Divisor summatory function
Encyclopedia
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, the Divisor summatory function is a function that is a sum over the divisor function
Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...
. 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
Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetical function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships...
. 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
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
. This allows us to provide an alternative expression for D(x), and a simple way to compute it in
time:
, where 
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
Gauss circle problem
In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centred at the origin and with radius r. The first progress on a solution was made by Carl Friedrich Gauss, hence its name....
.
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. Dirichlet demonstrated that
where
is the Euler-Mascheroni constantEuler-Mascheroni constant
The Euler–Mascheroni constant is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter ....
, and the non-leading term is
Here,
denotes Big-O notation. The Dirichlet divisor problem, precisely stated, is to find the infimumInfimum
In mathematics, the infimum of a subset S of some partially ordered set T is the greatest element of T that is less than or equal to all elements of S. Consequently the term greatest lower bound is also commonly used...
of all values
for which
holds true, for any
. , 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 Theorysurveys what is known and not known about these problems.
- In 1904, G. Voronoi proved that the error term can be improved to
- In 1916, G.H. Hardy showed that
. In particular, he demonstrated that for some constant 
, there exist values of x for which 
and values of x for which 
. - In 1922, J. van der Corput improved Dirichlet's bound to
- In 1928, J. van der Corput proved that
- In 1950, Chih Tsung-tao and independently in 1953 H. E. Richert proved that
- In 1969, Grigori Kolesnik demonstrated that
. - In 1973, Grigori Kolesnik demonstrated that
. - In 1982, Grigori Kolesnik demonstrated that
. - In 1988, H. Iwaniec and C. J. Mozzochi proved that
- In 2003, M.N. HuxleyMartin HuxleyMartin Neil Huxley is a British mathematician, working in the field of analytic number theory.He was awarded a PhD from the University of Cambridge in 1970, the year after his supervisor Harold Davenport had died...
improved this to show that
So, the true value of
lies somewhere between 1/4 and 131/416; it is widely conjectured to be exactly 1/4. Direct evaluation of 
lends credence to this conjecture, since 
appears to be approximately normally distributed with the standard deviation of 1 for x up to at least 1016.Generalized divisor problem
In the generalized case, one has
where
is a polynomial of degreeDegree (mathematics)
In mathematics, there are several meanings of degree depending on the subject.- Unit of angle :A degree , usually denoted by ° , is a measurement of a plane angle, representing 1⁄360 of a turn...
. Using simple estimates, it is readily shown that
for integer
. As in the 
case, the infimum of the bound is not known. Defining the order 
as the smallest value for which
holds, for any
, one has the following results:- Voronoi and Landau,
for 
- Hardy and Littlewood,
for 
- Hardy showed that
for 
- E.C. Titchmarsh conjectures that
Mellin transform
Both portions may be expressed as Mellin transformMellin transform
In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform...
s:
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 residueResidue (complex analysis)
In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities...
, by Cauchy's integral formula
Cauchy's integral formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all...
. In general, one has
and likewise for
, for 
.