Dirichlet divisor problem exponent

Let d(n) be the divisor function. The Dirichlet divisor problem concerns the error term Δ(x) := Σ_n≤ x d(n) - x(log x + 2γ -1), where γ is Euler's constant. <a href="#Tsa2010-def-Delta">[Tsa2010-def-Delta]</a> Define the divisor-problem exponent α := infBigla≥ 0: Δ(x)=O(x^a+ε) for all ε>0Bigr.

Current state (tracked automatically)

Best known bounds for constant 63a: 0.25 ≤ C ≤ 0.31490384615384615. Upper bound ([Hux2003], Record exponent (as stated in survey literature)); lower bound ([Tsa2010], Omega results imply ≥ 1/4).

All tracked facts and sources

Fact	Value	Source
Best lower bound with an explicit value (constant 63a)	0.25 [Tsa2010] — Omega results imply ≥ 1/4.	Tao's optimization constants
Best upper bound with an explicit value (constant 63a)	0.31490384615384615 [Hux2003] — Record exponent (as stated in survey literature).	Tao's optimization constants

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Cite

@misc{cairn-constant-63a-dirichlet-divisor-problem-exponent,
  title        = {Dirichlet divisor problem exponent},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/constant-63a-dirichlet-divisor-problem-exponent}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

Description of constant

Let $d (n)$ be the divisor function. The Dirichlet divisor problem concerns the error term $Δ (x) := n \leq x \sum d (n) - x (lo g x + 2 γ - 1),$ where $γ$ is Euler's constant. <a href="#Tsa2010-def-Delta">[Tsa2010-def-Delta]</a>

Define the divisor-problem exponent $α := in f {a \geq 0 : Δ (x) = O (x^{a + ε}) for all ε > 0} .$ <a href="#Tsa2010-def-alpha">[Tsa2010-def-alpha]</a>

We define $C_{63} := α,$ the Dirichlet divisor problem exponent.

It is conjectured that $α = 1/4$ . <a href="#Tsa2010-conj-1-4">[Tsa2010-conj-1-4]</a> The best known upper bound is due to Huxley, $α \leq \frac{131}{416} .$ <a href="#Tsa2010-ub-131-416">[Tsa2010-ub-131-416]</a>

Hardy's omega result implies that $Δ (x) = Ω (x^{1/4})$ (up to logarithmic factors), hence $α \geq 1/4$ . <a href="#Tsa2010-omega-1-4">[Tsa2010-omega-1-4]</a>

The best established range currently is $\frac{1}{4} \leq C_{63} \leq \frac{131}{416} .$

Known upper bounds

Bound	Reference	Comments
$131/416$	<a href="#Hux2003">[Hux2003]</a>	Record exponent (as stated in survey literature). <a href="#Tsa2010-ub-131-416">[Tsa2010-ub-131-416]</a>

Known lower bounds

Bound	Reference	Comments
$0$		Trivial from the definition $C_{63} \geq 0$ .
$1/4$	<a href="#Tsa2010">[Tsa2010]</a>	Omega results imply $α \geq 1/4$ . <a href="#Tsa2010-omega-1-4">[Tsa2010-omega-1-4]</a>

Additional comments and links

Conjectural value. The conjecture $C_{63} = 1/4$ is often stated as $Δ (x) = O (x^{1/4 + ε})$ . <a href="#Tsa2010-conj-1-4">[Tsa2010-conj-1-4]</a>

Wikipedia page on the Dirichlet divisor problem

References

<a id="Hux2003"></a>[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

<a id="Tsa2010"></a>[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
<a id="Tsa2010-def-Delta"></a>[Tsa2010-def-Delta] loc: PDF p.1, Introduction quote: "Let $Δ (x) = D (x) - x lo g x - (2 γ - 1) x$ be the error term in the above asymptotic formula for $D (x)$ ."
<a id="Tsa2010-def-alpha"></a>[Tsa2010-def-alpha] loc: PDF p.1, Introduction quote: "Dirichlet's divisor problem consists of determining the smallest $α$ for which $Δ (x) ≪_{ε} x^{α + ε}$ holds for any $ε > 0$ ."
<a id="Tsa2010-conj-1-4"></a>[Tsa2010-conj-1-4] loc: PDF p.1, Introduction quote: "It is widely conjectured that $α = 1/4$ is admissible, which is then the best possible."
<a id="Tsa2010-ub-131-416"></a>[Tsa2010-ub-131-416] loc: PDF p.1, Introduction quote: "The best estimate to-date is $α \leq 131/416$ , due to Huxley."
<a id="Tsa2010-omega-1-4"></a>[Tsa2010-omega-1-4] loc: PDF p.6, Section 3 (Omega-results) quote: "Hardy ... showed that $Δ (x) = Ω^{+} ((x lo g x)^{1/4} lo g_{2} x)$ and $Δ (x) = Ω^{-} (x^{1/4})$ ."

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 87b4ebbd426f, Apache License 2.0; reformatted for this page. New records should also be reported there.