Divisor summatory function

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The summatory function, with leading terms removed, for x < 104
The summatory function, with leading terms removed, for x < 104
The summatory function, with leading terms removed, for x < 107
The summatory function, with leading terms removed, for x < 107
The summatory function, with leading terms removed, for x < 107, graphed as a distribution or histogram. The vertical scale is not constant left to right; click on image for a detailed description.
The summatory function, with leading terms removed, for x < 107, graphed as a distribution or histogram. The vertical scale is not constant left to right; click on image for a detailed description.

In mathematics, in the area of 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.

Contents

[edit] Definition

The divisor summatory function is defined as

D(x)=\sum_{n\le x} d(n) = \sum_{j,k \atop jk\le 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 n can be written as a product of two integers. More generally, one defines

D_k(x)=\sum_{n\le x} d_k(n)=\sum_{mn\le x} d_{k-1}(n)

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.

[edit] 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

D(x) = xlogx + x(2γ − 1) + Δ(x)

where γ is the Euler-Mascheroni constant, and the non-leading term is

\Delta(x) = \mathcal{O}\left(\sqrt{x}\right)

Here, \mathcal{O} denotes Big-O notation. The Dirichlet divisor problem, precisely stated, is to find the infimum of all values θ for which

\Delta(x) = \mathcal{O}\left(x^{\theta+\epsilon}\right)

holds true, for any ε > 0. As of 2006, this problem remains unsolved. Progress has been slow.

  • In 1903, G. Voronoi proved that the error term can be improved to \mathcal{O}(x^{1/3}\log x).
  • In 1915, G.H. Hardy and E. Landau showed that \inf \theta \ge 1/4. In particular, they demonstrated that for any constant K, there exist values of x for which Δ(x) > Kx1 / 4 and Δ(x) < − Kx1 / 4.
  • In 1922, J. van der Corput improved Dirichlet's bound to \inf \theta \le 33/100.
  • In 1969, Kolesnik demonstrated that \inf \theta \le 12/37.
  • In 1988, H. Iwaniec and Mozzochi proved that \inf \theta \leq 7/22.
  • In 2003, M.N. Huxley improved this to show that \inf \theta \leq 131/416.

So, the true value of \inf \theta lies somewhere between 1/4 and 131/416; it is widely conjectured to be exactly 1/4.

[edit] Generalized divisor problem

In the generalized case, one has

D_k(x) = xP_k(\log x)+\Delta_k(x) \,

where Pk is a polynomial of degree k − 1. Using simple estimates, it is readily shown that

\Delta_k(x)=\mathcal{O}\left( x^{1-1/k} \log^{k-2} x\right)

for integer k\ge 2. As in the k = 2 case, the infimum of the bound is not known. Defining the order θk as the smallest value for which

\Delta_k(x)=\mathcal{O}\left( x^{\theta_k+\epsilon}\right)

holds, for any ε > 0, one has the following results:

  • Voronoi and Landau, \theta_k \le \frac{k-1}{k+1} for k=2,3,\ldots
  • Hardy and Littlewood, \theta_k \le \frac{k-1}{k+2} for k=4,5,\ldots
  • Hardy showed that \theta_k \ge \frac{k-1}{2k} for k=2,3,\ldots
  • E.C. Titchmarsh conjectures that \theta_k =\frac{k-1}{2k}

[edit] Mellin transform

Both portions may be expressed as Mellin transforms:

D(x)=\frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} \zeta^2(w) \frac {x^w}{w} dw

for c > 1. Here, ζ(s) is the Riemann zeta function. Similarly, one has

\Delta(x)=\frac{1}{2\pi i} \int_{c^\prime-i\infty}^{c^\prime+i\infty} \zeta^2(w) \frac {x^w}{w} dw

with 0<c^\prime<1. The leading term of D(x) is obtained by shifting the countour past the double pole at w = 1: the leading term is just the residue, by Cauchy's integral formula. In general, one has

D_k(x)=\frac{1}{2\pi i} \int_{c-i\infty}^{c+i\infty} \zeta^k(w) \frac {x^w}{w} dw

and likewise for Δk(x), for k\ge 2.

[edit] References

  • H.M. Edwards, Riemann's Zeta Function, (1974) Dover Publications, ISBN 0-486-41740-9
  • 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)
  • T. M. Apostol. Introduction to Analytic Number Theory, Springer-Verlag, 1976. (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
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