Source-linked AI summary
Numerical experiments on the Hardy conjecture for the Gauss circle problem
Satoshi Yamaguchi, Shoichi Fujima, Shigehiko Kuratsubo, Eiichi Nakai, Tsuyoshi Yoneda
TL;DR
The paper studies the open Gauss circle problem and seeks numerical evidence for the Hardy conjecture. Large-scale supercomputer computations extend prior calculations to t = 1.22 × 10^18, indicating θ < 0.261 and suggesting bounds for E(t), while revealing clustered extreme values whose locations lack a known general rule.
Problem
Determining the best possible exponent θ in E(t) = O(t^θ) remains an open problem in the Gauss circle problem, despite earlier bounds such as θ ≤ 1/3.
Method
The paper computes E(t) explicitly using a supercomputer, refining Tromp’s method and extending prior computations to t = 1.22 × 10^18.
Results
The computations indicate θ < 0.261 over the examined range and provide further numerical evidence supporting the Hardy conjecture.
Takeaways & Limitations
The paper conjectures E(t) = O(t^1/4(log t)^2/5) or E(t) = O(t^1/4(log t)^1/3), while extreme values concentrate in narrow subintervals.
Takeaways & Limitations
The authors were unable to identify an underlying pattern in the sequences Smax and Sinf.
Abstract
from arXiv · showhide
The classical unsolved Gauss circle problem concerns estimating the error between the number of lattice points inside a circle and the area of the circle as its radius tends to infinity. About a century ago, Hardy proposed a conjecture concerning this problem. In this paper, we attempt to provide numerical evidence in support of the Hardy conjecture through large-scale numerical computations.
1 Introduction
The paper frames the Gauss circle problem as determining the sharp error-growth exponent for lattice-point counts and uses large-scale computation to test Hardy’s conjecture. Its computations extend previous ranges, support θ < 0.261, suggest logarithmic refinements, and reveal clustered extreme values.
- Problem: The Gauss circle problem asks for the best exponent θ such that E(t) = O(t^θ) as t →∞.Here E(t) measures the discrepancy between the lattice-point count and the circle’s area approximation.
- Approach: The paper computes E(t) explicitly with a supercomputer to provide further numerical evidence for the Hardy conjecture.The study builds on earlier probabilistic evidence concerning lattice points near the circle boundary.
- Computational scope: t = 1.22 × 10^18 extends prior computations reaching t = 58,956,361,256 and t = 2.29 × 10^12.The computational method is described as a refinement of Tromp’s method.
- Findings: θ < 0.261 over the computed range provides further numerical evidence supporting the Hardy conjecture.This is an empirical indication over the stated range, not a proof of the conjecture.
- Findings: The computations conjecture E(t) = O(t^1/4(log t)^2/5) or E(t) = O(t^1/4(log t)^1/3) as t →∞.These proposed bounds refine the power t^1/4 with logarithmic factors.
- Findings: Extreme values of E(t) concentrate in several narrow subintervals rather than being uniformly distributed across the computed interval.The paper reports that it has no general rule for locating these subintervals.
2 Formulation
The paper studies extreme values of the Gauss circle error E(t)=A(t)−πt by isolating the discontinuity and derivative-transition points of two envelope functions. Computations examine these candidate points over increasingly large t.
- Extreme values: The analysis focuses on the extreme values of E(t)=A(t)−πt.A(t) counts lattice points inside a circle, while E(t) measures the deviation from πt.
- Envelope functions: emax(t) is a step function, whereas einf(t) is continuous and piecewise linear.
- Candidate points: T+ contains discontinuities of emax(t), while T− contains points where einf(t) changes from left derivative −π to right derivative 0.
- Candidate points: T± is the union T+∪T−, and Figure 5 evaluates both functions at all computed points in this set.
- Computed range: 9,573 points from T+ and 17,261 points from T− are shown in the computed range.
3 Program
The computational program modifies Tromp’s tracker-and-counter approach to count A(t) over intervals, detect candidate extreme points, verify continuity across subintervals, and reduce work through coarse-graining. The computation reached t=1.2218 in March 2026.
- 3.1 Count of A(t): The modified Tromp method computes A(t0),A(t0+1),…,A(t1) in one pass for arbitrary t0 and t1.
- 3.1 Count of A(t): Main and subtrackers traverse lattice points between circles C(t0) and C(t1), preserving x^2+y^2−t0−1 to test positions exactly.
- 3.1 Count of A(t): The tracker path yields A(t0), while counters inc[i] accumulate the subsequent values through t1.
- 3.3 Coarse-graining: 2^k consecutive counters can be merged into one counter, reducing the counter-array size to (t1−t0)2^−k with NSHIFT=k.
- 3.2 Detection of T±: EXPLORE searches only t where inc[t−t0]>0, because T± points can occur only when A(t)−A(t−0)>0.
- 3.2 Detection of T±: Exact matches between A(t0) and the previous subinterval’s LAST_A exposed unexpected errors and fixed program bugs.
- 3.3 Coarse-graining: Coarse-graining uses inequalities between sampled E(t), emax(t), and einf(t) to identify T±-able intervals for later exploration.
- 3.5 Job data table: The calculation progressed seamlessly to t=1.2218 in March 2026.
4 Conjecture within the range of our computations
Large-scale computations produced thousands of extreme-value points and fitted exponents below the conjectured power-law thresholds. Comparisons with logarithmic normalizations supported the paper’s conjectures within the computed range.
- 9,573 elements of T+ and 17,261 elements of T− were obtained from the computations.
- Least-squares fits gave a+ < 0.259 and a− < 0.261 for emax(t) ∼ t^a+ and einf(t) ∼ −t^a−.
- The fitted logarithmic exponents satisfied b+ < 0.33 and b− < 0.4 in the corresponding asymptotic estimates.
- Graphs compared emax(t) and einf(t) after normalization by 1 + t^1/4(log t)^2/5 and 1 + t^1/4(log t)^1/3.
- A further comparison used the normalization 1 + t^1/4(log t)^3/10 for both extreme-value functions.
5 The tendency of the points in T± to cluster
The computed extreme-value sets T+ and T− cluster in narrow subintervals rather than being uniformly distributed across the tested range. The authors identify these regions computationally but do not find a general rule for their locations.
- T+ and T− are concentrated in relatively narrow subintervals of [0, 1.22×10^18], rather than uniformly distributed.
- The computations detected all T± over [0, 1.22×10^18] and used them to determine the integer sets Smax and Sinf.
- The paper reports no underlying pattern in the sequences Smax and Sinf.
- For each s in Smax or Sinf, the corresponding points are concentrated in still narrower intervals described by functions κi±(s).
- For s = 1087048060, 223 elements of T+ concentrate in one reported interval, while for s = 954266569, 1423 elements of T− concentrate in reported intervals.
- Smax contains 394 integers and Sinf contains 310 integers, with their complete listings provided in the appendix.
Appendix
The appendix supplies Fortran programs and implementation details for generating the computed extreme-value sets over the tested interval.
- Listing 5 provides the Fortran program that outputs elements of T+ below 1.22×10^18.
- The T+ program stores Smax values and selects κ1 and κ2 ranges according to the size of s.
- Listing 6 provides the Fortran program that outputs elements of T− below 1.22×10^18.
Electric Appendix
The electronic appendix contains the programs corresponding to Listings 1–6 and gives a pre-publication access location.
- The electronic appendix includes programs corresponding to Listings 1 through 6.
- The programs were made available before publication at the listed Gauss circle problem website.