On the Waring-Goldbach problem for fourth and fifth powers. (English) Zbl 1016.11046

Let \(H(k)\) be the smallest integer \(s\) such that every sufficiently large number satisfying the natural congruence restrictions is a sum of the \(k\)th powers of \(s\) primes. Then it is known, for example, that \(H(1)\leq 3\), \(H(2)\leq 5\), \(H(3)\leq 9\), \(H(4)\leq 15\) and \(H(5)\leq 23\). The goal of the present paper is to improve the last two results, so as to have \(H(4)\leq 14\) and \(H(5)\leq 22\).
The argument depends upon the circle method, with estimates for Weyl sums over primes playing a key rôle. In addition the authors use mean value estimates for ordinary Weyl sums over “diminishing ranges”, derived from work of K. Thanigasalam [Bull. Calcutta Math. Soc. 81, 279-294 (1989; Zbl 0641.10037)] and R. C. Vaughan [Proc. Lond. Math. Soc. (3) 52, 445-463 (1986; Zbl 0601.10035)]. These tools suffice for the treatment of 4th powers, but the result \(H(5)\leq 21\) requires more effort. Here the authors use Iwaniec’s linear sieve, and encounter bilinear and trilinear sums, for which appropriate mean value estimates have to be developed. In addition, the contribution from certain almost-primes has to be allowed for, and this is achieved via Chen’s reversal of rôles technique.


11P32 Goldbach-type theorems; other additive questions involving primes
11P05 Waring’s problem and variants
11L20 Sums over primes
11N36 Applications of sieve methods
11P55 Applications of the Hardy-Littlewood method
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