On Vinogradov's mean value theorem. II (Q1313103)

Let \(J_{s,k} (P)\) denote the number of solutions of \(\sum^ s_{i = 1} (x^ j_ i-y^ j_ i) = 0\) \((1 \leq j \leq k)\) with \(1 \leq x_ i, y_ i \leq P\). Bounds of the form \[ J_{rk,k} (P) \leq D (k,r)P^{2rk-{1 \over 2} k(k + 1) + {1 \over 2} k^ 2(1 - 1/k)^ r} \tag{*} \] are of importance in both additive and multiplicative number theory, and are known as Vinogradov's mean value theorem. In [Part I, Mathematika 39, 379-399 (1992; Zbl 0769.11036), Corrections ibid. 40, 152 (1993; Zbl 0782.11026)] the author has treated \(J_{s,k}(P)\) by means of an efficient differencing technique which is a novel point of view in this area. In this way the exponent of \(P\) could be reduced. Here the same idea is used with emphasis on the size of \(D(k,r)\); this is of interest in the theory of the Riemann zeta function. It is shown that \((*)\) holds with \[ D(k,r) = \exp (C_ 2 rk \log k) \] provided only that \(1 \leq r \leq k^ 2\) and \(P \geq \exp(C_ 1 k(1 - 1/k)^{-r}) \log k)\); here \(C_ 1\), \(C_ 2\) are certain absolute constants. The proof is surprisingly simple.