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An improvement on Gauss's Circle Problem and Dirichlet's Divisor Problem

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arxiv 2308.14859 v2 pith:BUIHVG7T submitted 2023-08-28 math.NT math.CA

Xiaochun Li , Xuerui Yang This is my paper · ORCID
classification math.NTmath.CA
keywords problemcircledirichletdivisorgaussimprovementspacingbombieri-iwaniec
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Using Bombieri-Iwaniec method, we establish an improvement on both Gauss's Circle Problem and Dirichlet's Divisor Problem. More precisely, we derive a new estimate for the first spacing problem and combine it with Huxley's results on the second spacing problem.

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Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Fibonacci, Dirichlet, and Gauss in a single sum

    math.NT 2026-07 accept novelty 7.0 of 10

    For odd n, the remainder of ∑_{k≤n}{F_n/F_k} is governed by the Gauss circle error; for even n, by the Dirichlet divisor error, making either optimal-exponent conjecture equivalent to a single Fibonacci sum.

  2. Counterexamples of Friedlander--Iwaniec dual sums conjecture

    math.NT 2026-07 accept novelty 6.0 of 10

    For A(s)=B(s)=ζ(s)^m with m≥4, the sharply truncated dual sum can be made as large as a positive power of N, disproving Friedlander–Iwaniec Conjecture 1 as stated.

  3. Lattice point counting in Cygan--Kor\'anyi balls on Heisenberg groups

    math.NT 2026-07 accept novelty 6.0 of 10

    For q≥4 the lattice-point error in Cygan-Koranyi balls on the Heisenberg group satisfies |E_q(t)| ≲ t^{2q-1+241/753}, improving Gath's exponent 1/3 and recovering his q=3 bound up to a log factor.

  4. On a Smoothed Dirichlet Divisor Problem

    math.NT 2026-01 conditional novelty 6.0 of 10

    This paper proves Δ(x)-xδ(x)=1/4+O(x^{-1/4}log x) explicitly, and unconditionally settles the positivity of ∫_x^∞ Δ(u)/u^2 du.

  5. The Piltz divisor Problem in Number Fields Using The Resonance Method

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    For the Piltz divisor problem over number fields, the paper proves an improved Omega lower bound with a larger third-log exponent than previous results.

  6. Bessel functions and Weyl's law for balls and spherical shells

    math.CA 2024-12 conditional novelty 6.0 of 10

    The authors improve Weyl remainder estimates for balls and spherical shells in all dimensions and prove new facts about zeros of Bessel cross-products.

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