Divisor summatory function
It frequently occurs in the study of the asymptotic behaviour of the Riemann zeta 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 dk(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
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.
Sequence of D(n) (sequence A006218 in the OEIS): 0, 1, 3, 5, 8, 10, 14, 16, 20, 23, 27, 29, 35, 37, 41, 45, 50, 52, 58, 60, 66, 70, 74, 76, 84, 87, 91, 95, 101, 103, 111, ... Finding a closed form for this summed expression seems to be beyond the techniques available, but it is possible to give approximations.
The leading behavior of the series is given by where
denotes Big-O notation.
[1]: 37–38, 69 The Dirichlet divisor problem, precisely stated, is to improve this error bound by finding the smallest value of
As of today, this problem remains unsolved.
Theoretical evidence lends credence to this conjecture, since
has a (non-Gaussian) limiting distribution.
[6] The value of 1/4 would also follow from a conjecture on exponent pairs.
Using simple estimates, it is readily shown that for integer
case, the infimum of the bound is not known for any value of
Computing these infima is known as the Piltz divisor problem, after the name of the German mathematician Adolf Piltz (also see his German page).
, one has the following results (note that
Both portions may be expressed as Mellin transforms: for
is the Riemann zeta function.
is obtained by shifting the contour past the double pole at
: the leading term is just the residue, by Cauchy's integral formula.
