@DavidRoberts I looked at academia.SE several times in the past, and I found the amount of pointless comments and answers (pointless because the site is not maths-specific and a lot of people feel that they should offer their opinion even if it is not relevant) very frustrating. So I totally understand at least one reason to post this question on MO.

I sincerely hope that you and the OP meant something like $o(1/\log x)$. In particular, $O(\log(x))$ makes the presence of $\log\log x+M$ in (*) completely pointless?

Thanks Fedya, this is very nice. For my immediate purposes Neil's answer might be more immediately useful but I like this one very much too.

@Z.M Ah, I see - you suggest to expand each $q^\alpha$ around $q=1$. Yes, that is a perfectly valid point, thanks!

Thanks Neil! This is lovely. Though with my conventions the identity is correct (since my $(q;q)_k$ evaluates to $1$ at $q=0$): the cardinality of $GL_k$ is $(-1)^kq^{\binom{k}2}(q;q)_k$ ;)

@Z.M can you elaborate? I mean, these are not quite polynomials in $a,b$ as written.

That's quite remarkable that $\binom{2n}{n}-\binom{2n}{n-2}$ is also a Catalan number - I did not know it either!

Do you know any non-exterior example?

@DavidRoberts every good research paper I saw on RG was actually published elsewhere (in a reputable journal or on the arXiv). I strongly believe that if one looks at instances where the only published copy is on RG, it is as much bottom of the barrel as vixra. In any case, seeing it on the same list as HAL both amused and frightened me.

@Favst there are many examples of predatory publishers and predatory journals that waive fees on particular occasions. I find it quite odd that you (in the original post) were so relaxed about accusing moderators of a crucial service of harassment but feel the need to be so protective of the journal name.

@DavidRoberts in all fairness, ResearchGate also has an awful lot of truly bizarre preprints, as I discovered recently in the process of being harassed by someone demanding me to read their rather hopeless writings aiming to solve most Millenium problems...

Also, may I ask where it was published? The general criterion of arXiv has always been "publishable in a non-predatory journal".

Perhaps you might find the answers here useful: mathoverflow.net/questions/357077/…

I do not think there is any hope for any very general statements. For some related results, see sciencedirect.com/science/article/pii/S0021869312002475 and references therein.

If there is a need for a resource of that kind (which I am not sure about), it should not be hosted by a private person.

@Carl-FredrikNybergBrodda it is more than a slight exaggeration. Basically, there is a selection of good Russian language journals for which all papers were digitized. For quite a few others, nothing was digitized, unfortunately.