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. <a href="#Tsa2010-def-Delta">[Tsa2010-def-Delta]</a>

Define the divisor-problem exponent $$\alpha :=\inf \{a\ge 0:\Delta (x)=O(x^{a+\epsilon })\text{for all }\epsilon >0\}.$$ <a href="#Tsa2010-def-alpha">[Tsa2010-def-alpha]</a>

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

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

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

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

Known upper bounds

Known lower bounds

Additional comments and links

• Conjectural value. The conjecture $C_{63}=1/4$ is often stated as $\Delta (x)=O(x^{1/4+\epsilon })$. <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 $\Delta (x)=D(x)-x\log x-(2\gamma -1)x$ 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 $\alpha$ for which $\Delta (x){\ll }_{\epsilon }{x}^{\alpha +\epsilon }$ holds for any $\epsilon >0$."
• <a id="Tsa2010-conj-1-4"></a>[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."
• <a id="Tsa2010-ub-131-416"></a>[Tsa2010-ub-131-416] loc: PDF p.1, Introduction quote: "The best estimate to-date is $\alpha \le 131/416$, 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 $\Delta (x)={\Omega }^{+}((x\log x{)}^{1/4}{\log }_{2}x)$ and $\Delta (x)={\Omega }^{-}(x^{1/4})$."

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 2c1968cd520b, Apache License 2.0; reformatted for this page. New records should also be reported there.