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Is there a way to realize that, at a given moment, how many users are (MO) on line? Thank you

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As far as I know, there is no such thing. See also this post on Mathematics Meta. Number of people online. On main meta you can find some feature requests such as View approximate number of users online or How many SO users are online?.

Depending on what purpose you want this for, perhaps some other measures might be useful for you.

On Math.SE you can see patterns of chat messages posted per day/per week in the main chatroom - which might correlate with users online. This will not work on MathOverflow, since chat here is rather inactive, but somebody with sufficient knowledge of SQL should be able to make a SEDE query showing number of posts in various time slots - which also seems a reasonable approximation. Maybe you can use query from answer to this question on meta.SE: What is the best time to ask questions? (Do not forget to switch site to MathOverflow, when you try the query.)

You can also find some already existing queries (and try them also for MathOverflow) in these posts from Mathematics Meta: Best time to ask a question, Depending on the tag is there a possibility to know on which days and hours are there more answers?, When is peak time for this website?.

Users with 25k+ rep have access to site analytics. Here you can learn a bit about traffic on site, but AFAIK you do not see there breakdown by the time of day/day of week.

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  • $\begingroup$ Very great information, Thanks. $\endgroup$
    – Ali Taghavi
    Commented Aug 21, 2017 at 9:06
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Here is a snapshot of the site analytics to which Martin refers:

Traffic
So, roughly $40,000$ page views per day (in the summer). Might be interesting to know what happened on May 14th and on August 8th...

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    $\begingroup$ Very interesting. is it about MO? $\endgroup$
    – Ali Taghavi
    Commented Aug 21, 2017 at 12:39
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    $\begingroup$ @AliTaghavi: Yes, this is MathOverflow specific data. $\endgroup$
    – Joseph O'Rourke
    Commented Aug 21, 2017 at 13:36
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    $\begingroup$ I just noticed Stack Exchange says: "Since this is a restricted privilege, we'd prefer you not share the raw data." Not sure if the snapshot violates this injunction. $\endgroup$
    – Joseph O'Rourke
    Commented Aug 21, 2017 at 13:38
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    $\begingroup$ Perhaps the question Mathematically interesting screensavers has to do with the peak in August? It has over 20k view, it definitely appeared in the network-wide hot questions list and it was very probably shared on various websites. It was asked on August 8. $\endgroup$
    – Martin Sleziak
    Commented Aug 22, 2017 at 7:46
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    $\begingroup$ @MartinSleziak: I think you have pinpointed it. 20k views---Wow! Likely May 14th was a similar highly popular post. I did a bit of searching without, however, hitting on such a post. $\endgroup$
    – Joseph O'Rourke
    Commented Aug 22, 2017 at 12:34
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    $\begingroup$ When I try to see look at questions with high number of views posted around that May 14, there is only one with more that 5k views: Hironaka's proof of resolution of singularities in positive characteristics. This probably not enough to explain the difference. $\endgroup$
    – Martin Sleziak
    Commented Aug 22, 2017 at 12:43
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    $\begingroup$ @MartinSleziak: A mystery. Perhaps an old post resuscitated. $\endgroup$
    – Joseph O'Rourke
    Commented Aug 22, 2017 at 16:04
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    $\begingroup$ Re: Not sure if the snapshot violates this injunction. I think that sharing a picture is rather different from sharing the raw data, so IMO this is perfectly fine. (I think I have seen similar screenshots before on the main meta and Mathematics meta.) $\endgroup$
    – Martin Sleziak
    Commented Aug 27, 2017 at 7:37
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    $\begingroup$ I also find the periodicity very interesting. It seems that weekends really have lower views than weekdays. That seems to tell a pretty positive story about work–life balance. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Aug 30, 2017 at 23:36
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    $\begingroup$ @R.vanDobbendeBruyn: The weekly min is invariably Friday, but likely that includes part of Saturday in most parts of the world. Sunday-Wednesday are more or less constant. Thursday starts to drop toward the min, then Saturday sees a rise. Quite predictable. Mind you, this is a summer snapshot. When classes are in session there might be a slightly different pattern. My own usage is the inverse: I primarily only have time for MO on the weekends. $\endgroup$
    – Joseph O'Rourke
    Commented Aug 30, 2017 at 23:42
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    $\begingroup$ Here are two snapshots from the Wayback Machine: 5/14 and 5/15. I see: "A book you would like to write," "Intuition behind harmonic analysis in number theory," and "Solutions of the Continuum Hypothesis" are questions with more than 1k views. Perhaps they all contributed to a spike in attention. $\endgroup$
    – Neal
    Commented Sep 19, 2017 at 13:13
  • $\begingroup$ @Neal: Excellent! I think it was "Solutions of the Continuum Hypothesis": with ~100 votes. $\endgroup$
    – Joseph O'Rourke
    Commented Sep 19, 2017 at 13:44

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