On the Waring-Goldbach problem for seventh powers. (English) Zbl 1125.11055

Let \(H(k)\) denote the least integer \(s\) such that the equation \[ p_1^k + \ldots +ps^k = n \] is soluble in primes \(p_1, \ldots, p_s\) for all but finitely many \(n\). Recently, K. Kawada and T. D. Wooley [Proc. Lond. Math. Soc. (3) 83, 1–50 (2001; Zbl 1016.11046)] proved that \(H(4) \leq 14\) and \(H(5) \leq 21\).
In this paper, the author proves
Theorem 1: \(H(7) \leq 46\).
The previous best bound was \(H(7) \leq 47\). Theorem 1 is deduced from
Theorem 2: Let \(23\leq s \leq 45\) and let \(E_s(x)\) denote the number of integers \(n\leq x\) with \(n\equiv s \pmod 2\) and such that \(n\) cannot be represented as a sum of \(s\) seventh powers of primes. Then \(E_{23} \ll x(\log x)^{-A}\) for any \(A>0\). When \(s\geq 24\), there exists an absolute constant \(\theta < 1\) such that \(E_s(x) \ll x^{\theta - (s-23)/672}\).
The main novelty in the argument is the estimate for \(E_{23}(x)\). The proof of this estimate uses some exponential sum estimates, due to the author (preprint), a result of K. Thanigasalam [Bull. Calcutta Math. Soc. 86, No. 2, 175–178 (1994; Zbl 0812.11055)], and a primitive form of the sieve method of G. Harman [Proc. Lond. Math. Soc. (3) 72, No. 2, 241–260 (1996; Zbl 0874.11052)].


11P32 Goldbach-type theorems; other additive questions involving primes
11L20 Sums over primes
11N36 Applications of sieve methods
11P05 Waring’s problem and variants
Full Text: DOI