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

Kumchev, Angel V.

Mich. Math. J. 54, No. 2, 243-268 (2006).

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.

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

Keywords:

Weyl sums over primes; Waring-Goldbach problem

Citations:

Zbl 0751.11037; Zbl 1016.11046; Zbl 1007.11058; Zbl 0874.11052; Zbl 1142.11071

References:

[1]	R. C. Baker and G. Harman, On the distribution of \(\alpha p^k\) modulo one, Mathematika 38 (1991), 170–184. · Zbl 0751.11037
[2]	R. C. Baker, G. Harman, and J. Pintz, The exceptional set for Goldbach’s problem in short intervals, Sieve methods, exponential sums, and their applications in number theory (Cardiff, 1995), London Math. Soc. Lecture Note Ser., 237, pp. 1–54, Cambridge Univ. Press, 1997. · Zbl 0929.11042
[3]	G. Harman, Trigonometric sums over primes. I, Mathematika 28 (1981), 249–254. · Zbl 0465.10029
[4]	——, Trigonometric sums over primes. II, Glasgow Math. J. 24 (1983), 23–37. · Zbl 0504.10017
[5]	——, On averages of exponential sums over primes, Analytic number theory and Diophantine problems (Stillwater, 1984), Progr. Math., 70, pp. 237–246, Birkhäuser, Boston, 1987. · Zbl 0626.10035
[6]	——, On the distribution of \(\alpha p\) modulo one. II, Proc. London Math. Soc. (3) 72 (1996), 241–260. · Zbl 0874.11052
[7]	——, Trigonometric sums over primes. III, J. Théor. Nombres Bordeaux 15 (2003), 727–740. · Zbl 1165.11332
[8]	G. Harman and A. V. Kumchev, On sums of squares of primes, Math. Proc. Cambridge Philos. Soc. 140 (2006), 1–13. · Zbl 1114.11079
[9]	D. R. Heath-Brown, Prime numbers in short intervals and a generalized Vaughan identity, Canad. J. Math. 34 (1982), 1365–1377. · Zbl 0478.10024
[10]	L. K. Hua, Additive theory of prime numbers, Transl. Math. Monogr., 13, Amer. Math. Soc., Providence, RI, 1965. · Zbl 0192.39304
[11]	K. Kawada and T. D. Wooley, On the Waring–Goldbach problem for fourth and fifth powers, Proc. London Math. Soc. (3) 83 (2001), 1–50. · Zbl 1016.11046
[12]	A. V. Kumchev, The difference between consecutive primes in an arithmetic progression, Quart. J. Math. Oxford Ser. (2) 53 (2002), 479–501. · Zbl 1045.11061
[13]	——, On the Waring–Goldbach problem: Exceptional sets for sums of cubes and higher powers, Canad. J. Math. 57 (2005), 298–327. · Zbl 1080.11070
[14]	——, On the Waring–Goldbach problem for seventh powers, Proc. Amer. Math. Soc. 133 (2005), 2927–2937. · Zbl 1125.11055
[15]	Yu. V. Linnik, On the possibility of a unique method in certain problems of “additive” and “distributive” prime number theory, Dokl. Akad. Nauk SSSR 49 (1945), 3–7. · Zbl 0063.03587
[16]	J. Y. Liu and M. C. Liu, The exceptional set in the four prime squares problem, Illinois J. Math. 44 (2000), 272–293. · Zbl 0942.11044
[17]	J. Y. Liu and T. Zhan, Distribution of integers that are sums of three squares of primes, Acta Arith. 98 (2001), 207–228. · Zbl 0972.11099
[18]	——, The exceptional set in Hua’s theorem for three squares of primes, Acta Math. Sinica (N.S.) 21 (2005), 335–350. · Zbl 1142.11071
[19]	H. L. Montgomery, Topics in multiplicative number theory, Lecture Notes in Math., 227, Springer-Verlag, Berlin, 1971. · Zbl 0216.03501
[20]	H. L. Montgomery and R. C. Vaughan, The exceptional set in Goldbach’s problem, Acta Arith. 27 (1975), 353–370. · Zbl 0301.10043
[21]	X. M. Ren, The Waring–Goldbach problem for cubes, Acta Arith. 94 (2000), 287–301. · Zbl 0967.11041
[22]	R. C. Vaughan, On Goldbach’s problem, Acta Arith. 22 (1972), 21–48. · Zbl 0216.31603
[23]	——, Mean value theorems in prime number theory, J. London Math. Soc. (2) 10 (1975), 153–162. · Zbl 0314.10028
[24]	——, The Hardy–Littlewood method, 2nd ed., Cambridge Tracts in Math., 125, Cambridge Univ. Press, 1997. · Zbl 0868.11046
[25]	I. M. Vinogradov, Representation of an odd number as the sum of three primes, Dokl. Akad. Nauk SSSR 15 (1937), 291–294 (Russian). · Zbl 0016.29101
[26]	——, An introduction to the theory of numbers, 6th ed., Pergamon Press, London, 1955. · Zbl 0067.42403
[27]	——, Special variants of the method of trigonometric sums, Nauka, Moscow, 1976 (Russian). · Zbl 0429.10023
[28]	——, The method of trigonometric sums in number theory, 2nd ed., Nauka, Moscow, 1980 (Russian).
[29]	T. D. Wooley, Slim exceptional sets for sums of cubes, Canad. J. Math. 54 (2002), 417–448. · Zbl 1007.11058