Exponential sums with multiplicative coefficients

Bachman, Gennady¹

¹Department of Mathematical Sciences, University of Nevada, Las Vegas, 4505 Maryland Parkway, Las Vegas, Nevada 89154-4020

Electronic research announcements of the American Mathematical Society, Tome 05 (1999), pp. 128-135

Résumé

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.

Détail

DOI : 10.1090/S1079-6762-99-00071-2

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