Extremal orders of an arithmetic function

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.[1]:80 The subject was first studied systematically by Ramanujan starting in 1915.[1]:87

Examples

because always σ(n) ≥ n and for primes σ(p) = p + 1. We also have
proved by Gronwall in 1913.[1]:86[2]:Theorem 323[3] Therefore n is a minimal order and e−γ n ln ln n is a maximal order for σ(n).
because always φ(n) ≤ n and for primes φ(p) = p  1. We also have
proved by Landau in 1903.[1]:84[2]:Theorem 328

See also

Notes

  1. 1 2 3 4 5 6 7 Tenenbaum, Gérald (1995). Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advanced mathematics. 46. Cambridge University Press. ISBN 0-521-41261-7.
  2. 1 2 3 Hardy, G. H.; Wright, E. M. (1979). An Introduction to the Theory of Numbers (5th ed.). Oxford: Clarendon Press. ISBN 0-19-853171-0.
  3. Gronwall, T. H. (1913). "Some asymptotic expressions in the theory of numbers". Transactions of the American Mathematical Society. 14 (4): 113–122. doi:10.1090/s0002-9947-1913-1500940-6.

Further reading

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