Dirichlet divisor problem exponent
Description of constant
Let $d(n)$ be the divisor function. The Dirichlet divisor problem concerns the error term \(\Delta(x)\ :=\ \sum_{n\le x} d(n)\ -\ x(\log x + 2\gamma -1),\) where $\gamma$ is Euler’s constant. [Tsa2010-def-Delta]
Define the divisor-problem exponent \(\alpha\ :=\ \inf\Bigl\{a\ge 0:\ \Delta(x)=O(x^{a+\varepsilon})\ \text{for all }\varepsilon>0\Bigr\}.\) [Tsa2010-def-alpha]
We define \(C_{63}\ :=\ \alpha,\) the Dirichlet divisor problem exponent.
It is conjectured that $\alpha=1/4$. [Tsa2010-conj-1-4] The best known upper bound is due to Huxley, \(\alpha\ \le\ \frac{131}{416}.\) [Tsa2010-ub-131-416]
Hardy’s omega result implies that $\Delta(x)=\Omega(x^{1/4})$ (up to logarithmic factors), hence $\alpha\ge 1/4$. [Tsa2010-omega-1-4]
The best established range currently is \(\frac14\ \le\ C_{63}\ \le\ \frac{131}{416}.\)
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $131/416$ | [Hux2003] | Record exponent (as stated in survey literature). [Tsa2010-ub-131-416] |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $0$ | Trivial from the definition $C_{63}\ge 0$. | |
| $1/4$ | [Tsa2010] | Omega results imply $\alpha\ge 1/4$. [Tsa2010-omega-1-4] |
Additional comments and links
-
Conjectural value. The conjecture $C_{63}=1/4$ is often stated as $\Delta(x)=O(x^{1/4+\varepsilon})$. [Tsa2010-conj-1-4]
References
-
[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 $\Delta(x)=\Omega^-(x^{1/4})$.”
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.