Dirichlet divisor problem

Dirichlet divisor problem

In number theory, one of the arithmetic functions that has been extensively studied over many years is the divisor function d(n) which is the number of positive divisors of n. The behaviour of d(n) is rather erratic as n increases, as can be seen from the examples:

d(151199) = 4

d(151200) = 144

d(151201) = 2

As a consequence of this erratic growth of d(n), mathematicians have studied the average value of d(m), over the values m = 1, 2, ... n, in mathematical notation:

${\frac {1}{n}}\sum _{i\leq n}d(i)$.

Finding a closed form for the value of this expression seems to be beyond the techniques available so far, but it is possible to find approximations, and the first step in this direction is

${\frac {1}{n}}\sum _{i\leq n}d(i)\approx \log(n)$,

which may be written as

$\sum _{i\leq n}d(i)\approx n\log(n)$.

Dirichlet proved that

$\sum _{i\leq n}d(i)=n\log(n)+(2\gamma -1)n+O({\sqrt {n}}),$

where $\gamma$ is the Euler-Mascheroni constant.

The proof of this result is quite elementary, and may be found in many textbooks on number theory.

The Dirichlet divisor problem is the problem of improving the term $O({\sqrt {n}}),$ which is known as the 'error term'.

The progress towards finding better and better expressions for the error term maybe summed up as follows:

• In 1903, Voronoi proved that the error term can be improved to $O(x^{1/3}\log x).$
• In the opposite direction, Hardy and Landau showed that the error term can not be taken to be $O(x^{1/4}).$
• In 1922, van der Corput improved the error term to $O(x^{33/100}).$
• In 1969, Kolesnik showed that the error term can be taken to be $O(x^{(12/37)+\epsilon }),$ for any $\epsilon \ >0.$
• In 1988, Iwaniec and Mozzochi proved that if $\Theta$ is the infimum of the values of $\theta$ for which $\sum _{i\leq n}d(i)=n\log(n)+(2\gamma -1)n+O(n^{\theta }),$ then $\Theta \leq 7/22.$

So, the true value of $\Theta$ lies somewhere between 1/4 and 7/22.

References

• T. M. Apostol. Introduction to Analytic Number Theory, Springer-Verlag, 1976.
• H. E. Rose. A Course in Number Theory., Oxford, 1988.