Examples
- An average order of d(n), the number of divisors of n, is log(n);
- An average order of σ(n), the sum of divisors of n, is nπ2 / 6;
- An average order of φ(n), Euler's totient function of n, is 6n / π2;
- An average order of r(n), the number of ways of expressing n as a sum of two squares, is π;
- An average order of ω(n), the number of distinct prime factors of n, is log log n;
- An average order of Ω(n), the number of prime factors of n, is log log n;
- The prime number theorem is equivalent to the statement that the von Mangoldt function Λ(n) has average order 1;
- An average order of μ(n), the Möbius function, is zero; this is again equivalent to the prime number theorem.
Read more about this topic: Average Order Of An Arithmetic Function
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