On the number of summands in the asymptotic formula for the number of solutions of Waring's equation (Q1282538)

I. M. Vinogradov proved an asymptotic formula for the number of solutions of Waring's equation \(x_1^n +\cdots +x_k^n=N\) for \(n \geq 4\) and \(k \geq 2[n^2(2\log n + \log \log n +5)]\). In the present paper, the lower bound for \(k\) here is improved to \(k \geq 2[n^2(\log n +\log \log n +6)]\). The key result is a new bound for the number of solutions of Hardy's equation \(x_1^n +\cdots + x_k^n - y_1^n - \cdots - y_k^n=0\), where the variables run over a given interval \([0, P)\).