Sums with the Möbius function twisted by characters with powerful moduli
HTML articles powered by AMS MathViewer
- by William D. Banks and Igor E. Shparlinski PDF
- Trans. Amer. Math. Soc. 373 (2020), 249-272 Request permission
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.References
- W. D. Banks and I. E. Shparlinski, Bounds on short character sums and $L$-functions for characters with a smooth modulus, J. d’Analyse Math., to appear (available from http://arxiv.org/abs/1605.07553).
- J. Bourgain, Möbius-Walsh correlation bounds and an estimate of Mauduit and Rivat, J. Anal. Math. 119 (2013), 147–163. MR 3043150, DOI 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 304327, DOI 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, DOI 10.1017/S0963548312000284
- Glyn Harman and Imre Kátai, Primes with preassigned digits. II, Acta Arith. 133 (2008), no. 2, 171–184. MR 2417463, DOI 10.4064/aa133-2-5
- D. R. Heath-Brown, Hybrid bounds for Dirichlet $L$-functions. II, Quart. J. Math. Oxford Ser. (2) 31 (1980), no. 122, 157–167. MR 576334, DOI 10.1093/qmath/31.2.157
- Ghaith A. Hiary, An explicit hybrid estimate for $L(1/2+it,\chi )$, Acta Arith. 176 (2016), no. 3, 211–239. MR 3580112, DOI 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 602191, DOI 10.4064/aa-38-3-209-254
- H. Iwaniec, On zeros of Dirichlet’s $L$ series, Invent. Math. 23 (1974), 97–104. MR 344207, DOI 10.1007/BF01405163
- Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214, DOI 10.1090/coll/053
- 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, DOI 10.1145/174130.174138
- Hugh L. Montgomery and Robert C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007. MR 2378655
- 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 $L$-series with the character modulus equal to the power of a prime number, J. Indian Math. Soc. (N.S.) 20 (1956), 217–226. MR 84010
- 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, DOI 10.1515/CRELLE.2009.044
- Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Mathematische Forschungsberichte, XV, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963 (German). MR 0220685
Additional Information
- William D. Banks
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 336964
- Email: bankswd@missouri.edu
- Igor E. Shparlinski
- Affiliation: Department of Pure Mathematics, University of New South Wales, Sydney, New South Wales 2052, Australia
- MR Author ID: 192194
- Email: igor.shparlinski@unsw.edu.au
- 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. - © Copyright 2019 American Mathematical Society
- 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
- MathSciNet review: 4042874