Abstract
We estimate weighted character sums with determinants ad − bc of 2 × 2 matrices modulo a prime p with entries a, b, c and d varying over the interval [1, N]. Our goal is to obtain non-trivial bounds for values of N as small as possible. In particular, we achieve this goal, with a power saving, for N ⩾ p1/8+ε with any fixed ε > 0, which is very likely to be the best possible unless the celebrated Burgess bound is improved. By other techniques, we also treat more general sums but sometimes for larger values of N.
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Acknowledgements
The authors are very grateful to Satadal Ganguly for posing the question, which has been the starting point of this work. The authors also thank him for his remarks concerning a previous version of this paper. This work started during a very enjoyable visit by the second author to Institut de Mathématiques de Jussieu whose hospitality and support are very much appreciated. During the preparation of this work the second author was also supported in part by the Australian Research Council (Grant No. DP200100355).
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On the 50th Anniversary of Chen’s Theorem
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Fouvry, É., Shparlinski, I.E. On character sums with determinants. Sci. China Math. 66, 2693–2714 (2023). https://doi.org/10.1007/s11425-022-2122-0
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DOI: https://doi.org/10.1007/s11425-022-2122-0