Upper bound on the maximal number of prime factors

Question

I would like to prove

$$\omega(n) \le \frac{\ln{n}}{\ln\ln{n}}$$

This is a quite standard result, but I haven't been able to find a proof. Here's what I've tried doing:

$$n=\prod\limits_{i=1}^kp_i^{a_i}\ge \prod\limits_{i=1}^kp_i\approx 2\prod\limits_{i=2}^ki\ln{i}=2P_1P_2,$$ where $P_1=\prod\limits_{i=2}^ki$ and $P_2=\prod\limits_{i=2}^k\ln{i}$. $$P_1=k!$$ $$\ln(P_2)=\sum\limits_{i=2}^k \ln\ln{i} \approx \int_2^k \! \ln\ln{x} \, \mathrm{d}x \approx k\ln\ln{k}.$$ $$P_2=\ln^kk.$$ $$n\ge2P_1P_2 \approx 2\sqrt{2\pi}k^{k+\frac{1}{2}}e^{-k}\ln^kk$$ $$\ln{n} \approx \ln2+\ln{\sqrt{n\pi k}}+k\ln{k}-k+k\ln\ln{k}$$

but not really sure where to go from here.

Answer 1

It appears you have the inequality backwards. Also, all I find is an asymptotic result for the primorial numbers, in Hardy and Wright. Therefore, i am not entirely sure the inequality indicated by the computer run holds forever, maybe sometimes it goes the other way. Below is data for the first 25 primorials. I indicate the largest prime factor with lower case p, the primorial with upper case P.

Afterthought: maybe for all numbers $n \geq 2,$ primorial and otherwise, we get something like $\omega(n) < 2 \log n / \log \log n.$

Found it, there is an explicit reference in the comments: Effective Upper Bound for the Number of Prime Divisors with the result by G. Robin. Comment by Gerry Myerson: Estimation de la fonction de Tchebychef θ sur le k-ieme nombre premier et grandes valeurs de la fonction ω(n) nombre de diviseurs premiers de n, Acta Arith 42 (1983) 367-389, MR0736719 (85j:11109).

omega  p
  1     2    log P / log log P    -1.891194393528896 Primorial P    2
  2     3    log P / log log P     3.072300009669941 Primorial P    6
  3     5    log P / log log P     2.778466514782393 Primorial P    30
  4     7    log P / log log P     3.189340693812293 Primorial P    210
  5    11    log P / log log P     3.783498689132461 Primorial P    2310
  6    13    log P / log log P     4.418974532899663 Primorial P    30030
  7    17    log P / log log P     5.102354851231404 Primorial P    510510
  8    19    log P / log log P     5.790972042777756 Primorial P    9699690
  9    23    log P / log log P     6.50283030965765  Primorial P    223092870
 10    29    log P / log log P     7.246259778056221
 11    31    log P / log log P     7.985305729097139
 12    37    log P / log log P     8.744643009669849
 13    41    log P / log log P     9.50916568672571
 14    43    log P / log log P     10.26873760689452
 15    47    log P / log log P     11.03275458447593
 16    53    log P / log log P     11.80791815799024
 17    59    log P / log log P     12.59204609226844
 18    61    log P / log log P     13.37154899993717
 19    67    log P / log log P     14.15847947390451
 20    71    log P / log log P     14.94653596430098
 21    73    log P / log log P     15.73068925382913
 22    79    log P / log log P     16.52071947436941
 23    83    log P / log log P     17.31157778776554
 24    89    log P / log log P     18.10720534877059
 25    97    log P / log log P     18.9106523419625