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Transactions of the American Mathematical Society

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Sums with the Möbius function twisted by characters with powerful moduli


Authors: William D. Banks and Igor E. Shparlinski
Journal: Trans. Amer. Math. Soc. 373 (2020), 249-272
MSC (2010): Primary 11L40; Secondary 11L26, 11M06, 11M20
DOI: https://doi.org/10.1090/tran/7914
Published electronically: September 23, 2019
MathSciNet review: 4042874
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Abstract: In a recent work, the authors (2016) have combined classical ideas of A. G. Postnikov (1956) and N. M. Korobov (1974) to derive improved bounds on short character sums for certain nonprincipal characters with powerful moduli. In the present paper, these results are used to bound sums with the Möbius function twisted by characters of the same type, which complements and improves some earlier work of B. Green (2012). To achieve this, we obtain a series of results about the size and zero-free region of $ L$-functions with the same class of moduli.


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Additional Information

William D. Banks
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: bankswd@missouri.edu

Igor E. Shparlinski
Affiliation: Department of Pure Mathematics, University of New South Wales, Sydney, New South Wales 2052, Australia
Email: igor.shparlinski@unsw.edu.au

DOI: https://doi.org/10.1090/tran/7914
Keywords: M{\"o}bius function, character sums, exponential sums
Received by editor(s): October 30, 2018
Received by editor(s) in revised form: May 12, 2019
Published electronically: September 23, 2019
Additional Notes: The first author was supported in part by a grant from the University of Missouri Research Board.
The second author was supported in part by Australian Research Council Grant DP170100786.
Article copyright: © Copyright 2019 American Mathematical Society