Gao, Peng; Zhao, Liangyi The large sieve with power moduli in imaginary quadratic number fields. (English) Zbl 1500.11070 Int. J. Number Theory 18, No. 8, 1713-1733 (2022). Let \(K\) be any imaginary quadratic number field, \(D_K\) denotes the discriminant of \(K\), \(\mathcal{O}_K\) denotes the ring of integers in \(K\), \(N(q)\) and \(\mathrm{Tr}(q)\) denote the norm and the trace, respectively, of \(q\in\mathcal{O}_K\), and \(\widetilde{e}_K(z)=\exp(\mathrm{Tr}(z/\sqrt{D_K}))\). Also, let \(k\geq 1\) be an integer, \(\kappa=2^{k-1}\), \(N\geq 1\) and \((a_n)_{n\in\mathcal{O}_K}\) be any sequence of complex numbers. In the paper under review, the authors obtain the following large sieve inequality \[ \sum_{\substack{q\in\mathcal{O}_K\setminus\{0\}\\ N(q)\leq Q}}\sum_{\substack{r\,\mathrm{mod}\,q^k\\ (r,q)=1}}\left|\sum_{\substack{n\in\mathcal{O}_K\\ N(n)\leq N}}a_n\,\widetilde{e}_K\left(\frac{nr}{q^k}\right)\right|^2\ll_{\varepsilon,k}C_{Q,N,\varepsilon,k}\sum_{\substack{n\in\mathcal{O}_K\\ N(n)\leq N}}|a_n|^2, \] where \(C_{Q,N,\varepsilon,k}=(QN)^\varepsilon(Q^{k+1}+NQ^{1-1/\kappa}+N^{1-1/\kappa}Q^{1+k/\kappa})\). The authors improve on the above result for the case of all imaginary quadratic number fields of class number one. Reviewer: Mehdi Hassani (Zanjan) MSC: 11N35 Sieves 11L40 Estimates on character sums Keywords:large sieve; number fields; power moduli; prime moduli Citations:Zbl 1446.11152; Zbl 1461.11113 PDFBibTeX XMLCite \textit{P. Gao} and \textit{L. Zhao}, Int. J. Number Theory 18, No. 8, 1713--1733 (2022; Zbl 1500.11070) Full Text: DOI arXiv References: [1] Baier, S., On the large sieve with sparse sets of moduli, J. Ramanujan Math. Soc.21 (2006) 279-295. · Zbl 1152.11041 [2] Baier, S. and Bansal, A., The large sieve with power moduli for \(\Bbb Z[i]\), Int. J. Number Theory14(10) (2018) 2737-2756. · Zbl 1446.11152 [3] Baier, S. and Bansal, A., Large sieve with sparse sets of moduli for \(\Bbb Z[i]\), Acta Arith.196(1) (2020) 17-34. · Zbl 1461.11113 [4] Baier, S., Lynch, S. B. and Zhao, L., Elliptic curves in isogeny classes, Bull. Aust. Math. Soc.100(2) (2019) 225-229. · Zbl 1457.11130 [5] Baier, S. and Zhao, L., Large sieve inequality with characters for powerful moduli, Int. J. Number Theory1(2) (2005) 265-279. · Zbl 1083.11060 [6] Baier, S. and Zhao, L., An improvement for the large sieve for square moduli, J. Number Theory128(1) (2008) 154-174. · Zbl 1225.11121 [7] Dodson, M. M. and Kristensen, S., Hausdorff dimension and diophantine approximation, in Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot, Part 1, , Vol. 1 (American Mathematical Society, Providence, RI, 2004), 305-347. · Zbl 1196.11104 [8] Gao, P. and Zhao, L., One level density of low-lying zeros of families of L-functions, Compos. Math.147(1) (2011) 1-18. · Zbl 1230.11111 [9] Gao, P. and Zhao, L., Large sieve inequalities for quartic character sums, Q. J. Math.63 (2012) 891-917. · Zbl 1282.11104 [10] Halupczok, K., A new bound for the large sieve inequality with power moduli, Int. J. Number Theory8 (2012) 689-695. · Zbl 1270.11096 [11] Huxley, M. N., The large sieve inequality for algebraic number fields, Mathematika15 (1968) 178-187. · Zbl 0174.08201 [12] Iwaniec, H. and Kowalski, E., Analytic Number Theory, , Vol. 53 (American Mathematical Society, Providence, 2004). · Zbl 1059.11001 [13] Linnik, J. V., The large sieve, Doklady Akad. Nauk SSSR36 (1941) 119-120. · Zbl 0024.29302 [14] Parshin, A. N. and Shafarevich, I. R., Number Theory IV: Transcendental Numbers, , Vol. 44 (Springer, Berlin, 1998). · Zbl 1155.00324 [15] Shparlinski, I. E. and Zhao, L., Elliptic curves in isogeny classes, J. Number Theory191 (2018) 194-212. · Zbl 1444.11122 [16] Wolke, D., On the large sieve with primes, Acta Math. Acad. Sci. Hungar.22 (1971/1972) 239-247. · Zbl 0231.10027 [17] Zhao, L., Large sieve inequality for characters to square moduli, Acta Arith.112 (2004) 297-308. · Zbl 1060.11055 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.