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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
:<math> \liminf_{n \to \infty} \frac{f(n)}{m(n)} = 1 </math>
we say that ''m'' is a '''minimal order''' for ''f''. Similarly if ''M''(''n'') is a non-decreasing function that is ultimately positive and
:<math> \limsup_{n \to \infty} \frac{f(n)}{M(n)} = 1 </math>
we say that ''M'' is a '''maximal order''' for ''f''.<ref name=Tenenbaum>
{{cite book | title=Introduction to Analytic and Probabilistic Number Theory | last=Tenenbaum | first=Gérald | series=Cambridge studies in advanced mathematics | volume=46 | publisher=Cambridge University Press | year=1995 | isbn=0-521-41261-7 }}
</ref>{{Rp|80}}
The subject was first studied systematically by [[Srinivasa Ramanujan|Ramanujan]] starting in 1915.<ref name=Tenenbaum />{{Rp|87}}

==Examples==

* For the [[sum-of-divisors function]] σ(''n'') we have the trivial result
::<math>\liminf_{n \to \infty} \frac{\sigma(n)}{n} = 1</math>
:because always σ(''n'') ≥ ''n'' and for primes σ(''p'') = ''p'' + 1. We also have
::<math>\limsup_{n \to \infty} \frac{\sigma(n)}{n \ln \ln n} = e^\gamma,</math>
:proved by [[Thomas Hakon Grönwall|Gronwall]] in 1913.<ref name=Tenenbaum />{{Rp|86}}<ref name=HW>
{{cite book | last1 = Hardy | first1 = G. H. |authorlink1 = G. H. Hardy |last2 = Wright |first2 = E. M. |authorlink2 = E. M. Wright | title = An Introduction to the Theory of Numbers | publisher = Clarendon Press | location = Oxford | year = 1979 | edition=5th | isbn = 0-19-853171-0 }}
</ref>{{Rp|Theorem 323}}<ref>{{cite journal|last=Gronwall|first=T. H.|title=Some asymptotic expressions in the theory of numbers|journal=Transactions of the American Mathematical Society|volume=13|issue=4|year=1913|pages=113–122}}</ref> Therefore ''n'' is a minimal order and ''e''<sup>−γ</sup> ''n'' ln ln ''n'' is a maximal order for σ(''n'').
* For the [[Euler totient]] φ(''n'') we have the trivial result
::<math>\liminf_{n \to \infty} \frac{\phi(n)}{n} = 1</math>
:because always φ(''n'') ≤ ''n'' and for primes φ(''p'') = ''p'' − 1. We also have
::<math> \liminf_{n \to \infty} \frac{\phi(n) \ln \ln n}{n} = e^{-\gamma}, </math>
:proved by [[Edmund Landau|Landau]] in 1903.<ref name=Tenenbaum />{{Rp|84}}<ref name=HW />{{Rp|Theorem 328}}
* For the [[number of divisors]] function ''d''(''n'') we have the trivial lower bound 2 ≤ ''d''(''n''), in which equality occurs when ''n'' is prime, so 2 is a minimal order. For ln ''d''(''n'') we have a maximal order ln 2 ln ''n'' / ln ln ''n'', proved by Wigert in 1907.<ref name=Tenenbaum />{{Rp|82}}<ref name=HW />{{Rp|Theorem 317}}
* For the number of distinct [[prime factors]] ω(''n'') we have a trivial lower bound 1 ≤ ω(''n''), in which equality occurs when ''n'' is a prime power. A maximal order for ω(''n'') is ln ''n'' / ln ln ''n''.<ref name=Tenenbaum />{{Rp|83}}
* For the number of prime factors counted with multiplicity Ω(''n'') we have a trivial lower bound 1 ≤ Ω(''n''), in which equality occurs when ''n'' is prime. A maximal order for Ω(''n'') is ln ''n'' / ln 2.<ref name=Tenenbaum />{{Rp|83}}

==See also==

* [[Average order of an arithmetic function]]
* [[Normal order of an arithmetic function]]

==Notes==
{{reflist}}

==Further reading==
* {{cite book |last1=Nicolas |first1=J.-L. |editor1-first=G. E. |editor1-last=Andrews |editor1-link=George Andrews (mathematician) |editor2-first=R. A. |editor2-last=Askey |editor2-link=Richard Askey |editor3-first=B. C. |editor3-last=Berndt |editor3-link=Bruce Berndt |editor4-first=K. G. |editor4-last=Ramanathan |editor4-link= |displayeditors=4 |title=Ramanujan Revisited |year=1988 |publisher=Academic Press |location= |isbn=978-0-12-058560-1 |pages=215–244 |chapter=On Highly Composite Numbers}} A survey of extremal orders, with an extensive bibliography.

[[Category:Arithmetic functions]]

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