Divisor Function - Approximate Growth Rate

Approximate Growth Rate

In little-o notation, the divisor function satisfies the inequality (see page 296 of Apostol’s book)

More precisely, Severin Wigert showed that

On the other hand, since there are infinitely many prime numbers,

In Big-O notation, Dirichlet showed that the average order of the divisor function satisfies the following inequality (see Theorem 3.3 of Apostol’s book)

where is Euler's constant. Improving the bound in this formula is known as Dirichlet's divisor problem

The behaviour of the sigma function is irregular. The asymptotic growth rate of the sigma function can be expressed by:

where lim sup is the limit superior. This result is Grönwall's theorem, published in 1913 (Grönwall 1913). His proof uses Mertens' 3rd theorem, which says that

where p denotes a prime.

In 1915, Ramanujan proved that under the assumption of the Riemann hypothesis, the inequality:

(Robin's inequality)

holds for all sufficiently large n (Ramanujan 1997). In 1984 Guy Robin proved that the inequality is true for all n ≥ 5,041 if and only if the Riemann hypothesis is true (Robin 1984). This is Robin's theorem and the inequality became known after him. The largest known value that violates the inequality is n=5,040. If the Riemann hypothesis is true, there are no greater exceptions. If the hypothesis is false, then Robin showed there are an infinite number of values of n that violate the inequality, and it is known that the smallest such n ≥ 5,041 must be superabundant (Akbary & Friggstad 2009). It has been shown that the inequality holds for large odd and square-free integers, and that the Riemann hypothesis is equivalent to the inequality just for n divisible by the fifth power of a prime (Choie et al. 2007).

A related bound was given by Jeffrey Lagarias in 2002, who proved that the Riemann hypothesis is equivalent to the statement that

for every natural number n, where is the nth harmonic number, (Lagarias 2002).

Robin also proved, unconditionally, that the inequality

holds for all n ≥ 3.