What is the origin of radioactivity that appears *sometimes* in rivers close to a nuclear plant? What is the origin of radioactivity that appears sometimes in rivers close to a nuclear plant? (due to nuclear plant activity) ?
In particular, is it due to primary circuit that would contaminate the tertiary circuit ?
Or is it due to nuclear waste that is sent on purpose to the river once its radioactivity is lower to the maximum threshold authorized by authorities ?
 A: Your question is very vague.  There could be numerous reasons.

*

*Natural occurring

*Minor releases from the reactor

Very little radioactive material can be transferred from the primary to the tertiary loops.  They are designed to not allow material transfer.
Also the steam used to turn turbines is not released into rivers.  The steam that has passed through the turbines will be released into the air as steam after it passes through a cooling tower or condensers.
A: Thank you for providing the link to the article, which discusses measured tritium contamination in a French river.
Tritium can come from many different places/reactions, but in a PWR it is usually from boron absorption and decay in the primary loop.  Tritium is a very light nuclei and is very difficult to contain, it can even pass through steel piping.  The tritium is produced in the primary loop, but then it can go to cleanup loops and escape into the environment, which is usually groundwater.  From groundwater, it can get to the river.
Tritium is very difficult to contain, but it is also very benign biologically (meaning no measured adverse affects).  The body quickly rejects tritium, so it doesn't stay in the body long enough to do much damage.  It is interesting that the news article had a quote a "'flagrant' lack of data on the carcinogenic effects of tritium".  The reason for the lack of data is that it is very difficult to show that tritium has a carcinogenic effect!
Since there is not much data showing the carcinogenic effect of tritium, international standards vary considerably.  The article implies that the environmental limits in France are 100 Bq/L, which means you can measure 100 tritium decays in a liter of water.  In the US, the limit is 740 Bq/L  (20,000 picoCuries per liter).  The values measured were 310 Bq/L.
Additional information about tritium can be found on the NRC website.
In summary, tritium is very hard to contain, easy to measure, and relatively harmless.
