What to do if a paper by respected authors has obvious problems, and I have told them so already? I have recently found a paper of a very respected research group in my field. In this paper, they have some theoretical calculations, and they are obviously completely wrong. Not in the sense that the results are wrong - the formulae themselves have nothing to do with what they want to describe. Imagine something equivalent to a paper in which they want to use the Poisson distribution, but their theoretical derivation is that of a Gaussian distribution, while in the text they claim that this is how the Poisson distribution is derived. It makes no sense if you know about it, but is still close enough that readers might not realize the difference. (Personally, I am really wondering how it got through the review process, but that's a different topic.)

What is interesting, however, is that the results they show are correct, i.e. they behave as if they had actually used (the equivalent of) the Poisson distribution instead of the equations they actually wrote down. It's possible that the actual work was done by someone else than the person writing the paper (who is the senior researcher), or maybe they tortured the wrong formulae until they behaved similarly to the correct ones, it's hard to say. But the experimental results are fitted correctly by their simulations, so technically one could argue that everything is fine.

However, recently the same authors published a review paper in which they basically copied the whole theoretical part of the old paper. Of course, it is wrong again. And I am really worried about this idea that completely wrong formulae get through the review process again and again, just because the author is well-known or the review process is a sham.

So I have contacted the authors and told them that I think there is a mistake there (of course I also suggested that I might have misunderstood something, but I know that I did not misunderstand anything). The response that I received had nothing to do with my original points, however. I replied again and emphasized that I believe that there is a mistake there, and the next response was very brief and did not really give a straight answer again. I feel like any further correspondence would only make them angry at me, if they aren't already.

So what should I do now? The papers are not widely cited, but my field is also something of a niche. They might still have some impact and might mislead others. The journal does not have a clear guideline with regards to errata, and since the fits are correct and only the theoretical derivation is wrong, I am also not sure how "relevant" this really is. For me, something like this is really important and annoying, but the journal might brush it off as unimportant details.

By the way, I do not actually want to start a fight with these researchers. If I write the journal now, and the journal then contacts them, I'm pretty sure they can put one and one together. This kind of bothers me. But this is not the most important point. For me, the death of science starts in the small things, and if I find an error, I want to do something about it.

(I should also mention that, yes, I did read the other questions that were similar. The answers were not satisfactory I thought, because they mostly assumed that the person asking the question might have misunderstood something. The questions were also about errors that invalidate the whole paper, while mine is a bit more subtle. The results are fine, but the foundation that is given is shaky, and they keep using that foundation in all their papers.)
Just a guess here, but I think you have done everything you can be expected to do and it up to the authors now. If they do something positive by your standards then all is well. But if they don't you aren't likely to get them to change unless they reach out to you in some way. 

You could, of course, produce a study with the correct methodology and attempt to publish it as a confirmation of the results of the other group. That is worth doing in any case. But how much you want to criticize the "respected authors" in your work is a judgement call that you should make only rather late in your process, not before you begin. 

I think that would be a good outcome for everyone, actually. 

But don't let the inaction of others frustrate you. It is out of your hands what other people do and going up against an establishment might not be a good career move. 



I'll add here that you might not be the only one to have informed them and that they are taking time dealing with it somehow. Anything from ignoring it to fixing it.