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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 | References | Similar Articles | Additional Information

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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Additional Information

Gennady Bachman
Affiliation: Department of Mathematical Sciences, University of Nevada, Las Vegas, 4505 Maryland Parkway, Las Vegas, Nevada 89154-4020
Email: bachman@nevada.edu

DOI: https://doi.org/10.1090/S1079-6762-99-00071-2
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

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