Exponential sums with multiplicative coefficients
Author:
Gennady Bachman
Journal:
Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 128-135
MSC (1991):
Primary 11L07, 11N37
DOI:
https://doi.org/10.1090/S1079-6762-99-00071-2
Published electronically:
October 29, 1999
MathSciNet review:
1716573
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Abstract: We provide estimates for the exponential sum \begin{equation*}F(x,\alpha )=\sum _{n\le x} f(n)e^{2\pi i\alpha n}, \end{equation*} where $x$ and $\alpha$ are real numbers and $f$ is a multiplicative function satisfying $|f|\le 1$. Our main focus is the class of functions $f$ which are supported on the positive proportion of primes up to $x$ in the sense that $\sum _{p\le x}|f(p)|/p\gg \log \log x$. For such $f$ we obtain rather sharp estimates for $F(x,\alpha )$ by extending earlier results of H. L. Montgomery and R. C. Vaughan. Our results provide a partial answer to a question posed by G. Tenenbaum concerning such estimates.
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- ---, On exponential sums with multiplicative coefficients, II, in preparation.
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- ---, Private communication.
- G. Bachman, On the coefficients of cyclotomic polynomials, Mem. Amer. Math. Soc. 510 (1993).
- ---, On exponential sums with multiplicative coefficients, Number theory (Halifax, NS, 1994), CMS Conf. Proc., 15, Amer. Math. Soc., Providence, RI, 1995, pp. 29–38.
- ---, On exponential sums with multiplicative coefficients, II, in preparation.
- ---, Some remarks on non-negative multiplicative functions on arithmetic progressions, J. Number Theory 73 (1998), 72–91.
- H. Daboussi, Fonctions multiplicatives presque périodiques B. D’après un travail commun avec Hubert Delange, Journées Arithmétique de Bordeaux (Conf. Univ. Bordeaux, 1974), Astérisque 24–25 (1975), 321–324.
- ---, On some exponential sums, Analytic number theory (Allerton Park, IL, 1989), Progr. Math., 85 (B. C. Berndt, H. G. Diamond, H. Halberstam and A. Hildebrand, eds.), Birkhäuser, Boston, MA, 1990, pp. 111–118.
- H. Daboussi and H. Delange, Quelques propriétés des fonctions multiplicatives de module au plus égal à 1, C.R. Acad. Sci. Paris Ser. A 278 (1974), 657–660.
- ---, On multiplicative arithmetic functions whose modulus does not exceed one, J. London Math. Soc. (2) 26 (1982), 245–264.
- H. Davenport, On some infinite series involving arithmetic functions II, Quart. J. Math. Oxford 8 (1937), 313–320.
- Y. Dupain, R. R. Hall et G. Tenenbaum, Sur l’équirépartition modulo 1 de certaines fonctions de diviseurs, J. London Math. Soc. (2) 26 (1982), 397–411.
- P. D. T. A. Elliott, Multiplicative functions on arithmetic progressions. VI. More middle moduli, J. Number Theory 44 (1993), 178-208.
- E. Fouvry et G. Tenenbaum, Entiers sans grand facteur premier en progressions arithmé- tiques, Proc. London. Math. Soc. 63 (1991), 449–494.
- H. Halberstam and H.-E. Richert, On a result of R. R. Hall, J. Number Theory 11 (1979), 76–89.
- R. R. Hall, Halving an estimate obtained from Selberg’s upper bound method, Acta Arith. 25 (1974), 347–351.
- A. Hildebrand, Multiplicative functions on arithmetic progressions, Proc. Amer. Math. Soc. 108 (1990), 307–318.
- A. Hildebrand and G. Tenenbaum, Integers without large prime factors, J. Théorie des Nombres de Bordeaux 5 (1993), 411–484.
- L. Goubin, Sommes d’exponentielles et principe de l’hyperbole, Acta Arith. 73 (1995), 303–324.
- H. L. Montgomery and R. C. Vaughan, The order of magnitude of the mth coefficients of cyclotomic polynomials, Glasgow Math. J. 27 (1985), 143–159.
- ---, Exponential sums with multiplicative coefficients, Invent. Math. 43 (1977), 69–82.
- P. Shiu, A Brun-Titchmarsh Theorem for multiplicative functions, J. reine angew. Math. 313 (1980), 161–170.
- G. Tenenbaum, Facteurs premiers de sommes d’entiers, Proc. Amer. Math. Soc. 106 (1989), 287–296.
- R.C. Vaughan, A new iterative method in Waring’s problem, Acta Math. 162 (1989), 1–71.
- ---, Private communication.
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Gennady Bachman
Affiliation:
Department of Mathematical Sciences, University of Nevada, Las Vegas, 4505 Maryland Parkway, Las Vegas, Nevada 89154-4020
Email:
bachman@nevada.edu
Received by editor(s):
June 22, 1998
Received by editor(s) in revised form:
October 11, 1999
Published electronically:
October 29, 1999
Additional Notes:
The author would like to thank Professors Andrew Granville and Gérald Tenenbaum for helpful discussions about various topics related to this project. He especially wishes to thank Professor Adolf Hildebrand for suggesting this problem in the first place, and for numerous discussions on this and related topics over the course of this project.
Communicated by:
Hugh Montgomery
Article copyright:
© Copyright 1999
American Mathematical Society
