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
we say that m is a minimal order for f. Similarly if M(n) is a non-decreasing function that is ultimately positive and
we say that M is a maximal order for f. The subject was first studied systematically by Ramanujan starting in 1915.
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References
Extremal orders of an arithmetic function Wikipedia(Text) CC BY-SA