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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.

Copyright 1999 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 05 (1999), pp. 128-135
• Publisher Identifier: S 1079-6762(99)00071-2
• 1991 Mathematics Subject Classification. Primary 11L07, 11N37
• Received by the editors June 22, 1998 and, in revised form, October 11, 1999
• Posted on October 29, 1999
• Communicated by Hugh Montgomery
• Comments (When Available)

Gennady Bachman

Department of Mathematical Sciences, University of Nevada, Las Vegas, 4505 Maryland Parkway, Las Vegas, Nevada 89154-4020

E-mail address: bachman@nevada.edu

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.