On sums of two \(k\)-th powers of numbers in residue classes. II
[For part I, cf. ibid. 60, 249-256 (1990; Zbl 0731.11055).] This paper studies the average order of the arithmetic function \(r_ k (l_ 1,m_ 1; l_ 2,m_ 2; n)\) which counts the number of ways to write a natural number \(n\) in the form \(u^ k+ v^ k\) where \(u\), \(v\) are positive integers which lie in prescribed congruence classes \(l_ 1\), \(l_ 2\), modulo \(m_ 1\), \(m_ 2\), respectively (\(k\geq 3\) a fixed natural number). For the Dirichlet summatory function, the author derives a rather intricate asymptotic formula which in particular implies that \[ \begin{multlined} \sum_{n\leq x} r_ k (l_ 1,m_ 1; l_ 2,m_ 2; n)= {{\Gamma^ 2(1/k)} \over {2k\Gamma (2/k)}} {{x^{2/k}} \over {m_ 1 m_ 2}}+\\ +({\textstyle {1\over 2}} (m_ 1+ m_ 2)- l_ 1- l_ 2) {{x^{1/k}} \over {m_ 1 m_ 2}}+ \begin{cases} O(m_ 1^{1/k} m_ 2^{- 1} x^{1/k- 1/k^ 2}),\\ \Omega(m_ 1^{1/k} m_ 2^{-1} x^{1/k- 1/k^ 2}),\end{cases} \end{multlined} \] uniformly in \(1\leq l_ 1\leq m_ 1\), \(1\leq l_ 2\leq m_ 2\leq m_ 1\leq x^{1/k} (\log x)^{- 60}\). The proof is substantially based on \textit{M. N. Huxley}'s `Discrete Hardy- Littlewood method' [Exponential sums and lattice points. II, Proc. Lond. Math. Soc., III. Ser. 66, 279-301 (1993)] which, in the present refined form, provides the bound \[ \sum_{M<m\leq M'} \psi \bigl( {\textstyle {T\over M}} F\bigl( {\textstyle {m\over M}} \bigr)\bigr)\ll T^{23/73} (\log T)^{315/146} \tag{*} \] for real functions \(F\) whose first four derivatives satisfy certain nonvanishing conditions on \([1,2]\); \(\psi\) is the first (periodic) Bernoulli function and \(M'\leq 2M\). A considerable amount of difficulties arises from the fact that uniformity in the moduli \(m_ 1\), \(m_ 2\) is aimed at. This is carried out successfully because (*) is not only true for \(M\ll T^{1/2}\) (as was required at earlier stages of Huxley's research) but for a somewhat larger range of \(M\).
scientific article; zbMATH DE number 492585