Extremal orders of an arithmetic function

In mathematics, specifically 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 }{\frac {f(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 }{\frac {f(n)}{M(n)}}=1

we say that M is a maximal order for f.^[1]^: 80 Here, $\liminf _{n\to \infty }$ and $\limsup _{n\to \infty }$ denote the limit inferior and limit superior, respectively.

The subject was first studied systematically by Ramanujan starting in 1915.^[1]^: 87

^ ^a ^b Tenenbaum, Gérald (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advanced mathematics. Vol. 46. Cambridge University Press. ISBN 0-521-41261-7.

[Tenenbaum-1] Tenenbaum, Gérald (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advanced mathematics. Vol. 46. Cambridge University Press. ISBN 0-521-41261-7.

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