β(πΏβπ)) / π)) |
11 | 10 | cbvsumv 15648 | . . . . . . . . . . . . . . . 16 β’ Ξ£π β β ((π¦β(πΏβπ)) / π) = Ξ£π β β ((π¦β(πΏβπ)) / π) |
12 | 11 | eqeq1i 2731 | . . . . . . . . . . . . . . 15 β’ (Ξ£π β β ((π¦β(πΏβπ)) / π) = 0 β Ξ£π β β ((π¦β(πΏβπ)) / π) = 0) |
13 | 12 | rabbii 3432 | . . . . . . . . . . . . . 14 β’ {π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0} = {π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0} |
14 | simpr 484 | . . . . . . . . . . . . . 14 β’ ((π β§ π β {π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0}) β π β {π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0}) | |
15 | 1, 2, 4, 5, 6, 7, 13, 14 | dchrisum0 27408 | . . . . . . . . . . . . 13 β’ Β¬ (π β§ π β {π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0}) |
16 | 15 | imnani 400 | . . . . . . . . . . . 12 β’ (π β Β¬ π β {π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0}) |
17 | 16 | eq0rdv 4399 | . . . . . . . . . . 11 β’ (π β {π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0} = β ) |
18 | 17 | fveq2d 6889 | . . . . . . . . . 10 β’ (π β (β―β{π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0}) = (β―ββ )) |
19 | hash0 14332 | . . . . . . . . . 10 β’ (β―ββ ) = 0 | |
20 | 18, 19 | eqtrdi 2782 | . . . . . . . . 9 β’ (π β (β―β{π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0}) = 0) |
21 | 20 | oveq2d 7421 | . . . . . . . 8 β’ (π β (1 β (β―β{π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0})) = (1 β 0)) |
22 | 1m0e1 12337 | . . . . . . . 8 β’ (1 β 0) = 1 | |
23 | 21, 22 | eqtrdi 2782 | . . . . . . 7 β’ (π β (1 β (β―β{π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0})) = 1) |
24 | 23 | adantr 480 | . . . . . 6 β’ ((π β§ π₯ β β+) β (1 β (β―β{π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0})) = 1) |
25 | 24 | oveq2d 7421 | . . . . 5 β’ ((π β§ π₯ β β+) β ((logβπ₯) Β· (1 β (β―β{π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0}))) = ((logβπ₯) Β· 1)) |
26 | relogcl 26464 | . . . . . . . 8 β’ (π₯ β β+ β (logβπ₯) β β) | |
27 | 26 | adantl 481 | . . . . . . 7 β’ ((π β§ π₯ β β+) β (logβπ₯) β β) |
28 | 27 | recnd 11246 | . . . . . 6 β’ ((π β§ π₯ β β+) β (logβπ₯) β β) |
29 | 28 | mulridd 11235 | . . . . 5 β’ ((π β§ π₯ β β+) β ((logβπ₯) Β· 1) = (logβπ₯)) |
30 | 25, 29 | eqtrd 2766 | . . . 4 β’ ((π β§ π₯ β β+) β ((logβπ₯) Β· (1 β (β―β{π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0}))) = (logβπ₯)) |
31 | 30 | oveq2d 7421 | . . 3 β’ ((π β§ π₯ β β+) β (((Οβπ) Β· Ξ£π β ((1...(ββπ₯)) β© π)((Ξβπ) / π)) β ((logβπ₯) Β· (1 β (β―β{π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0})))) = (((Οβπ) Β· Ξ£π β ((1...(ββπ₯)) β© π)((Ξβπ) / π)) β (logβπ₯))) |
32 | 31 | mpteq2dva 5241 | . 2 β’ (π β (π₯ β β+ β¦ (((Οβπ) Β· Ξ£π β ((1...(ββπ₯)) β© π)((Ξβπ) / π)) β ((logβπ₯) Β· (1 β (β―β{π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0}))))) = (π₯ β β+ β¦ (((Οβπ) Β· Ξ£π β ((1...(ββπ₯)) β© π)((Ξβπ) / π)) β (logβπ₯)))) |
33 | eqid 2726 | . . 3 β’ {π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0} = {π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0} | |
34 | rpvmasum.u | . . 3 β’ π = (Unitβπ) | |
35 | rpvmasum.b | . . 3 β’ (π β π΄ β π) | |
