## 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

### Citations:

Zbl 0465.10029; Zbl 0504.10017
Full Text:

### References:

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