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I'm not sure "research" is the word I'd use, as I'm just talking about finding the original and possibly watching it. It's not the kind of thing I'd expect to find in real life, anyway, because it usually seems to take a runaway hit to creep over into the offline world much.
But really, I'd be stunned if rentals of Let the Right One In (on Netflix, especially) don't go way up after this version comes out. How could they not? Millions of people are going to see Let Me In before all's said and done. The idea that even a fraction of them won't Google it and take some interest in the source is inconceivable.
We can speculate about the degree of this difference, but it has to make a difference. Googling the thing you just saw, or just heard about, is something that inevitably happens. Trending search topics on all the major sites almost invariably reflect new movie releases, celebrity news, etc. For an increasing number of people, finding more information on the thing you just saw is just standard operating procedure. And I would guess your average horror fan is younger than your average moviegoer, and those more predisposed to do this.
I'll comment more on Ringu after I see it, but horror movies are probably poor examples of this sort of thing, because even the best of them aren't as likely to be as scary the second time around, which is kind of what it's like to watch a remake. I suspect the order in which we view these things has a greater influence than most of us (myself included) would care to admit. But, technically speaking, there's really nothing about the order of their release that renders the second version inherently inferior. Even if you want to take points away for originality, this would only withhold some praise for the second screenwriter.