Hardy–Landau conjecture on error terms in the divisor and circle problems

Determine whether the error terms in the Dirichlet divisor problem and the Gauss circle problem satisfy the bounds Δ_d(x) = O(x^{1/4+ε}) and Δ_r(x) = O(x^{1/4+ε}) for every ε > 0, where Δ_d(x) = \sum_{n\le x} d(n) − x\log x − (2γ − 1)x and Δ_r(x) = \left|\sum_{n\le x} r_2(n) − πx\right|.

Background

The paper recalls Voronoï’s summation formulas and their role in improving bounds on the error terms in the Dirichlet divisor and Gauss circle problems from O(x{1/2}) to O(x{1/3}). Using these formulas, Hardy and Landau showed the error terms cannot be O(x{1/4}), which led them to conjecture O(x{1/4+ε}) bounds for both problems.

This conjecture is a classical benchmark in analytic number theory and is included by the authors in their introduction to motivate the significance of Voronoï-type summation identities.

References

By using eq:vsd and eq:vsr, Hardy and Landau observed in that $\Delta_d(x)$ and $\Delta_r(x)$ cannot be $O(x{1/4})$. Consequently, they conjectured that both $\Delta_d(x)$ and $\Delta_r(x)$ are $O_(x{1/4+\epsilon})$ for any $>0$.

— Equivalence between the Functional Equation and Voronoï-type summation identities for a class of $L$-Functions  (2604.02803 - Roy et al., 3 Apr 2026) in Section 1 (Introduction)