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Trigonometric sums over primes. III. (English) Zbl 1165.11332

From the introduction: Throughout this paper we write \(e(x)= \exp(2\pi ix)\) and let \(p\) denote a prime variable. Sums of the form \(\sum_{p\leq P}e(\alpha p^k)\) arise in applications of the Hardy-Littlewood circle method to the Waring-Goldbach problem, or in problems involving the distribution of \(\alpha p^k\) modulo one. The aim of this paper is to give improved bounds for these sums when \(k\geq 5\) (although Theorem 3 does give a better bound in certain ranges than any previously published explicit result for \(k=4\) as well). The first unconditional bounds for these sums were given by Vinogradov. The case \(k=1\) was required for his celebrated proof of the ternary Goldbach problem. In 1981 the author [Mathematika 28, 249–254 (1981; Zbl 0465.10029)] showed that, if
\[ |q\alpha-a|< q^{-1}, \quad q\geq 1,\quad (a,q)=1, \tag \(*\) \]
then, for \(k\geq 2\),
\[ \bigg|\sum_{p\leq P}e(\alpha p^k)\bigg|\ll P^{1+\varepsilon} \bigg(\frac1q+ \frac{1}{P^{1/2}}+ \frac{q}{P^k}\bigg)^\gamma, \]
with \(\gamma= 4^{1-k}\), \(\varepsilon>0\), and constants implied by the \(\ll\) notation depend at most on \(k\) and \(\varepsilon\). This improved the value \(\gamma= (4^{k+1}(k+1))^{-1}\) obtained by Vinogradov, and generalised the case \(k=2\) given by Ghosh. Improved bounds for \(P^{1-1/k}\leq q\leq P^{k/2}\), \(|\alpha q-a|< P^{-k/2}\) were given in [Glasg. Math. J. 24, 23–37 (1983; Zbl 0504.10017)].
Our main result is as follows.
Theorem 1. Let \(k\in\mathbb N\), \(k\geq 5\), \(\varepsilon>0\) and let \(P,H\in\mathbb R\), \(P\geq 2\),
\[ 2P^{k\sigma(k)}\leq H\leq P^{k/2+2k\sigma(k)}. \]
Suppose that \(\alpha\) is a real number, and there exist integers \(a\), \(q\) with
\[ |q\alpha-a|<H^{-1}, \quad 1\leq q\leq H, \quad (a,q)=1. \]
Then
\[ \bigg|\sum_{p\sim P}e(\alpha p^k)\bigg|\ll_{k,\varepsilon} P^{1-\sigma(k)+ \varepsilon}+ \frac{q^\varepsilon w_k(q)^{1/2} P(\log P)^{20}} {(1+P^k|\alpha-a/q|)^{1/2}}. \]
The restriction to \(k\geq 5\) is caused by parts of the proof which require \(\sigma(k)\leq (8k+2)^{-1}\). For larger \(k\) one should be able to obtain an exponent \(\sigma(k)\simeq (4.5k^2\log k)^{-1}\).
Theorem 2. Suppose that \(k\geq 5\), \(P\geq 2\), \(\varepsilon>0\) and \(\alpha\in\mathbb R\) such that \((*)\) holds. Then
\[ \bigg|\sum_{p\sim P}e(\alpha p^k)\bigg|\ll_{k,\varepsilon} P^{1+\varepsilon- \sigma(k)}+ P(\log P)^{17} \big( q^{-\tau_1(k)+ \varepsilon}+ (q/P^k)^{\tau_2(k)-\varepsilon}\big). \]
Here
\[ \tau_1(k)= \frac16+\frac{1}{3k}, \qquad \tau_2(k)= \frac18+ \frac{4}{8k-2}. \]
Moreover, the bound remains valid for \(k=4\) with \(\sigma(k)\) replaced by \(\frac{1}{32}\).

MSC:

11L20 Sums over primes
11L07 Estimates on exponential sums
11L15 Weyl sums
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References:

[1] Baker, R.C., Harman, G., On the distribution of αpk modulo one. Mathematika38 (1991), 170-184. · Zbl 0751.11037
[2] Fouvry, E., Michel, P., Sur certaines sommes d’exponentielles sur les nombres premiers. Ann. Sci. Ec. Norm. Sup. IV Ser.31 (1998), 93-130. · Zbl 0915.11045
[3] Ghosh, A., The distribution of αp2 modulo one. Proc. London Math. Soc. (3) 42 (1981), 252-269. · Zbl 0447.10035
[4] Harman, G., Trigonometric sums over primes I. Mathematika28 (1981), 249-254. · Zbl 0465.10029
[5] Harman, G., Trigonometric sums over primes II. Glasgow Math. J.24 (1983), 23-37. · Zbl 0504.10017
[6] Heath-Brown, D.R., Prime numbers in short intervals and a generalized Vaughan identity. Can. J. Math.34 (1982), 1365-1377. · Zbl 0478.10024
[7] Kawada, K., Wooley, T.D., On the Waring-Goldbach problem for fourth and fifth powers. Proc. London Math. Soc. (3) 83 (2001), 1-50. · Zbl 1016.11046
[8] Vaughan, R.C., Mean value theorems in prime number theory. J. London Math. Soc. (2) 10 (1975), 153-162. · Zbl 0314.10028
[9] Vaughan, R.C., The Hardy-Littlewood Method second edition. Cambridge University Press, 1997. · Zbl 0868.11046
[10] Vinogradov, I.M., Some theorems concerning the theory of primes. Rec. Math. Moscow N.S. 2 (1937), 179-194. · JFM 63.0131.05
[11] Vinogradov, I.M., A new estimation of a trigonometric sum containing primes. Bull. Acad. Sci. URSS Ser. Math.2 (1938), 1-13. · JFM 64.0983.01
[12] Wong, K.C., On the distribution of αpk modulo one. Glasgow Math. J.39 (1997), 121-130. · Zbl 0880.11052
[13] Wooley, T.D., New estimates for Weyl sums. Quart. J. Math. Oxford (2) 46 (1995), 119-127. · Zbl 0855.11043
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