2vmadivsum - Metamath Proof Explorer

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Theorem 2vmadivsum 26966

Description: The sum Σ𝑚𝑛 ≤ 𝑥, Λ(𝑚)Λ(𝑛) / 𝑚𝑛 is asymptotic to log↑2(𝑥) / 2 + 𝑂(log𝑥). (Contributed by Mario Carneiro, 30-May-2016.)

**Assertion**
Ref	Expression
2vmadivsum	⊢ (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)

Distinct variable group: 𝑚,𝑛,𝑥

Proof of Theorem 2vmadivsum Dummy variables 𝑐 𝑖 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vmadivsumb 26908	. 2 ⊢ ∃𝑐 ∈ ℝ⁺ ∀𝑦 ∈ (1[,)+∞)(abs‘(Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) / 𝑖) − (log‘𝑦))) ≤ 𝑐
2		simpl 483	. . . 4 ⊢ ((𝑐 ∈ ℝ⁺ ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘(Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) / 𝑖) − (log‘𝑦))) ≤ 𝑐) → 𝑐 ∈ ℝ⁺)
3		simpr 485	. . . 4 ⊢ ((𝑐 ∈ ℝ⁺ ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘(Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) / 𝑖) − (log‘𝑦))) ≤ 𝑐) → ∀𝑦 ∈ (1[,)+∞)(abs‘(Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) / 𝑖) − (log‘𝑦))) ≤ 𝑐)
4	2, 3	2vmadivsumlem 26965	. . 3 ⊢ ((𝑐 ∈ ℝ⁺ ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘(Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) / 𝑖) − (log‘𝑦))) ≤ 𝑐) → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1))
5	4	rexlimiva 3146	. 2 ⊢ (∃𝑐 ∈ ℝ⁺ ∀𝑦 ∈ (1[,)+∞)(abs‘(Σ𝑖 ∈ (1...(⌊‘𝑦))((Λ‘𝑖) / 𝑖) − (log‘𝑦))) ≤ 𝑐 → (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1))
6	1, 5	ax-mp 5	1 ⊢ (𝑥 ∈ (1(,)+∞) ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))(((Λ‘𝑛) / 𝑛) · Σ𝑚 ∈ (1...(⌊‘(𝑥 / 𝑛)))((Λ‘𝑚) / 𝑚)) / (log‘𝑥)) − ((log‘𝑥) / 2))) ∈ 𝑂(1)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 396 ∈ wcel 2106 ∀wral 3060 ∃wrex 3069 class class class wbr 5138 ↦ cmpt 5221 ‘cfv 6529 (class class class)co 7390 1c1 11090 · cmul 11094 +∞cpnf 11224 ≤ cle 11228 − cmin 11423 / cdiv 11850 2c2 12246 ℝ⁺crp 12953 (,)cioo 13303 [,)cico 13305 ...cfz 13463 ⌊cfl 13734 abscabs 15160 𝑂(1)co1 15409 Σcsu 15611 logclog 25987 Λcvma 26518

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 ax-inf2 9615 ax-cnex 11145 ax-resscn 11146 ax-1cn 11147 ax-icn 11148 ax-addcl 11149 ax-addrcl 11150 ax-mulcl 11151 ax-mulrcl 11152 ax-mulcom 11153 ax-addass 11154 ax-mulass 11155 ax-distr 11156 ax-i2m1 11157 ax-1ne0 11158 ax-1rid 11159 ax-rnegex 11160 ax-rrecex 11161 ax-cnre 11162 ax-pre-lttri 11163 ax-pre-lttrn 11164 ax-pre-ltadd 11165 ax-pre-mulgt0 11166 ax-pre-sup 11167 ax-addf 11168 ax-mulf 11169

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 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 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-tp 4624 df-op 4626 df-uni 4899 df-int 4941 df-iun 4989 df-iin 4990 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-se 5622 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 6286 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-isom 6538 df-riota 7346 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7650 df-om 7836 df-1st 7954 df-2nd 7955 df-supp 8126 df-frecs 8245 df-wrecs 8276 df-recs 8350 df-rdg 8389 df-1o 8445 df-2o 8446 df-oadd 8449 df-er 8683 df-map 8802 df-pm 8803 df-ixp 8872 df-en 8920 df-dom 8921 df-sdom 8922 df-fin 8923 df-fsupp 9342 df-fi 9385 df-sup 9416 df-inf 9417 df-oi 9484 df-dju 9875 df-card 9913 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11425 df-neg 11426 df-div 11851 df-nn 12192 df-2 12254 df-3 12255 df-4 12256 df-5 12257 df-6 12258 df-7 12259 df-8 12260 df-9 12261 df-n0 12452 df-xnn0 12524 df-z 12538 df-dec 12657 df-uz 12802 df-q 12912 df-rp 12954 df-xneg 13071 df-xadd 13072 df-xmul 13073 df-ioo 13307 df-ioc 13308 df-ico 13309 df-icc 13310 df-fz 13464 df-fzo 13607 df-fl 13736 df-mod 13814 df-seq 13946 df-exp 14007 df-fac 14213 df-bc 14242 df-hash 14270 df-shft 14993 df-cj 15025 df-re 15026 df-im 15027 df-sqrt 15161 df-abs 15162 df-limsup 15394 df-clim 15411 df-rlim 15412 df-o1 15413 df-lo1 15414 df-sum 15612 df-ef 15990 df-e 15991 df-sin 15992 df-cos 15993 df-tan 15994 df-pi 15995 df-dvds 16177 df-gcd 16415 df-prm 16588 df-pc 16749 df-struct 17059 df-sets 17076 df-slot 17094 df-ndx 17106 df-base 17124 df-ress 17153 df-plusg 17189 df-mulr 17190 df-starv 17191 df-sca 17192 df-vsca 17193 df-ip 17194 df-tset 17195 df-ple 17196 df-ds 17198 df-unif 17199 df-hom 17200 df-cco 17201 df-rest 17347 df-topn 17348 df-0g 17366 df-gsum 17367 df-topgen 17368 df-pt 17369 df-prds 17372 df-xrs 17427 df-qtop 17432 df-imas 17433 df-xps 17435 df-mre 17509 df-mrc 17510 df-acs 17512 df-mgm 18540 df-sgrp 18589 df-mnd 18600 df-submnd 18645 df-mulg 18920 df-cntz 19144 df-cmn 19611 df-psmet 20865 df-xmet 20866 df-met 20867 df-bl 20868 df-mopn 20869 df-fbas 20870 df-fg 20871 df-cnfld 20874 df-top 22320 df-topon 22337 df-topsp 22359 df-bases 22373 df-cld 22447 df-ntr 22448 df-cls 22449 df-nei 22526 df-lp 22564 df-perf 22565 df-cn 22655 df-cnp 22656 df-haus 22743 df-cmp 22815 df-tx 22990 df-hmeo 23183 df-fil 23274 df-fm 23366 df-flim 23367 df-flf 23368 df-xms 23750 df-ms 23751 df-tms 23752 df-cncf 24318 df-limc 25307 df-dv 25308 df-ulm 25813 df-log 25989 df-cxp 25990 df-atan 26294 df-em 26419 df-cht 26523 df-vma 26524 df-chp 26525 df-ppi 26526

This theorem is referenced by: selberg4r 26995

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