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**On the Waring-Goldbach problem: exceptional sets of sums of cubes and higher powers.**
*(English)*
Zbl 1080.11070

This paper treats estimation for densities of sets of natural numbers that are not represented by sums of powers of primes, and is mainly devoted to the case of cubes.

Let \({\mathcal N}_5\) be the set of odd natural numbers \(n\) satisfying \(n\not\equiv0\), \(\pm2\) (mod 9) and \(n\not\equiv0\) (mod 7), \({\mathcal N}_6\) be the set of even natural numbers \(n\) with \(n\not\equiv\pm1\) (mod 9), \({\mathcal N}_7\) be the set of odd natural numbers \(n\) with \(n\not\equiv0\) (mod 9), and let \({\mathcal N}_8\) be the set of even natural numbers. And, for \(5\leq s\leq8\), define \(E_s(x)\) as the number of \(n\in{\mathcal N}_s\) with \(n\leq x\) such that \(n\) cannot be written as the sum of \(s\) cubes of primes. One may account the definitions of the sets \({\mathcal N}_s\) natural, in view of the fact that all the sums of \(s\) cubes of primes greater than 7 belong to \({\mathcal N}_s\). In this paper, the bounds \(E_s(x)\ll x^{\theta_s}\) are established with \(\theta_5=79/84\), \(\theta_6=31/35\), \(\theta_7=17/28\), and \(\theta_8=23/84\). These bounds supersede the corresponding results of T. D. Wooley [“Slim exceptional sets for sums of cubes”, Can. J. Math. 54, 417–448 (2002; Zbl 1007.11058)].

As the author points out, the methods of this paper can provide minutely smaller values for \(\theta_s\) than the above ones. The best values of \(\theta_s\) attained in this paper are given as solutions of certain equations involving multiple integrals, and perhaps unable to be expressed by elementary functions only.

The proof is based on the Hardy-Littlewood method. Minor arc integrals are handled by applying the method of the aforementioned work of Wooley, and the author takes advantage of the new estimates for relevant exponential sums established in his recent paper [“On Weyl sums over primes and almost primes”, to appear], combining with a skilful use of the ideas of sieve methods.

In addition to the above results, the paper includes several conclusions concerning sums of \(k\)th powers of primes for \(4\leq k\leq10\).

Let \({\mathcal N}_5\) be the set of odd natural numbers \(n\) satisfying \(n\not\equiv0\), \(\pm2\) (mod 9) and \(n\not\equiv0\) (mod 7), \({\mathcal N}_6\) be the set of even natural numbers \(n\) with \(n\not\equiv\pm1\) (mod 9), \({\mathcal N}_7\) be the set of odd natural numbers \(n\) with \(n\not\equiv0\) (mod 9), and let \({\mathcal N}_8\) be the set of even natural numbers. And, for \(5\leq s\leq8\), define \(E_s(x)\) as the number of \(n\in{\mathcal N}_s\) with \(n\leq x\) such that \(n\) cannot be written as the sum of \(s\) cubes of primes. One may account the definitions of the sets \({\mathcal N}_s\) natural, in view of the fact that all the sums of \(s\) cubes of primes greater than 7 belong to \({\mathcal N}_s\). In this paper, the bounds \(E_s(x)\ll x^{\theta_s}\) are established with \(\theta_5=79/84\), \(\theta_6=31/35\), \(\theta_7=17/28\), and \(\theta_8=23/84\). These bounds supersede the corresponding results of T. D. Wooley [“Slim exceptional sets for sums of cubes”, Can. J. Math. 54, 417–448 (2002; Zbl 1007.11058)].

As the author points out, the methods of this paper can provide minutely smaller values for \(\theta_s\) than the above ones. The best values of \(\theta_s\) attained in this paper are given as solutions of certain equations involving multiple integrals, and perhaps unable to be expressed by elementary functions only.

The proof is based on the Hardy-Littlewood method. Minor arc integrals are handled by applying the method of the aforementioned work of Wooley, and the author takes advantage of the new estimates for relevant exponential sums established in his recent paper [“On Weyl sums over primes and almost primes”, to appear], combining with a skilful use of the ideas of sieve methods.

In addition to the above results, the paper includes several conclusions concerning sums of \(k\)th powers of primes for \(4\leq k\leq10\).

Reviewer: Koichi Kawada (Morioka)

### MSC:

11P32 | Goldbach-type theorems; other additive questions involving primes |

11P05 | Waring’s problem and variants |

11P55 | Applications of the Hardy-Littlewood method |

11N36 | Applications of sieve methods |