Some applications of the double large sieve

Abstract

We sharpen a procedure of Cao and Zhai (J Theorie Nombres Bordeaux,11: 407–423, 1999) to estimate the sum $$\begin{aligned} \sum _{m\sim M} \sum _{n\sim N} a_m b_n \, e\left(\frac{F m^\alpha n^\beta }{M^\alpha N^\beta }\right) \end{aligned}$$ with \(|a_m|,\ |b_n| \le 1\). We apply this to give bounds for the discrepancy (mod 1) of the sequence \(\{p^c: p\le X\}\) where \(p\) is a prime variable, in the range \(\frac{130}{79}\le c \le \frac{11}{5}\). An alternative strategy is used for the range \(1.48 \le c \le \frac{130}{79}\). We use further exponential sum estimates to show that for large \(R>0\), and a small constant \(\eta >0\), the inequality $$\begin{aligned} \left| p_1^c+p_2^c+p_3^c+p_4^c+p_5^c - R\right| < R^{-\eta } \end{aligned}$$ holds for many prime tuples, provided \(2<c\le 2.041\). This improves work of Cao and Zhai (Monatsh Math, 150:173–179, 2007) and a theorem claimed by Shi and Liu (Monatsh Math, published online, 2012).