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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 26458. (Contributed by Mario Carneiro, 31-May-2016.) |
Ref | Expression |
---|---|
pntsval.1 | ⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖))))) |
Ref | Expression |
---|---|
pntsval | ⊢ (𝐴 ∈ ℝ → (𝑆‘𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝐴 / 𝑛))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6738 | . . . . 5 ⊢ (𝑖 = 𝑛 → (Λ‘𝑖) = (Λ‘𝑛)) | |
2 | fveq2 6738 | . . . . . 6 ⊢ (𝑖 = 𝑛 → (log‘𝑖) = (log‘𝑛)) | |
3 | oveq2 7242 | . . . . . . 7 ⊢ (𝑖 = 𝑛 → (𝑎 / 𝑖) = (𝑎 / 𝑛)) | |
4 | 3 | fveq2d 6742 | . . . . . 6 ⊢ (𝑖 = 𝑛 → (ψ‘(𝑎 / 𝑖)) = (ψ‘(𝑎 / 𝑛))) |
5 | 2, 4 | oveq12d 7252 | . . . . 5 ⊢ (𝑖 = 𝑛 → ((log‘𝑖) + (ψ‘(𝑎 / 𝑖))) = ((log‘𝑛) + (ψ‘(𝑎 / 𝑛)))) |
6 | 1, 5 | oveq12d 7252 | . . . 4 ⊢ (𝑖 = 𝑛 → ((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))) = ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑎 / 𝑛))))) |
7 | 6 | cbvsumv 15290 | . . 3 ⊢ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))) = Σ𝑛 ∈ (1...(⌊‘𝑎))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑎 / 𝑛)))) |
8 | fveq2 6738 | . . . . 5 ⊢ (𝑎 = 𝐴 → (⌊‘𝑎) = (⌊‘𝐴)) | |
9 | 8 | oveq2d 7250 | . . . 4 ⊢ (𝑎 = 𝐴 → (1...(⌊‘𝑎)) = (1...(⌊‘𝐴))) |
10 | fvoveq1 7257 | . . . . . . 7 ⊢ (𝑎 = 𝐴 → (ψ‘(𝑎 / 𝑛)) = (ψ‘(𝐴 / 𝑛))) | |
11 | 10 | oveq2d 7250 | . . . . . 6 ⊢ (𝑎 = 𝐴 → ((log‘𝑛) + (ψ‘(𝑎 / 𝑛))) = ((log‘𝑛) + (ψ‘(𝐴 / 𝑛)))) |
12 | 11 | oveq2d 7250 | . . . . 5 ⊢ (𝑎 = 𝐴 → ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑎 / 𝑛)))) = ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝐴 / 𝑛))))) |
13 | 12 | adantr 484 | . . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑛 ∈ (1...(⌊‘𝑎))) → ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑎 / 𝑛)))) = ((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝐴 / 𝑛))))) |
14 | 9, 13 | sumeq12dv 15300 | . . 3 ⊢ (𝑎 = 𝐴 → Σ𝑛 ∈ (1...(⌊‘𝑎))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝑎 / 𝑛)))) = Σ𝑛 ∈ (1...(⌊‘𝐴))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝐴 / 𝑛))))) |
15 | 7, 14 | syl5eq 2792 | . 2 ⊢ (𝑎 = 𝐴 → Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖)))) = Σ𝑛 ∈ (1...(⌊‘𝐴))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝐴 / 𝑛))))) |
16 | pntsval.1 | . 2 ⊢ 𝑆 = (𝑎 ∈ ℝ ↦ Σ𝑖 ∈ (1...(⌊‘𝑎))((Λ‘𝑖) · ((log‘𝑖) + (ψ‘(𝑎 / 𝑖))))) | |
17 | sumex 15281 | . 2 ⊢ Σ𝑛 ∈ (1...(⌊‘𝐴))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝐴 / 𝑛)))) ∈ V | |
18 | 15, 16, 17 | fvmpt 6839 | 1 ⊢ (𝐴 ∈ ℝ → (𝑆‘𝐴) = Σ𝑛 ∈ (1...(⌊‘𝐴))((Λ‘𝑛) · ((log‘𝑛) + (ψ‘(𝐴 / 𝑛))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ↦ cmpt 5151 ‘cfv 6400 (class class class)co 7234 ℝcr 10755 1c1 10757 + caddc 10759 · cmul 10761 / cdiv 11516 ...cfz 13122 ⌊cfl 13392 Σcsu 15279 logclog 25472 Λcvma 26003 ψcchp 26004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 ax-cnex 10812 ax-resscn 10813 ax-1cn 10814 ax-icn 10815 ax-addcl 10816 ax-addrcl 10817 ax-mulcl 10818 ax-mulrcl 10819 ax-mulcom 10820 ax-addass 10821 ax-mulass 10822 ax-distr 10823 ax-i2m1 10824 ax-1ne0 10825 ax-1rid 10826 ax-rnegex 10827 ax-rrecex 10828 ax-cnre 10829 ax-pre-lttri 10830 ax-pre-lttrn 10831 ax-pre-ltadd 10832 ax-pre-mulgt0 10833 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3711 df-csb 3828 df-dif 3885 df-un 3887 df-in 3889 df-ss 3899 df-pss 3901 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4836 df-iun 4922 df-br 5070 df-opab 5132 df-mpt 5152 df-tr 5178 df-id 5471 df-eprel 5477 df-po 5485 df-so 5486 df-fr 5526 df-we 5528 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ima 5581 df-pred 6178 df-ord 6236 df-on 6237 df-lim 6238 df-suc 6239 df-iota 6358 df-fun 6402 df-fn 6403 df-f 6404 df-f1 6405 df-fo 6406 df-f1o 6407 df-fv 6408 df-riota 7191 df-ov 7237 df-oprab 7238 df-mpo 7239 df-om 7666 df-1st 7782 df-2nd 7783 df-wrecs 8070 df-recs 8131 df-rdg 8169 df-er 8414 df-en 8650 df-dom 8651 df-sdom 8652 df-pnf 10896 df-mnf 10897 df-xr 10898 df-ltxr 10899 df-le 10900 df-sub 11091 df-neg 11092 df-nn 11858 df-n0 12118 df-z 12204 df-uz 12466 df-fz 13123 df-seq 13604 df-sum 15280 |
This theorem is referenced by: selbergs 26484 selbergsb 26485 pntsval2 26486 |
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