Divisor summatory function


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.

Definition

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 dk 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 = D2 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, 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:

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,...

Dirichlet's divisor 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
where is the Euler–Mascheroni constant, and the non-leading term is
Here, denotes Big-O notation. The Dirichlet divisor problem, precisely stated, is to find the smallest value of for which
holds true, for any. As of today, this problem remains unsolved. Progress has been slow. Many of the same methods work for this problem and for Gauss's circle problem, another lattice-point counting problem. Section F1 of Unsolved Problems in Number Theory
surveys what is known and not known about these problems.
So, lies somewhere between 1/4 and 131/416 ; it is widely conjectured to be 1/4. Theoretical evidence lends credence to this conjecture, since has a limiting distribution. The value of 1/4 would also follow from a conjecture on exponent pairs.

Piltz divisor problem

In the generalized case, one has
where is a polynomial of degree. Using simple estimates, it is readily shown that
for integer. As in the 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. Defining the order as the smallest value for which holds, for any, one has the following results :
Both portions may be expressed as Mellin transforms:
for. Here, is the Riemann zeta function. Similarly, one has
with. The leading term of is obtained by shifting the contour past the double pole at : the leading term is just the residue, by Cauchy's integral formula. In general, one has
and likewise for, for.