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Some applications of the double large sieve

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Abstract

We sharpen a procedure of Cao and Zhai (J Théorie 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).

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Acknowledgments

This work was partially supported by a grant from NSA.

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Authors and Affiliations

  1. Department of Mathematics, Brigham Young University, Provo, UT, 84602, USA

    Roger Baker

  2. Department of Mathematics, Southern Utah University, Cedar City, UT, 84720, USA

    Andreas Weingartner

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  1. Roger Baker
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  2. Andreas Weingartner
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Correspondence to Andreas Weingartner.

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Communicated by J. Schoißengeier.

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Baker, R., Weingartner, A. Some applications of the double large sieve. Monatsh Math 170, 261–304 (2013). https://doi.org/10.1007/s00605-012-0447-0

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  • DOI: https://doi.org/10.1007/s00605-012-0447-0

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