2022-07-16

I wanted to define a similar expression, and search for its properties:

just looking at how the domain explodes even when $n<8$ baffles me, as I obtained:

This may be not the only "strange" property of this particular function, but even if I think it can be exploited, I've never found any source (neither textbooks nor online); so in conclusion, is there any application of such function (or functions' class)?

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Expert

Recently, it was established by Kevin Ford, Ben Green, Sergei Konyagin, Terence Tao, and also by James Maynard, that there are prime gaps below $N$ as large as
$c\frac{\mathrm{log}N\mathrm{log}\mathrm{log}N\mathrm{log}\mathrm{log}\mathrm{log}\mathrm{log}N}{\left(\mathrm{log}\mathrm{log}\mathrm{log}N{\right)}^{2}}$
when $N$ is large enough, and they showed that $c$ could be as large as you like.

Do you have a similar question?