Open problem, Arithmetic and number theory, Analysis, posed 1849

Dirichlet divisor problem

Open

Let $d(n)$ be the number of positive divisors of $n$ , and write $D(x) = \sum_{n \le x} d(n) = x\log x + (2\gamma - 1)x + \Delta(x)$ , where $\gamma$ is the Euler–Mascheroni constant. Determine the infimum $\theta$ of all exponents such that $\Delta(x) = O(x^{\theta+\varepsilon})$ as $x \to \infty$ for every $\varepsilon > 0$ ; it is conjectured that $\theta = \tfrac{1}{4}$ .

Research frontier as of 2026

As of 2026, the Dirichlet divisor problem remains open, with the true exponent $\theta$ known to lie in $[\tfrac{1}{4}, \tfrac{131}{416}]$ (exactly half the exponent range of the Gauss circle problem under the parameter change $x = r^2$ ). The best peer-reviewed upper bound is Martin Huxley's 2003 theorem $\theta \le \tfrac{131}{416} \approx 0.31490$ (with subsequent preprint work by Bourgain and Watt exploring decoupling approaches toward $\tfrac{517}{1648} \approx 0.31371$ ). On average, Tong (1956) and Heath-Brown (1992) established that $\int_1^T \Delta(x)^2\,dx \sim c T^{3/2}$ , so $|\Delta(x)|$ has root-mean-square order $x^{1/4}$ , while Soundararajan (2003) proved the oscillation bound $\Delta(x) = \Omega\!\left(x^{1/4}(\log x)^{1/4}(\log\log x)^{\frac{3}{4}(2^{4/3}-1)}(\log\log\log x)^{-5/8}\right)$ .

Best known results

Published upper bound: $\Delta(x) = O\!\left(x^{131/416}(\log x)^{26947/8320}\right)$ , so $\theta \le \tfrac{131}{416} \approx 0.31490$ (Huxley, 2003).
Mean-square and limiting distribution: $\int_1^T \Delta(x)^2\,dx \sim c T^{3/2}$ and $x^{-1/4}\Delta(x)$ has a continuous non-Gaussian limiting distribution (Heath-Brown, 1992, 1999).
Oscillation lower bound: $\Delta(x) = \Omega\!\left(x^{1/4}(\log x)^{1/4}(\log\log x)^{\frac{3}{4}(2^{4/3}-1)}(\log\log\log x)^{-5/8}\right)$ (Soundararajan, 2003).

Tools and where they stop

Tool	Achieved	Where it stops
Voronoi summation formula and Perron contour integration	Represents $\Delta(x)$ either as a contour integral of $\zeta(s)^2 x^s / s$ or as a truncated Bessel-function series $\frac{x^{1/4}}{\pi\sqrt{2}} \sum_{n \le N} d(n) n^{-3/4} \cos(4\pi\sqrt{nx} - \tfrac{\pi}{4}) + O(x^{1/2+\varepsilon}N^{-1/2})$ .	Even assuming the Lindelöf hypothesis ( $\zeta(\tfrac{1}{2}+it) = O(t^{\varepsilon})$ ), shifting the Perron contour to $\mathrm{Re}(s) = \tfrac{1}{2}$ only yields $\Delta(x) = O(x^{1/2+\varepsilon})$ without exploiting phase cancellation in $x^{it}$ .
Exponent pairs and the Bombieri–Iwaniec–Huxley discrete Hardy–Littlewood method	Bounds double exponential sums with phase $\sqrt{nx}$ to establish $\theta \le \tfrac{131}{416}$ .	The exponent-pair conjecture $(k, l) = (\varepsilon, \tfrac{1}{2}+\varepsilon)$ would imply $\theta = \tfrac{1}{4}$ , but known A- and B-processes and decoupling bounds remain far from $(\varepsilon, \tfrac{1}{2}+\varepsilon)$ .

Open questions

Does $\Delta(x) = O(x^{1/4+\varepsilon})$ hold for every $\varepsilon > 0$ ?
In the Piltz divisor problem for $k \ge 3$ , does $\Delta_k(x) = O(x^{(k-1)/(2k)+\varepsilon})$ hold for every $\varepsilon > 0$ ?

References

G. H. Hardy (1916). On Dirichlet's divisor problem · DOI:10.1112/plms/s2_15.1.1
Henryk Iwaniec, C. J. Mozzochi (1988). On the divisor problem, and the circle problem · DOI:10.1016/0022-314X(88)90025-X
Martin N. Huxley (2003). Exponential sums and lattice points III · DOI:10.1112/S0024611503014485
K. Soundararajan (2003). Omega results for the divisor and circle problems · DOI:10.1155/S1073792803130929