## 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.

### 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

### Keywords:

Weyl sums over primes; Waring-Goldbach problem
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### References:

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