Divisor summatory function

Definition

The divisor summatory function is defined as

$D(x)\equiv \sum _{n\leq x}d(n)=\sum _{j,k \atop jk\leq x}1\,$

where

$d(n)=\sigma _{0}(n)=\sum _{j,k \atop jk=n}1\,$

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

$D_{k}(x)=\sum _{n\leq x}d_{k}(n)=\sum _{mn\leq x}d_{k-1}(n)\,$

where $\scriptstyle d_{k}(n)\,$ counts the number of ways that $\scriptstyle n\,$ can be written as a product of $\scriptstyle k\,$ numbers. This quantity can be visualized as the count of the number of lattice points fenced off by a hyperbolic surface in $\scriptstyle k\,$ dimensions. Thus, for $\scriptstyle k\,=\,2,\,D(x)\,=\,D_{2}(x)\,$ 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 $\scriptstyle jk\,=\,x\,$. Roughly, this shape may be envisioned as a hyperbolic simplex. 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.