[Paper Review] A Successive Approximation Algorithm for Computing the Divisor Summatory Function

Richard Sladkey|arXiv (Cornell University)|Jun 15, 2012

Numerical Methods and Algorithms2 references3 citations

TL;DR

This paper presents a novel successive approximation algorithm that computes the divisor summatory function $ T(n) = \sum_{k=1}^n \tau(k) $ in $ O(n^{1/3}) $ time and $ O(\log n) $ space using a geometric approach based on Farey neighbors and coordinate transformation. By recursively decomposing the hyperbolic region into tangent lines and triangular subregions, the method achieves sub-square-root complexity, breaking the long-standing $ O(n^{1/2}) $ barrier of traditional methods.

ABSTRACT

An algorithm is presented to compute isolated values of the divisor summatory function in O(n^(1/3)) time and O (log n) space. The algorithm is elementary and uses a geometric approach of successive approximation combined with coordinate transformation.

Motivation & Objective

• To break the $ O(n^{1/2}) $ time complexity barrier of standard methods for computing the divisor summatory function.
• To develop an efficient, elementary algorithm that computes isolated values of $ T(n) $ with minimal space usage.
• To extend the geometric approach of Voronoï’s error decomposition into a constructive, recursive lattice point counting algorithm.
• To achieve sub-square-root time complexity by leveraging symmetry, Farey sequences, and recursive region subdivision.

Proposed method

• The algorithm uses a geometric decomposition of the hyperbola $ xy = n $ into successive tangent lines with slopes corresponding to Farey neighbors.
• Each region bounded by two tangent lines and a hyperbolic arc is transformed into a new coordinate system to simplify lattice point counting.
• Lattice points in triangular regions are counted using a formula based on triangular numbers: $ \Delta(i) = i(i+1)/2 $, adjusted for boundary exclusions.
• The method recursively processes subregions by introducing a third tangent line to split the upper region into smaller curvilinear triangles.
• A coordinate transformation maps the original $ xy = n $ hyperbola into a general quadratic form, enabling efficient area and lattice point computation.
• The algorithm combines partial sums $ S(n, x_1, x_2) = \sum_{x=x_1}^{x_2} \lfloor n/x \rfloor $ with recursive region processing to avoid redundant computation.

Experimental results

Research questions

• RQ1 Can the divisor summatory function be computed in $ O(n^{1/3}) $ time using a geometric, recursive approach based on Farey sequences?
• RQ2 How can the symmetry and lattice structure of the hyperbola $ xy \leq n $ be exploited to reduce time complexity below $ O(n^{1/2}) $?
• RQ3 What role do Farey neighbors and coordinate transformations play in enabling efficient lattice point counting in curvilinear regions?
• RQ4 Can recursive subdivision of the hyperbolic region using tangent lines yield a provably sub-square-root algorithm with low space complexity?
• RQ5 How does the performance scale when extending the method to higher-order summatory functions $ T_k(n) $?

Key findings

• The algorithm computes $ T(n) $ in $ O(n^{1/3}) $ time and $ O(\log n) $ space, representing a significant improvement over the standard $ O(n^{1/2}) $ method.
• The method uses a recursive region decomposition based on Farey neighbors and tangent lines to approximate the hyperbolic region with polygons.
• By transforming coordinates and counting lattice points in triangles, the algorithm avoids direct summation over $ O(n) $ terms.
• The approach generalizes to higher-order summatory functions, yielding $ O(n^{1 - 4/(3k)}) $ complexity for $ T_k(n) $, with $ O(n^{5/9}) $ for $ T_3(n) $.
• The algorithm is amenable to parallelization, particularly in the summation and region-processing phases, enabling scalable performance on multi-core systems.
• The method avoids double and triple counting through careful shell-based decomposition and recursive refinement of subregions.