e/Extremal orders of an arithmetic function

Information
has gloss	eng: In mathematics, in number theory, the extremal orders of an arithmetic function are best possible bounds of the given arithmetic function. Specifically, if f(n) is an arithmetic function and m(n) is a non-decreasing function that is ultimately positive and : \liminf_n \to \infty} \fracf(n)}m(n)} = 1 we say that m is a minimal order for f. Similarly if M(n) is a non-decreasing function that is ultimately positive and : \limsup_n \to \infty} \fracf(n)}M(n)} = 1 we say that M is a maximal order for f. The subject was first studied systematically by Ramanujan starting in 1915. Therefore n is a minimal order and e−γ n ln ln n is a maximal order for σ(n). * For the Euler totient φ(n) we have the trivial result ::\liminf_n \to \infty} \frac\phi(n)}n} = 1 :because always φ(n) ≤ n and for primes φ(p) = p − 1.
lexicalization	eng: Extremal orders of an arithmetic function
instance of	e/Arithmetic function