Diplomacy when reporting errors I am a young researcher and sometimes I face an uncomfortable situation : I find an error in a research paper! Of course, most of the time, it is all just my misunderstanding but it happens that after checking tens of times, there is indeed an error and therefore I am not sure what I should do. 
"Am I the first one that notices this mistake or someone has already reported it? and then where?" "Hass this error been corrected in a later version? How can I know? ", "Should I send an email to the authors ? But still how could I be diplomatic enough not to bother them (and maybe the mistake is still mine, I am not an expert of the field)? " 
The situation gets worse when 
-The paper is already old.  Personally I would find annoying if someone asked me questions on a project finished years ago. And for very old papers the authors can just be retired already.
-It comes from a different field like physics or chemistry. Where they use a different language. How could I explain myself correctly and without creating a "diplomatic crisis"?
So I am here to ask advice to the experimented researchers how to deal with this awkward situation. 
 A: Of course, reporting an error benefits the author, so I advise to write to him/her. A polite form is: Dear X, I am reading your interesting paper Y, and I have difficulty in understanding why A implies B on p. C. I will appreciate it very much if you explain this to me (help me to understand). 
I received such letters several times. In most cases I was able to explain.
In one case I published a correction. And of course I am thankful to all who wrote these letters.
When I was a student I wrote to one famous mathematician: Dear X. Let me bring to your attention that conjecture W which you state in your paper Y seems to be proved by Picard in 1880 in his paper Z. Do you think his proof is correct?  
A: I have discussed about this at the following link: 
https://sites.google.com/site/wrongmathematicalpapers/
In particular, I propose the idea that we can establish a journal whose sole purpose is about reporting significant errors in papers already published in recognised journals (so the referee process in those journals is supposed to be rigorous, so the supposed journal - which does not exist yet - will not consider predator journals with bad referee procedures) and/or giving corrections which save the claimed results in those papers.
This journal is also a good bookkeeper about wrong published results in mathematics, so a novice will not mistakenly use them without finding an alternative proofs of the claimed results. Such a journal will potentially solve much of the problems you mentioned. I also proposed that the journal can harmoniously cooperate with other normal journals. 
This journal will be of great benefits to mathematics, I can imagine, and I hope that mathematicians will seriously consider about it. I myself am contacting some publishers about the ideas to see if they will be interested in. 
Best regards,
Tuyen  
Addition: This is meant to be a comment to Alexander Eremenko's answer, but I do not have 50 points so cannot do so. Yes, I also encourage first to contact the author. However, if it turns out that either the published paper was wrong (with no way to save the results from the ideas in the paper) or the author was not responsive, then you can consider the journal I proposed (of course, I also encourage to submit to the journal I proposed only after you contacted the journal which published the original paper if they are willing to publish an erratum and they refused to do so). Even if other journals decided to publish an erratum, then still the journal I proposed is still good, it can post a short announcement on the erratum, so it is as intended a bookkeeper of wrong published papers. More detail can be read in the link.  
Addition 2: This is meant to be a comment to Adam Epstein's comment, again I cannot give a direct comment because of low points. Maybe not many people notice the mistakes in the published papers, in particular if they are not in the same research field. Anyway, when a mistake is found, I think it is easy to sort out how to present it, for example in the journal I proposed. You are worry about what the person who made the mistakes feel, but you can also turn the cards and think what the same person feels if his/her (wrong) publications get him/her some advantages over peers. There have been people in science whose careers have been very negatively affected by having wrong publications. As far as I know, mathematics seems to be more tolerant about this issue. 
