A mean-value theorem on sums of two k-th powers of numbers in residue classes
For \(k,\ell_ 1,\ell_ 2,m,n\in {\mathbb{N}}:k\geq 2\), \(\ell_ 1,\ell_ 2\leq m\) define \(r_ k(\ell_ 1,\ell_ 2,m;n):=\#\{(u_ 1,u_ 2)\in {\mathbb{N}}^ 2:\) \(u^ k_ 1+u^ k_ 2=n\), \(u_{1,2}\equiv \ell_{1,2} mod m\}\), the number of representations of n as a sum of two k-th powers each lying in a prescribed residue class modulo m. Moreover, for a large parameter \(x>0\) let \(R_ k(\ell_ 1,\ell_ 2,m;x):=\sum_{n\leq x}r_ k(\ell_ 1,\ell_ 2,m;n)\) and \(x_ m:=x/m^ k\). The author shows with certain constants \(A_ k,B_ k>0\), a sine-series \(F_ k()\), which is both O(1) and \(\Omega\) (1), that \[ (*)\;R_ k(\ell_ 1,\ell_ 2,m;x)=A_ k(m)x_ m^{2/k}+B_ k(\ell_ 1,\ell_ 2,m)x_ m^{1/k}+x_ m^{(k-1)/k^ 2}F_ k(\ell_ 1,\ell_ 2,m;x)+O(x_ m^{7/(11k)}(\log x_ m)^{45/22}). \] The implied O-constant depends only on k. Note that (*) is a generalization of a now classical result due to \textit{E. Krätzel} [Math. Ann. 179, 90-96 (1969; Zbl 0165.363)] in concert with the latest refinement of the O-term by \textit{W. Müller} and \textit{W. G. Nowak} [Lect. Notes Math. 1452, 139-164 (1990; Zbl 0715.11054)] based on the application of the so-called `Discrete Hardy- Littlewood method' due to M. N. Huxley, N. Watt and others, e.g. [\textit{M. N. Huxley}, Proc. Lond. Math. Soc., III. Ser. 60, 471-502 (1990; Zbl 0659.10057)]. The proof of (*) itself is following the lines of the above mentioned papers.
scientific article; zbMATH DE number 4208192