Dirichlet divisor problem

### What is the conjecture

Let $d(n)$ be the number of divisors of $n$ and $D(x) = \sum_{n \le x} d(n)$ the divisor summatory function. Dirichlet's hyperbola method gives
$$D(x) = x \log x + (2\gamma - 1) x + \Delta(x), \qquad \Delta(x) = O(x^{1/2}),$$
where $\gamma$ is the Euler–Mascheroni constant. The **Dirichlet divisor problem** asks for the infimum $\theta$ of exponents with $\Delta(x) = O(x^{\theta})$. It is conjectured that $\theta = 1/4$, i.e. $\Delta(x) = O(x^{1/4 + \varepsilon})$ for every $\varepsilon > 0$. Hardy showed $\theta \ge 1/4$ (so the conjecture would be sharp), and the best known upper bound is Huxley's $\theta \le 131/416 \approx 0.3149$. The problem is open.

References:
- [Wikipedia: Divisor summatory function](https://en.wikipedia.org/wiki/Divisor_summatory_function)
- [Di49] Dirichlet, *Über die Bestimmung der mittleren Werthe in der Zahlentheorie* (1849).
- [Ha16] Hardy, *On Dirichlet's divisor problem*, Proc. London Math. Soc. (1916).
- [Hu03] Huxley, *Exponential sums and lattice points III*, Proc. London Math. Soc. (2003).

### Prerequisites needed

All available in Mathlib: `Nat.divisors` (divisor count), `Real.eulerMascheroniConstant`, and `Asymptotics.IsBigO` / `IsLittleO` for the $O$ / $o$ estimates. No new Mathlib infrastructure required.

### [AMS categories](https://github.com/google-deepmind/formal-conjectures/labels?q=ams-)

* ams-11

### Choose either option
- [x] I plan on adding this conjecture to the repository
- [ ] This issue is up for grabs: I would like to see this conjecture added by somebody else

Start by reviewing the stated prerequisites: Nat.divisors, Real.eulerMascheroniConstant, and Asymptotics.IsBigO/IsLittleO. Formalize the Dirichlet divisor problem as a conjecture in Lean without attempting to prove the open result; done means the conjecture is added to the repository with its AMS-11 classification and references preserved.

Written by the indexing model from the issue text.