Number of distinct prime divisors of an integer $n$ is $O(\log n/\log\log n)$
Solution 1:
Yes, the paper by Robin you cite has, on page 368 (its second page):
la fonction $\omega(n)$: nombre de diviseurs premiers de l'entier $n$.
Which translates to:
the function $\omega(n)$: the number of prime divisors of the integer $n$.
(This is a standard definition, but it's good to check the paper is using the definition you want.)
Then, on the following page (369, the third page of the article) there are various upper bounds on $\omega(n)$ given. I'll just quote the first one; you can read them yourself with the note that pour means for and a comma is used instead of a decimal point:
$\omega(n) \leq 1.3841 \frac{\log n}{\log\log n}$ for $n \geq 3$
This confirms that $\omega(n)$ is $O(\frac{\log n}{\log \log n})$.
Solution 2:
The primorials are the products of the consecutive primes from 2 up to something, so they are $2, 6, 30, 210,$ and so on. A number $n$ cannot have more distinct prime divisors than the largest primorial that is no larger than $n.$ I suggest that you write a quick program to calculate the first ten or twenty primorials, and for each primorial $n$ calculate $$ \frac{\log n}{ \log \log n} $$ and compare with the number of prime factors, these being distinct by the definition of the primorials. It is not necessary to explicitly find the primorial itself (or print it), its logarithm is just the sum of the logs of its prime factors. Then take another log of that to get the denominator.
In turn, this observation can be made to quickly give the bound quoted from Robin using explicit bounds in Rosser and Schoenfeld (1962). Edit: looking again, and noticing that the optimum occurs at the primorial with largest prime factor 23, I suggest that Robin found that result, while Rosser and Schoenfeld might give a less precise constant, maybe 2.0 instead of 1.3841. I imagine Robin quotes their constant in his article in the first few pages.
omega 1 top prime 2 Robin ratio -1.891194393528896 Primorial 2
omega 2 top prime 3 Robin ratio 3.072300009669941 Primorial 6
omega 3 top prime 5 Robin ratio 2.778466514782393 Primorial 30
omega 4 top prime 7 Robin ratio 3.189340693812293 Primorial 210
omega 5 top prime 11 Robin ratio 3.783498689132461 Primorial 2310
omega 6 top prime 13 Robin ratio 4.418974532899663 Primorial 30030
omega 7 top prime 17 Robin ratio 5.102354851231403 Primorial 510510
omega 8 top prime 19 Robin ratio 5.790972042777755 Primorial 9699690
omega 9 top prime 23 Robin ratio 6.502830309657649 Primorial 223092870
omega 10 top prime 29 Robin ratio 7.246259778056221 Primorial 6469693230
omega 11 top prime 31 Robin ratio 7.985305729097138 Primorial 200560490130
omega 12 top prime 37 Robin ratio 8.744643009669849 Primorial 7420738134810
omega 13 top prime 41 Robin ratio 9.50916568672571 Primorial 304250263527210
omega 14 top prime 43 Robin ratio 10.26873760689452 Primorial 13082761331670030
omega 15 top prime 47 Robin ratio 11.03275458447593 Primorial 614889782588491410
omega 16 top prime 53 Robin ratio 11.80791815799024 Primorial 32589158477190044730
omega 17 top prime 59 Robin ratio 12.59204609226844 Primorial 1922760350154212639070
omega 18 top prime 61 Robin ratio 13.37154899993717 Primorial 117288381359406970983270
omega 19 top prime 67 Robin ratio 14.15847947390451 Primorial 7858321551080267055879090
omega 20 top prime 71 Robin ratio 14.94653596430098 Primorial 557940830126698960967415390
omega 21 top prime 73 Robin ratio 15.73068925382913 Primorial 40729680599249024150621323470
omega 22 top prime 79 Robin ratio 16.52071947436941 Primorial 3217644767340672907899084554130
omega 23 top prime 83 Robin ratio 17.31157778776554 Primorial 267064515689275851355624017992790
omega 24 top prime 89 Robin ratio 18.10720534877059 Primorial 23768741896345550770650537601358310
omega 25 top prime 97 Robin ratio 18.9106523419625 Primorial 2305567963945518424753102147331756070
omega 1 top prime 2 Robin ratio -1.891194393528896 compare 1.3841 -0.5287663729448977
omega 2 top prime 3 Robin ratio 3.072300009669941 compare 1.3841 0.6509780925381897
omega 3 top prime 5 Robin ratio 2.778466514782393 compare 1.3841 1.07973228543118
omega 4 top prime 7 Robin ratio 3.189340693812293 compare 1.3841 1.254177707562094
omega 5 top prime 11 Robin ratio 3.783498689132461 compare 1.3841 1.321528143874282
omega 6 top prime 13 Robin ratio 4.418974532899663 compare 1.3841 1.357781076883214
omega 7 top prime 17 Robin ratio 5.102354851231403 compare 1.3841 1.371915557443171
omega 8 top prime 19 Robin ratio 5.790972042777755 compare 1.3841 1.381460649594613
omega 9 top prime 23 Robin ratio 6.502830309657649 compare 1.3841 1.384012740826666
omega 10 top prime 29 Robin ratio 7.246259778056221 compare 1.3841 1.380022288226942
omega 11 top prime 31 Robin ratio 7.985305729097138 compare 1.3841 1.377530225288409
omega 12 top prime 37 Robin ratio 8.744643009669849 compare 1.3841 1.372268712025222
omega 13 top prime 41 Robin ratio 9.50916568672571 compare 1.3841 1.367102060083705
omega 14 top prime 43 Robin ratio 10.26873760689452 compare 1.3841 1.363361353259263
omega 15 top prime 47 Robin ratio 11.03275458447593 compare 1.3841 1.359587932927135
omega 16 top prime 53 Robin ratio 11.80791815799024 compare 1.3841 1.355022941886928
omega 17 top prime 59 Robin ratio 12.59204609226844 compare 1.3841 1.350058590592204
omega 18 top prime 61 Robin ratio 13.37154899993717 compare 1.3841 1.346141722255558
omega 19 top prime 67 Robin ratio 14.15847947390451 compare 1.3841 1.34195201080871
omega 20 top prime 71 Robin ratio 14.94653596430098 compare 1.3841 1.338102691337241
omega 21 top prime 73 Robin ratio 15.73068925382913 compare 1.3841 1.334970112316485
omega 22 top prime 79 Robin ratio 16.52071947436941 compare 1.3841 1.33166113220016
omega 23 top prime 83 Robin ratio 17.31157778776554 compare 1.3841 1.32859062772745
omega 24 top prime 89 Robin ratio 18.10720534877059 compare 1.3841 1.325439212607677
omega 25 top prime 97 Robin ratio 18.9106523419625 compare 1.3841 1.322006218924839