I wanted to define a similar expression, and search for its properties: f ( n...

Jadon Melendez

Jadon Melendez

Answered

2022-07-16

I wanted to define a similar expression, and search for its properties:
f ( n , x ) = n log x = log ( log ( log ( log x ) ) ) n  times , n N + , x R
just looking at how the domain explodes even when n < 8 baffles me, as I obtained:
D f ( n 2 , x ) : x > n 2 e D f ( 1 , x ) : x > 0 for example,  D f ( 7 , x ) : x 10 10 10 6.22
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)?

Answer & Explanation

ab8s1k28q

ab8s1k28q

Expert

2022-07-17Added 17 answers

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 log N log log N log log log log N ( log log log N ) 2
when N is large enough, and they showed that c could be as large as you like.

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