Conjectured one-quarter-plus-epsilon error bound in the Gauss circle problem

Establish the conjectured error bound for the Gauss circle problem by proving that, for x > 0, the discrepancy |C(√x) − πx| is O(x^(1/4+ε)) for every ε > 0, where C(√x) denotes the number of integer lattice points inside the circle of radius √x centered at the origin.

Background

The Gauss circle problem studies the discrepancy between the lattice-point count C(r) in a circle of radius r and the area πr2. Writing x = r2, the error term is commonly expressed as E(x) = C(√x) − πx. Over more than a century, successive improvements have lowered the exponent in bounds of the form E(x) = O(xα), culminating in results such as O(x37/112).

A long-standing conjecture predicts the optimal bound E(x) = O(x1/4+ε) for any ε > 0. The paper notes that Iwaniec and Mozzochi related this conjectured bound to the Riemann Hypothesis, underscoring its centrality and difficulty.

References

In 1988. Iwaniec and Mozzochi [4] showed the conjectured bound O(x1/4+€) for e > 0 is related to the Riemann hypothesis.

— Gauss Circle Primes  (2502.06804 - Ehrenborg, 2 Feb 2025) in Section 1 (Gauss Circle Problem), paragraph following Theorem 1.2

In 1916, Hardy conjectured that for every $\varepsilon>0$, \begin{align}\label{circleproblem} |N_0(R)-\pi R2|=O\left(R{\frac 12+\varepsilon}\right). \end{align} This conjecture remains open to date.

— On $\mathcal{B}^4$-almost periodicity for a class of arithmetical functions  (2608.13172 - Aggarwal et al., 13 Aug 2026) in Section 1, Introduction

Determining the best possible bound on $\theta$ such that $E(t)=O(t{\theta})$ is a famous open problem known as the Gauss circle problem.

— Numerical experiments on the Hardy conjecture for the Gauss circle problem  (2609.04725 - Yamaguchi et al., 4 Sep 2026) in Section 1, Introduction

Moreover, Hardy (1917) conjectured that \begin{equation}\label{Hardy conj} E(t)=O(t{1/4+}) \quad\text{for any}\quad >0. \end{equation}

Hardy conj:

E(t)=O(t1/4+)for any>0.E(t)=O(t^{1/4+}) \quad\text{for any}\quad >0.

— Numerical experiments on the Hardy conjecture for the Gauss circle problem  (2609.04725 - Yamaguchi et al., 4 Sep 2026) in Section 1, Introduction

Furthermore, we conjecture that \begin{equation} E(t)=O(t{1/4}(\log t){2/5}) \quad\text{or}\quad E(t)=O(t{1/4}(\log t){1/3}) \quad\text{as}\quad t\to\infty. \end{equation}

— Numerical experiments on the Hardy conjecture for the Gauss circle problem  (2609.04725 - Yamaguchi et al., 4 Sep 2026) in Section 1, Introduction; Section 4, "Conjecture within the range of our computations"