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(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 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) : 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 behavior of the series is given by

is the Euler–Mascheroni constant, and the error term is

denotes Big-O notation. This estimate can be proven using the Dirichlet hyperbola method, and was first established by Dirichlet in 1849. The Dirichlet divisor problem for which

. 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.

• In 1904, G. Voronoi

• In 1916, G. H. Hardy, there exist arbitrarily large values of x and arbitrarily large values of x.

• In 1922, J. van der Corput.

• .

• In 1950, Chih Tsung-tao and independently in 1953 H. E. Richert.

• In 1969, Grigori Kolesnik.

• .

• .

• In 1988, H. Iwaniec and C. J. Mozzochi.

• In 2003, M.N. Huxley.

has a (non-Gaussian) 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

is a polynomial of degree. Using simple estimates, it is readily shown that

. Computing these infima is known as the Piltz divisor problem, after the name of the German mathematician Adolf Piltz of the previous section):

1. and

2. E. C. Titchmarsh

Mellin transform

Both portions may be expressed as Mellin transforms:

is the Riemann zeta function. Similarly, one has

: the leading term is just the residue, by Cauchy's integral formula. In general, one has

.

Notes

References

• H.M. Edwards, Riemann's Zeta Function, (1974) Dover Publications,

• E. C. Titchmarsh, The theory of the Riemann Zeta-Function, (1951) Oxford at the Clarendon Press, Oxford. (See chapter 12 for a discussion of the generalized divisor problem)

• (Provides an introductory statement of the Dirichlet divisor problem.)

• H. E. Rose. A Course in Number Theory., Oxford, 1988.

• M.N. Huxley (2003) 'Exponential Sums and Lattice Points III', Proc. London Math. Soc. (3)87: 591–609