Starting this January, I observe more and more inconsistencies in the MathSciNet citation database. A typical situation is that the number of citations shown in the description of a paper differs from the number of matches that one actually gets by following the link. For example, take papers by Gerd Faltings (sorted by the number of citations). MR0718935 is listed with 347 citations, but has only 340 matches; MR1083353 is listed with 304 citations, but has only 303 matches; MR1463696 is listed with 133 citations, but has only 129 matches, and so on. Moreover, the number of matches depends on the mirror. The above numbers are provided by the Providence cite. The Bielefeld mirror gives, respectively, 340 (the same number), 302 (one less) and 131 (two more). Is there any good explanation for this phenomenon?

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    $\begingroup$ Have you tried asking someone at MathSciNet? $\endgroup$ – Gerry Myerson May 4 '17 at 23:09
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    $\begingroup$ If you could single out a few papers with such an inconsistency but with a very small number of citations, it could help identifying the problem. (But of course emailing MSN is the natural way to act. I emailed them a few times for wrong MR numbers in references - resulting in errors in citations - and they fixed it.) $\endgroup$ – YCor May 4 '17 at 23:33
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    $\begingroup$ blogs.ams.org/beyondreviews/2016/12/30/things-have-changed/… directly addresses your question. $\endgroup$ – Zach Teitler May 5 '17 at 1:38
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    $\begingroup$ @ZachTeitler I think that you should post that as an answer, and maybe copy-and-paste the relevant item from the blog (I am sure that Ed Dunne would be happy for you to do so). That way it can be accepted as an answer, rather than being buried in the comments. $\endgroup$ – Joe Silverman May 5 '17 at 1:50
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    $\begingroup$ I emailed MSN several times since January. Already on January 9 I posted a comment to the blog post mentioned by Zach Teitler. I've sent to MSN specific examples containing complete case analysis (that is, I pointed to specific citations that existed in the old version and disappeared in the new one). Their typical answer is "Dear Professor Vainshtein, Thank you for your message. We are looking into the incorrect reference link you mentioned and will let you know when it has been corrected.Thank you for your interest in the Mathematical Reviews Citation Database.", and no correction followed. $\endgroup$ – Alek Vainshtein May 5 '17 at 6:37

As @ZachTeitler pointed out by quoting my blog post, one of our new features acted a bit like a bug: related papers. We have made adjustments to avoid the "counting with multiplicity" issue, but some unexpected cases still slipped in.

One of the examples involving Faltings given by @AlekVainshtein had related papers involved. Some of Vainshtein's other examples uncovered a different issue, which had to do with timing of updates. Updates directly related to individual items in the database are done (almost) continuously. Some other tables are updated continuously at Math Reviews, but pushed out to the web less frequently. We have now synced them up.

As @AlexSuciu pointed out, there were some growing pains when we released the updates to MathSciNet. We completely changed the search software. We had tested it at both the Math Reviews offices and the AMS headquarters in Providence, but nothing provides a stress test like releasing it publicly. People gave us great feedback, especially about citations, that allowed us to make improvements.

If you find anything odd, be it citations or something else, you can write to us at either
msn-support@ams.org or mathrev@ams.org. The first address doubles for problems related to access, including subscription problems. The second address doubles as the address reviewers use for communicating with the editorial staff. Both addresses have some traffic direction that will get messages to the right person.

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Edward Dunne commented on the AMS Beyond Reviews Blog

We added a feature that is performing a bit like a bug. There are some linked items in Mathematical Reviews that we refer to as “related papers”. In the print days, we could print the two items right next to each other. However, in the database days, that is not possible. With the new version of MathSciNet, we began linking these papers so that if you found one, you would also find the other. A side effect of this has been that counts can be funny. One number counts with multiplicity; the other does not. Somewhere in the list of papers citing you is probably a pair of related papers. We have already addressed some instances of the counting with multiplicity, and will release a global fix in the near future. See also my blog post on citations.

This addresses at least the issue of inconsistencies between the citation counts and the number of matches. But unfortunately it doesn't explain why the number of matches differ between the mirrors.

(credit where it is due: this was originally a comment by Zach Teitler.)

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