Lamzouri’s maximal-order conjecture

Characterize the maximal order of the Gauss circle remainder by proving that there are constants c_1,c_2>0 such that, for either sign and all sufficiently large X, c_1\mathcal L(X)\le\sup_{X\le x\le2X}(\pm E(x))\le c_2\mathcal L(X), where \mathcal L(X)=(X\log X)^{1/4}(\log_2X)^{\frac34(2^{1/3}-1)}.

Background

The paper summarizes increasingly strong lower bounds for the Gauss circle remainder, culminating in Lamzouri’s result E(X)=\Omega_-(\mathcal L(X)). Here \mathcal L(X)=(X\log X)^{1/4}(\log_2X)^{\frac34(2^{1/3}-1)}.

The conjecture proposes that \mathcal L(X) gives the correct maximal order on every dyadic interval [X,2X], simultaneously providing upper and lower bounds for the supremum of either signed remainder. The paper notes that this conjecture implies Hardy’s conjecture, while the known negative \Omega-result only concerns an unbounded sequence and does not establish both signs on every dyadic interval.