##
**On Weyl sums over primes and almost primes.**
*(English)*
Zbl 1137.11054

In the paper under review, new estimates are established for exponential sums over primes of the form
\[
f(\alpha)=\sum\limits_{P\leq p<2P} e(\alpha p^k),
\]
where \(\alpha\) is a real number and \(k\) is a positive integer. These new estimates improve bounds of R. C. Baker and G. Harman [Mathematika 38, No. 1, 170–184 (1991; Zbl 0751.11037)] and K. Kawada and T. D. Wooley [Proc. Lond. Math. Soc. (3) 83, No. 1, 1–50 (2001; Zbl 1016.11046)] for \(f(\alpha)\). Clearly, all of these bounds depend on the Diophantine properties of \(\alpha\).

In particular, the author proves the following result (Theorem 3): Let \(k\geq 2\), \(\rho(2)=1/8\), \(\rho(3)=1/14\), and \(\rho(k)=\frac{2}{3}\cdot 2^{-k}\) if \(k\geq 4\). Further let \(Q=P^{3/2}\) if \(k=2\), and let \(Q=P^{(k^2-2k\rho(k))/(2k-1)}\) if \(k\geq 3\). If \(\alpha\) satisfies the inequality \(|q\alpha-a|<Q^{-1}\) for some integers \(a\), \(q\) with \(1\leq q\leq Q\) and \((a,q)=1\), then, for any fixed \(\varepsilon>0\), \[ f(\alpha)\ll P^{1-\rho(k)+\varepsilon}+\frac{q^{\varepsilon}P(\log P)^c}{(q+P^k|q\alpha-a|)^{1/2}}, \] where \(c>0\) is an absolute constant, and the implied \(\ll\)-constant depends at most on \(k\) and \(\varepsilon\).

The above bound for \(f(\alpha)\) is then employed to make progress on several questions related to the Waring-Goldbach problem. To be more precise, the author obtains new estimates for cardinalities of exceptional sets for sums of \(k\)-powers of primes. In particular, the author sharpens a result by J. Liu and T. Zhan [Acta Math. Sin., Engl. Ser. 21, No. 2, 335–350 (2005; Zbl 1142.11071)] on sums of three squares of primes and a result by T. D. Wooley [Can. J. Math. 54, No. 2, 417–448 (2002; Zbl 1007.11058)] on sums of cubes of primes.

Estimates for exceptional sets of the type described above are usually obtained by employing the Hardy-Littlewood circle method. As another result, the author significantly extends the standard set of major arcs appearing in the circle method when applied to the Waring-Goldbach problem.

The author’s method uses a sieve argument due to G. Harman [Proc. Lond. Math. Soc. (3) 72, No. 2, 241–260 (1996; Zbl 0874.11052)] to reduce the sums over primes to multilinear Weyl sums. The latter sums are then estimated in different ways. In particular, the author uses techniques from the above-mentioned works of Kawada-Wooley and Wooley, and bounds obtained from the large sieve. As a by-product, the author obtains estimates for Weyl sums over almost primes free of small prime divisors which are of independent interest.

In particular, the author proves the following result (Theorem 3): Let \(k\geq 2\), \(\rho(2)=1/8\), \(\rho(3)=1/14\), and \(\rho(k)=\frac{2}{3}\cdot 2^{-k}\) if \(k\geq 4\). Further let \(Q=P^{3/2}\) if \(k=2\), and let \(Q=P^{(k^2-2k\rho(k))/(2k-1)}\) if \(k\geq 3\). If \(\alpha\) satisfies the inequality \(|q\alpha-a|<Q^{-1}\) for some integers \(a\), \(q\) with \(1\leq q\leq Q\) and \((a,q)=1\), then, for any fixed \(\varepsilon>0\), \[ f(\alpha)\ll P^{1-\rho(k)+\varepsilon}+\frac{q^{\varepsilon}P(\log P)^c}{(q+P^k|q\alpha-a|)^{1/2}}, \] where \(c>0\) is an absolute constant, and the implied \(\ll\)-constant depends at most on \(k\) and \(\varepsilon\).

The above bound for \(f(\alpha)\) is then employed to make progress on several questions related to the Waring-Goldbach problem. To be more precise, the author obtains new estimates for cardinalities of exceptional sets for sums of \(k\)-powers of primes. In particular, the author sharpens a result by J. Liu and T. Zhan [Acta Math. Sin., Engl. Ser. 21, No. 2, 335–350 (2005; Zbl 1142.11071)] on sums of three squares of primes and a result by T. D. Wooley [Can. J. Math. 54, No. 2, 417–448 (2002; Zbl 1007.11058)] on sums of cubes of primes.

Estimates for exceptional sets of the type described above are usually obtained by employing the Hardy-Littlewood circle method. As another result, the author significantly extends the standard set of major arcs appearing in the circle method when applied to the Waring-Goldbach problem.

The author’s method uses a sieve argument due to G. Harman [Proc. Lond. Math. Soc. (3) 72, No. 2, 241–260 (1996; Zbl 0874.11052)] to reduce the sums over primes to multilinear Weyl sums. The latter sums are then estimated in different ways. In particular, the author uses techniques from the above-mentioned works of Kawada-Wooley and Wooley, and bounds obtained from the large sieve. As a by-product, the author obtains estimates for Weyl sums over almost primes free of small prime divisors which are of independent interest.

Reviewer: Stephan Baier (Bremen)

### MSC:

11L20 | Sums over primes |

11L15 | Weyl sums |

11N36 | Applications of sieve methods |

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

11P55 | Applications of the Hardy-Littlewood method |

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\textit{A. V. Kumchev}, Mich. Math. J. 54, No. 2, 243--268 (2006; Zbl 1137.11054)

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### References:

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