A form of this question is asked after virtually every large earthquake that makes the news, so I'm going to keep my answer generic with the futile hope that I can put this in our FAQs and maybe retire this question. To first cover the underlying geology and earthquake behavior, there are specific processes by which one earthquake can trigger others. This is discussed in more detail in one of our FAQs, but in short we can consider either static or dynamic triggering. Static triggering is where the permanent movement of the crust that results from a particular earthquake changes the stress state on neighboring faults pushing some of them closer (or past) failure through stress transfer. Static triggering occurs over a very limited distance (roughly X km away from anywhere along the portion of the fault that ruptures where X is the total length of the original rupture). Dynamic triggering is where passing seismic waves, and the temporary change in stress they induce, causes a portion of a fault to fail. Dynamic triggering can occur over long (i.e., teleseismic) distances, but it tends to mostly be associated with very large magnitude earthquakes as the generative event, is pretty rare, is temporally limited (i.e., we only consider dynamic triggering to be possible over a narrow time window after the original event), and is hard to demonstrate. Given the above, when this question gets asked, i.e., "There was a big earthquake in location X and then there were moderate magnitude earthquakes in distant locations in the days, weeks, or months following, are they related?" there is a vanishingly small probability that one or more of those events may be a dynamically triggered event, but the overwhelmingly vast majority of the time, the answer is firmly and unequivocally, "No, they are not related".

So what's going on and why is this question asked so often? Mostly cognitive biases. Specifically, some mixture of the frequency illusion and the clustering illusion. The frequency illusion (or Baader–Meinhof phenomenon) is basically the tendency for your brain to take note of similar events after you become aware of an event. So, a large magnitude earthquake hits a populated area and makes the news and for some period after that you (and the news media more broadly) take note of other even moderate magnitude events. This also brings in the second bias, i.e., the clustering illusion, or the tendency for us to see patterns in stochastic (random) things. Within this context, it's worth considering just how many earthquakes of a given magnitude there are, e.g., the global statistics from the USGS. You'll notice a rough logarithmic behavior in these, i.e., for M8+ we expect about 1 a year, for M7-7.9 ~15/year, for M6-6.9 ~150/year, for M5-5.9 ~1500/year, and so on. So in a scenario where one of the ~15 7-7.9 events happens in a populated place and you notice a tiny fraction of the hundreds to thousands of 6-6.9 or 5-5.9 magnitude events we expect every year that also occur in a different populated place (i.e., excluding aftershocks from the original event) in the following days/weeks, does that mean anything or imply any linkage? No, almost always, it represents nothing other than you paying attention to a small fraction of events because you are primed to and seeing patterns that aren't there.