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Mirrors > Home > MPE Home > Th. List > pntsval | Structured version Visualization version GIF version |
Description: Define the "Selberg function", whose asymptotic behavior is the content of selberg 27385. (Contributed by Mario Carneiro, 31-May-2016.) |
Ref | Expression |
---|---|
pntsval.1 | โข ๐ = (๐ โ โ โฆ ฮฃ๐ โ (1...(โโ๐))((ฮโ๐) ยท ((logโ๐) + (ฯโ(๐ / ๐))))) |
Ref | Expression |
---|---|
pntsval | โข (๐ด โ โ โ (๐โ๐ด) = ฮฃ๐ โ (1...(โโ๐ด))((ฮโ๐) ยท ((logโ๐) + (ฯโ(๐ด / ๐))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6881 | . . . . 5 โข (๐ = ๐ โ (ฮโ๐) = (ฮโ๐)) | |
2 | fveq2 6881 | . . . . . 6 โข (๐ = ๐ โ (logโ๐) = (logโ๐)) | |
3 | oveq2 7409 | . . . . . . 7 โข (๐ = ๐ โ (๐ / ๐) = (๐ / ๐)) | |
4 | 3 | fveq2d 6885 | . . . . . 6 โข (๐ = ๐ โ (ฯโ(๐ / ๐)) = (ฯโ(๐ / ๐))) |
5 | 2, 4 | oveq12d 7419 | . . . . 5 โข (๐ = ๐ โ ((logโ๐) + (ฯโ(๐ / ๐))) = ((logโ๐) + (ฯโ(๐ / ๐)))) |
6 | 1, 5 | oveq12d 7419 | . . . 4 โข (๐ = ๐ โ ((ฮโ๐) ยท ((logโ๐) + (ฯโ(๐ / ๐)))) = ((ฮโ๐) ยท ((logโ๐) + (ฯโ(๐ / ๐))))) |
7 | 6 | cbvsumv 15638 | . . 3 โข ฮฃ๐ โ (1...(โโ๐))((ฮโ๐) ยท ((logโ๐) + (ฯโ(๐ / ๐)))) = ฮฃ๐ โ (1...(โโ๐))((ฮโ๐) ยท ((logโ๐) + (ฯโ(๐ / ๐)))) |
8 | fveq2 6881 | . . . . 5 โข (๐ = ๐ด โ (โโ๐) = (โโ๐ด)) | |
9 | 8 | oveq2d 7417 | . . . 4 โข (๐ = ๐ด โ (1...(โโ๐)) = (1...(โโ๐ด))) |
10 | fvoveq1 7424 | . . . . . . 7 โข (๐ = ๐ด โ (ฯโ(๐ / ๐)) = (ฯโ(๐ด / ๐))) | |
11 | 10 | oveq2d 7417 | . . . . . 6 โข (๐ = ๐ด โ ((logโ๐) + (ฯโ(๐ / ๐))) = ((logโ๐) + (ฯโ(๐ด / ๐)))) |
12 | 11 | oveq2d 7417 | . . . . 5 โข (๐ = ๐ด โ ((ฮโ๐) ยท ((logโ๐) + (ฯโ(๐ / ๐)))) = ((ฮโ๐) ยท ((logโ๐) + (ฯโ(๐ด / ๐))))) |
13 | 12 | adantr 480 | . . . 4 โข ((๐ = ๐ด โง ๐ โ (1...(โโ๐))) โ ((ฮโ๐) ยท ((logโ๐) + (ฯโ(๐ / ๐)))) = ((ฮโ๐) ยท ((logโ๐) + (ฯโ(๐ด / ๐))))) |
14 | 9, 13 | sumeq12dv 15648 | . . 3 โข (๐ = ๐ด โ ฮฃ๐ โ (1...(โโ๐))((ฮโ๐) ยท ((logโ๐) + (ฯโ(๐ / ๐)))) = ฮฃ๐ โ (1...(โโ๐ด))((ฮโ๐) ยท ((logโ๐) + (ฯโ(๐ด / ๐))))) |
15 | 7, 14 | eqtrid 2776 | . 2 โข (๐ = ๐ด โ ฮฃ๐ โ (1...(โโ๐))((ฮโ๐) ยท ((logโ๐) + (ฯโ(๐ / ๐)))) = ฮฃ๐ โ (1...(โโ๐ด))((ฮโ๐) ยท ((logโ๐) + (ฯโ(๐ด / ๐))))) |
16 | pntsval.1 | . 2 โข ๐ = (๐ โ โ โฆ ฮฃ๐ โ (1...(โโ๐))((ฮโ๐) ยท ((logโ๐) + (ฯโ(๐ / ๐))))) | |
17 | sumex 15630 | . 2 โข ฮฃ๐ โ (1...(โโ๐ด))((ฮโ๐) ยท ((logโ๐) + (ฯโ(๐ด / ๐)))) โ V | |
18 | 15, 16, 17 | fvmpt 6988 | 1 โข (๐ด โ โ โ (๐โ๐ด) = ฮฃ๐ โ (1...(โโ๐ด))((ฮโ๐) ยท ((logโ๐) + (ฯโ(๐ด / ๐))))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โฆ cmpt 5221 โcfv 6533 (class class class)co 7401 โcr 11104 1c1 11106 + caddc 11108 ยท cmul 11110 / cdiv 11867 ...cfz 13480 โcfl 13751 ฮฃcsu 15628 logclog 26393 ฮcvma 26928 ฯcchp 26929 |
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 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-seq 13963 df-sum 15629 |
This theorem is referenced by: selbergs 27411 selbergsb 27412 pntsval2 27413 |
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