Sums of almost equal prime cubes
It is known that every sufficiently large integer satisfying the natural congruence conditions is a sum of the cubes of nine primes; and that for sums of 5, 6, 7, or 8 cubes of primes ``almost all'' large integers satisfying the appropriate natural congruences can be represented. The present paper shows that in the latter case one can restrict the primes involved to be almost equal. To be specific, let \(\theta<1/45\) for \(r=5\), or \(<1/30\) for \(r=6\), or \(<1/25\) for \(r=7\), or \(<2/45\) for \(r=8\). Then if \(5\leq r\leq 8\), ``almost all'' numbers \(N\) satisfying the relevant congruence conditions can be written as \(p_1^3+\ldots+p_r^3\) with \[ | p_j-(N/r)^{1/3}| \leq N^{1/3-\theta}. \] The proof uses the circle method. On the minor arcs an estimate for exponential sums over primes in short intervals, due to \textit{J. Liu, G. Lü} and \textit{T. Zhan} [Sci. China, Ser. A 49, 611--619 (2006; Zbl 1106.11033)], plays an important role.
scientific article; zbMATH DE number 5848590
scientific article