Dirichlet divisor problem

In number theory, the divisor summatory function is a function that is a sum over the divisor function. It frequently occurs in the study of the asymptotic behaviour of the Riemann zeta function. The various studies of the behaviour of the divisor function are sometimes called divisor problems.

The divisor summatory function is defined as

where

is the divisor function. The divisor function counts the number of ways that the integer n can be written as a product of two integers. More generally, one defines

where d_k(n) counts the number of ways that n can be written as a product of k numbers. This quantity can be visualized as the count of the number of lattice points fenced off by a hyperbolic surface in k dimensions. Thus, for k=2, D(x) = D₂(x) counts the number of points on a square lattice bounded on the left by the vertical-axis, on the bottom by the horizontal-axis, and to the upper-right by the hyperbola jk = x. Roughly, this shape may be envisioned as a hyperbolic simplex. This allows us to provide an alternative expression for D(x), and a simple way to compute it in ${\displaystyle O({\sqrt {x}})}$ time:

If the hyperbola in this context is replaced by a circle then determining the value of the resulting function is known as the Gauss circle problem.

Finding a closed form for this summed expression seems to be beyond the techniques available, but it is possible to give approximations. The leading behaviour of the series is not difficult to obtain. Peter Gustav Lejeune Dirichlet demonstrated that

...
Wikipedia