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on Gauss circle problem exponent ($C_{64}$)

The statement

Let

N(t)\ :=\ \#\{(m,n)\in\mathbb{Z}^2:\ m^2+n^2\le t^2\}

be the number of integer lattice points inside the (closed) disk of radius $t$ centered at the origin. The Gauss circle problem is to find the smallest exponent $\theta$ such that, for every $\varepsilon>0$ ,

N(t)\ =\ \pi t^2\ +\ O(t^{\theta+\varepsilon}).

[CRM2023-def-theta]

We define

C_{64}\ :=\ \theta,

the Gauss circle problem exponent.

The best established upper bound is due to Huxley:

\theta\ \le\ \frac{131}{208}.

[CRM2023-ub-131-208]

Hardy conjectured the optimal exponent $\theta=1/2$ . [CRM2023-conj-1-2]

The best established range currently is

0\ \le\ C_{64}\ \le\ \frac{131}{208}.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.