36 | rpvmasum.t | . . 3 β’ π = (β‘πΏ β {π΄}) | |
37 | 15 | pm2.21i 119 | . . 3 β’ ((π β§ π β {π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0}) β π΄ = (1rβπ)) |
38 | 1, 2, 3, 5, 6, 7, 33, 34, 35, 36, 37 | rpvmasum2 27400 | . 2 β’ (π β (π₯ β β+ β¦ (((Οβπ) Β· Ξ£π β ((1...(ββπ₯)) β© π)((Ξβπ) / π)) β ((logβπ₯) Β· (1 β (β―β{π¦ β ((Baseβ(DChrβπ)) β {(0gβ(DChrβπ))}) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0}))))) β π(1)) |
39 | 32, 38 | eqeltrrd 2828 | 1 β’ (π β (π₯ β β+ β¦ (((Οβπ) Β· Ξ£π β ((1...(ββπ₯)) β© π)((Ξβπ) / π)) β (logβπ₯))) β π(1)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 {crab 3426 β cdif 3940 β© cin 3942 β c0 4317 {csn 4623 β¦ cmpt 5224 β‘ccnv 5668 β cima 5672 βcfv 6537 (class class class)co 7405 βcr 11111 0cc0 11112 1c1 11113 Β· cmul 11117 β cmin 11448 / cdiv 11875 βcn 12216 β+crp 12980 ...cfz 13490 βcfl 13761 β―chash 14295 π(1)co1 15436 Ξ£csu 15638 Οcphi 16706 Basecbs 17153 0gc0g 17394 1rcur 20086 Unitcui 20257 β€RHomczrh 21386 β€/nβ€czn 21389 logclog 26443 Ξcvma 26979 DChrcdchr 27120 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 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 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-rpss 7710 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-oadd 8471 df-omul 8472 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 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ioc 13335 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-word 14471 df-concat 14527 df-s1 14552 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-o1 15440 df-lo1 15441 df-sum 15639 df-ef 16017 df-e 16018 df-sin 16019 df-cos 16020 df-tan 16021 df-pi 16022 df-dvds 16205 df-gcd 16443 df-prm 16616 df-numer 16680 df-denom 16681 df-phi 16708 df-pc 16779 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17457 df-qtop 17462 df-imas 17463 df-qus 17464 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-nsg 19051 df-eqg 19052 df-ghm 19139 df-gim 19184 df-ga 19206 df-cntz 19233 df-oppg 19262 df-od 19448 df-gex 19449 df-pgp 19450 df-lsm 19556 df-pj1 19557 df-cmn 19702 df-abl 19703 df-cyg 19798 df-dprd 19917 df-dpj 19918 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-cring 20141 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-invr 20290 df-dvr 20303 df-rhm 20374 df-subrng 20446 df-subrg 20471 df-drng 20589 df-lmod 20708 df-lss 20779 df-lsp 20819 df-sra 21021 df-rgmod 21022 df-lidl 21067 df-rsp 21068 df-2idl 21107 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-zring 21334 df-zrh 21390 df-zn 21393 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cld 22878 df-ntr 22879 df-cls 22880 df-nei 22957 df-lp 22995 df-perf 22996 df-cn 23086 df-cnp 23087 df-haus 23174 df-cmp 23246 df-tx 23421 df-hmeo 23614 df-fil 23705 df-fm 23797 df-flim 23798 df-flf 23799 df-xms 24181 df-ms 24182 df-tms 24183 df-cncf 24753 df-0p 25554 df-limc 25750 df-dv 25751 df-ply 26077 df-idp 26078 df-coe 26079 df-dgr 26080 df-quot 26181 df-ulm 26268 df-log 26445 df-cxp 26446 df-atan 26754 df-em 26880 df-cht 26984 df-vma 26985 df-chp 26986 df-ppi 26987 df-mu 26988 df-dchr 27121 |
This theorem is referenced by: rplogsum 27415 |
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