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.


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
Full Text: DOI


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