Numerical experiments on the Hardy conjecture for the Gauss circle problem

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.Source: §1 · p. 1

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.Source: §1 · p. 2

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.Source: §1 · p. 2

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.Source: §1 · p. 2
• 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.Source: §1 · p. 2
• 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.Source: §1 · p. 2

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.Source: §2 · p. 3

Envelope functions

• emax(t) is a step function, whereas einf(t) is continuous and piecewise linear.Source: §2 · p. 3

Candidate points

• T+ contains discontinuities of emax(t), while T− contains points where einf(t) changes from left derivative −π to right derivative 0.Source: §2 · p. 3
• T± is the union T+∪T−, and Figure 5 evaluates both functions at all computed points in this set.Source: §2 · p. 3, Figure 5 · p. 5

Computed range

• 9,573 points from T+ and 17,261 points from T− are shown in the computed range.Source: §2 · p. 3

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.Source: §3.1 · p. 5
• Main and subtrackers traverse lattice points between circles C(t0) and C(t1), preserving x^2+y^2−t0−1 to test positions exactly.Source: §3.1 · pp. 5–6
• The tracker path yields A(t0), while counters inc[i] accumulate the subsequent values through t1.Source: §3.1 · p. 6

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.Source: §3.1 · p. 6

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.Source: §3.2 · p. 8
• Exact matches between A(t0) and the previous subinterval’s LAST_A exposed unexpected errors and fixed program bugs.Source: §3.2 · pp. 8–9

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.Source: §3.3 · p. 10

3.5 Job data table

• The calculation progressed seamlessly to t=1.2218 in March 2026.Source: §3.5 · p. 13

• 9,573 elements of T+ and 17,261 elements of T− were obtained from the computations.Source: §4 · p. 14
• Least-squares fits gave a+ < 0.259 and a− < 0.261 for emax(t) ∼ t^a+ and einf(t) ∼ −t^a−.Source: §4 · p. 14, Figure 8 · p. 14, Figure 9 · p. 15
• The fitted logarithmic exponents satisfied b+ < 0.33 and b− < 0.4 in the corresponding asymptotic estimates.Source: §4 · p. 14, Figure 10 · p. 15, Figure 11 · p. 15
• 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.Source: §4 · p. 14, Figure 12 · p. 16, Figure 13 · p. 16
• A further comparison used the normalization 1 + t^1/4(log t)^3/10 for both extreme-value functions.Source: §4 · p. 14, Figure 14 · p. 16

• T+ and T− are concentrated in relatively narrow subintervals of [0, 1.22×10^18], rather than uniformly distributed.Source: §5 · p. 17
• The computations detected all T± over [0, 1.22×10^18] and used them to determine the integer sets Smax and Sinf.Source: §5 · p. 17
• The paper reports no underlying pattern in the sequences Smax and Sinf.Source: §5 · p. 17
• For each s in Smax or Sinf, the corresponding points are concentrated in still narrower intervals described by functions κi±(s).Source: §5 · p. 17
• For s = 1087048060, 223 elements of T+ concentrate in one reported interval, while for s = 954266569, 1423 elements of T− concentrate in reported intervals.Source: §5 · p. 18
• Smax contains 394 integers and Sinf contains 310 integers, with their complete listings provided in the appendix.Source: §5 · p. 18

• Listing 5 provides the Fortran program that outputs elements of T+ below 1.22×10^18.Source: Appendix · p. 20
• The T+ program stores Smax values and selects κ1 and κ2 ranges according to the size of s.Source: Appendix · p. 22
• Listing 6 provides the Fortran program that outputs elements of T− below 1.22×10^18.Source: Appendix · pp. 22–24

• The electronic appendix includes programs corresponding to Listings 1 through 6.Source: Electric Appendix · p. 24
• The programs were made available before publication at the listed Gauss circle problem website.Source: Electric Appendix · p. 24