Dirichlet divisor problem exponent ($C_{63}$)

• Conjectural value. The conjecture $C_{63}=1/4$ is often stated as $\Delta(x)=O(x^{1/4+\varepsilon})$. [Tsa2010-conj-1-4]

• Wikipedia page on the Dirichlet divisor problem

References (from the repository page)

• [Hux2003] Huxley, M. N. Exponential sums and lattice points III. Proceedings of the London Mathematical Society (3) 87 (2003), no. 3, 591-609. DOI: https://doi.org/10.1112/S0024611503014485. Google Scholar

• [Tsa2010] Tsang, K.-M. Recent progress on the Dirichlet divisor problem and the mean square of the Riemann zeta-function. IMR Preprint Series 2010-10 (2010). PDF: https://hkumath.hku.hk/~imr/IMRPreprintSeries/2010/IMR2010-10.pdf. Google Scholar

• [Tsa2010-def-Delta] loc: PDF p.1, Introduction quote: "Let $\Delta(x)=D(x)-x\log x-(2\gamma-1)x$ be the error term in the above asymptotic formula for $D(x)$."

• [Tsa2010-def-alpha] loc: PDF p.1, Introduction quote: "Dirichlet's divisor problem consists of determining the smallest $\alpha$ for which $\Delta(x)\ll_{\varepsilon} x^{\alpha+\varepsilon}$ holds for any $\varepsilon>0$."

• [Tsa2010-conj-1-4] loc: PDF p.1, Introduction quote: "It is widely conjectured that $\alpha=1/4$ is admissible, which is then the best possible."

• [Tsa2010-ub-131-416] loc: PDF p.1, Introduction quote: "The best estimate to-date is $\alpha\le 131/416$, due to Huxley."

• [Tsa2010-omega-1-4] loc: PDF p.6, Section 3 (Omega-results) quote: "Hardy ... showed that $\Delta(x)=\Omega^+((x\log x)^{1/4}\log_2 x)$ and $\Delta(x)=\Omega^-(x^{1/4})$."

Repository table (commit 2c1968c, 26 September 2026): best known lower bound $\frac{1}{4}$; best known upper bound $\frac{131}{416}$. Values marked with an asterisk have minimal verification.

Known upper bounds

Known lower bounds