Sums with the Möbius function twisted by characters with powerful moduli
Authors: William D. Banks and Igor E. Shparlinski
MSC (2010): Primary 11L40; Secondary 11L26, 11M06, 11M20
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 -functions with the same class of moduli.
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J. Bourgain, Möbius-Walsh correlation bounds and an estimate of Mauduit and Rivat, J. Anal. Math. 119 (2013), 147-163. MR 3043150, https://doi.org/10.1007/s11854-013-0005-2
S. Chowla, The Riemann hypothesis and Hilbert's tenth problem, Mathematics and Its Applications, Vol. 4, Gordon and Breach Science Publishers, New York-London-Paris, 1965. MR 0177943
P. X. Gallagher, Primes in progressions to prime-power modulus, Invent. Math. 16 (1972), 191-201. MR 0304327, https://doi.org/10.1007/BF01425492
Ben Green, On (not) computing the Möbius function using bounded depth circuits, Combin. Probab. Comput. 21 (2012), no. 6, 942-951. MR 2981162, https://doi.org/10.1017/S0963548312000284
Glyn Harman and Imre Kátai, Primes with preassigned digits. II, Acta Arith. 133 (2008), no. 2, 171-184. MR 2417463, https://doi.org/10.4064/aa133-2-5
D. R. Heath-Brown, Hybrid bounds for Dirichlet -functions. II, Quart. J. Math. Oxford Ser. (2) 31 (1980), no. 122, 157-167. MR 0576334, https://doi.org/10.1093/qmath/31.2.157
Ghaith A. Hiary, An explicit hybrid estimate for , Acta Arith. 176 (2016), no. 3, 211-239. MR 3580112, https://doi.org/10.4064/aa8433-7-2016
Jürgen G. Hinz, Eine Erweiterung des nullstellenfreien Bereiches der Heckeschen Zetafunktion und Primideale in Idealklassen, Acta Arith. 38 (1980/81), no. 3, 209-254 (German). MR 0602191, https://doi.org/10.4064/aa-38-3-209-254
H. Iwaniec, On zeros of Dirichlet's series, Invent. Math. 23 (1974), 97-104. MR 0344207, https://doi.org/10.1007/BF01405163
N. M. Korobov, The distribution of digits in periodic fractions, Mat. Sb. (N.S.) 89(131) (1972), 654-670, 672 (Russian). MR 0424660
Nathan Linial, Yishay Mansour, and Noam Nisan, Constant depth circuits, Fourier transform, and learnability, J. Assoc. Comput. Mach. 40 (1993), no. 3, 607-620. MR 1370363, https://doi.org/10.1145/174130.174138
A. G. Postnikov, On the sum of characters with respect to a modulus equal to a power of a prime number, Izv. Akad. Nauk SSSR. Ser. Mat. 19 (1955), 11-16 (Russian). MR 0068575
A. G. Postnikov, On Dirichlet -series with the character modulus equal to the power of a prime number, J. Indian Math. Soc. (N.S.) 20 (1956), 217-226. MR 0084010
Peter Sarnak, Mobius randomness and dynamics, Not. S. Afr. Math. Soc. 43 (2012), no. 2, 89-97. MR 3014544
K. Soundararajan, Partial sums of the Möbius function, J. Reine Angew. Math. 631 (2009), 141-152. MR 2542220, https://doi.org/10.1515/CRELLE.2009.044
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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
Keywords: M{\"o}bius function, character sums, exponential sums
Received by editor(s): October 30, 2018
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.