Lattice points in domains \(| x|\) \(p+| y|\) p\(\leq R\) p
For arbitrary real \(p\geq 2\), let \(A_ p(R)\) denote the number of lattice points in the planar domain \(D_ p(R):\) \(| x| ^ p+| y| ^ p\leq R^ p\). As a generalization of the classical circle problem \((p=2)\), one can ask for the infimum \(\theta\) (p) of all exponents \(\lambda\) for which the asymptotic \(A_ p(R)=a_ pR^ 2+O(R^{\lambda})\) is true \((a_ p\) the area of \(D_ p(1))\). It was proven that \((*)\quad \theta (p)=1-1/p\) for \(p>3\) by \textit{J. G. van der Corput} [Thesis Groningen 1919], for \(p=3\) by \textit{E. Krätzel} [Math. Ann. 179, 90-96 (1969; Zbl 0165.363)] and for \(p>41/14\) by the first author [Sitzungsber., Abt. II, Österr. Akad. Wiss., Math.-Naturwiss. Kl. 191, 125-132 (1982; Zbl 0486.10038)]. In the present paper, the recently developd ``discrete Hardy-Littlewood method'' in the version of \textit{M. Huxley} and \textit{N. Watt} [Exponential sums and the Riemann zeta-function (submitted to Proc. Lond. Math. Soc.)] is applied in order to verify (*) for any \(p>38/13=2,92307..\). In fact, the asymptotic formula \[ A_ p(R)=a_ pR^ 2+F_ p(R)R^{1- 1/p}+O(R^{25/38+\epsilon}) \] (with an explicitly given almost periodic function \(F_ p(R))\) is established for this range of p.
scientific article; zbMATH DE number 4019168