These pages are not updated anymore. They reflect the state of . For the current production of this journal, please refer to http://www.math.psu.edu/era/.
%_ ************************************************************************** %_ * The TeX source for AMS journal articles is the publishers TeX code * %_ * which may contain special commands defined for the AMS production * %_ * environment. Therefore, it may not be possible to process these files * %_ * through TeX without errors. To display a typeset version of a journal * %_ * article easily, we suggest that you either view the HTML version or * %_ * retrieve the article in DVI, PostScript, or PDF format. * %_ ************************************************************************** % Author Package %% Translation via Omnimark script a2l, October 15, 1999 (all in one %day!) \controldates{28-OCT-1999,28-OCT-1999,28-OCT-1999,29-OCT-1999} \documentclass{era-l} \issueinfo{5}{18}{}{1999} \dateposted{October 29, 1999} \pagespan{128}{135} \PII{S 1079-6762(99)00071-2} \def\copyrightyear{1999} %% Declarations: \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{corollary}{Corollary} \newtheorem{conjecture}{Conjecture} %% User definitions: \newcommand{\q}{\quad } \newcommand{\qq}{\qquad } \newcommand{\ff}{\frac } \newcommand{\mya}{\alpha } \newcommand{\ee}{\epsilon } \newcommand{\la}{\lambda } \newcommand{\vfi}{\varphi } \newcommand{\ffi}{\vfi (q)} \newcommand{\EF}{F(x,\alpha )} \newcommand{\xol}{\frac{x}{\log x}} \newcommand{\myP}{\mathcal{P}} \newcommand{\F}{\mathcal{F}} \newcommand{\G}{\mathcal{G}} \newcommand{\sumn}{\sum _{n\le x}} \newcommand{\sump}{\sum _{p\le x}} \begin{document} \title[Exponential sums with multiplicative coefficients]{Exponential sums\\ with multiplicative coefficients} \author{Gennady Bachman} \address{Department of Mathematical Sciences, University of Nevada, Las Vegas, 4505 Maryland Parkway, Las Vegas, Nevada 89154-4020} \email{bachman@nevada.edu} %\issueinfo{5}{1}{}{1999} %\copyrightinfo{1999}{American Mathematical Society} \commby{Hugh Montgomery} \subjclass{Primary 11L07, 11N37} \thanks{The author would like to thank Professors Andrew Granville and G\'{e}rald 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.} \date{June 22, 1998 and, in revised form, October 11, 1999} \begin{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. \end{abstract} \maketitle Estimating exponential sums \begin{equation*}\EF =\sumn f(n)e(\mya n)\qq (x\ge 3,\q \mya \in \mathbb{R}), \end{equation*} where $f$ is a multiplicative function and $e(t)$ stands for $e^{2\pi it}$, is an interesting problem which has received considerable attention in analytic number theory. Roughly, the research on this topic is split into two categories. On the one hand, there are estimates for $\EF $ concerned with the particular choices of the function $f$. On the other hand, there are estimates for $\EF $ valid uniformly for all $f$ belonging to some class of multiplicative functions. The main purpose of this note is to announce some new results of the latter kind. We begin, however, by surveying some known estimates for $\EF $. The simplest instance of our problem is the case when the function $f$ is identically equal to 1. In this case $\EF $ is a partial sum of a geometric series and we immediately get the bound \begin{equation*}\EF \ll \min \left (x,\ff 1{|e(\mya )-1|}\right )\ll \min \left (x,\ff 1{\|\mya \|}\right ), \end{equation*} where $\|\mya \|$ denotes the distance from $\mya $ to the nearest integer. It is also immediate that this upper bound is best possible, and so this case is, indeed, trivial. Any other ``natural'' choice of the function $f$ leads to non-trivial considerations. Perhaps the most thoroughly studied cases are when the function $f$ is closely related to the M\"{o}bius function $\mu $ or is the characteristic function of smooth numbers. A classic result of H. Davenport \cite{Da} states that \begin{equation*}\sumn \mu (n)e(\mya n)\ll _{A}\ff x{(\log x)^{A}}, \end{equation*} for any real number $A$. More recently this result has been extended as follows. Let $\myP $ denote a set of primes and let $u_{\myP }$ be the characteristic function of the set of natural numbers composed entirely of prime factors from ${\myP }$, i.e., $u_{\myP }$ is the completely multiplicative function whose values on primes is given by $u_{\myP }(p)=1$, if $p\in {\myP }$, and 0, otherwise. H. L. Montgomery and R. C. Vaughan \cite{MV1} have shown that \begin{equation*} \max _{{\myP },\mya }\left |\sumn (u_{\myP }\mu )(n)e(\mya n)\right |\asymp \ff {x}{\sqrt {\log x}}. \end{equation*} This was later strengthened by the author \cite{Ba1} to an asymptotic estimate \begin{equation*} \max _{{\myP },\mya }\left |\sumn (u_{\myP }\mu )(n)e(\mya n)\right |= B\ff {x}{\sqrt {\log x}}\left (1+O\left (\ff {\log \log x}{\sqrt {\log x}}\right )\right ), \end{equation*} for some constant $B>0$. The study of exponential sums over smooth numbers \begin{equation*}E(x,y;\mya )=\sumn u_{\myP }(n)e(\mya n)\qq \left ({\myP }={\myP }_{y}=\{p\le y\}\right ),\end{equation*} was initiated by Vaughan \cite{Va1} (as a special case of a more general exponential sum). Recently E. Fouvry and G. Tenenbaum \cite{FT} obtained sharp estimates for $E(x,y;\mya )$. In particular, they established the following bound. Let real numbers $\delta >0$ and $C>0$ be fixed, and let $x\ge 3$ and $y$ satisfy $\displaystyle x^{\delta \log _{3}x/\log _{2}x}\le y\le x$, where $\log _{k}x$, $k=2,3$, denotes the $k$-th iterate of the logarithm function. Then there exists a constant $D=D(\delta , C)$ such that if $Q=x(\log y)^{-D}$ and if $a$ and $q$ are coprime integers satisfying $2\le q\le Q$ and $|\mya -a/q|\le 1/(qQ)$, then we have \begin{equation*} E(x,y;\mya )\ll _{\delta ,C}\Psi (x,y)\left \{ \ff {2^{\omega (q)}\log q}{\ffi }\ff {\log \left (1+\log x/\log y\right )}{\log y} + \ff 1{(\log y)^{A}}\right \}, \end{equation*} where, as usual, $\displaystyle \Psi (x,y)=\sumn u_{\myP }(n)$. We remark that an estimate for $E(x,y;\mya )$ in terms of elementary functions of $x$ and $y$ is readily deduced from this by an appeal to known estimates for $\Psi (x,y)$ (the reader is referred to \cite{HT} for a wonderful survey of this topic). We now turn to the problem of obtaining estimates for $\EF $ valid uniformly for all $f$ belonging to some class of multiplicative functions. Such a problem was first considered by H. Daboussi for the class $\F $ of all complex-valued multiplicative functions $f$ satisfying $|f|\le 1$. He showed \cite{Da1} (see also \cite{DD1} and \cite{DD2}) that if $|\mya -s/r|\le 1/r^{2}$ and $3\le r\le (x/\log x)^{1/2}$, for some coprime integers $s$ and $r$, then \begin{equation}\EF \ll \ff x{\sqrt {\log _{2} r}}, \tag{1}\label{eq1} \end{equation} uniformly for all $f\in \F $. This implies, in particular, that for every irrational $\mya $ we have \begin{equation} \lim _{x\to \infty }\ff 1x\EF =0, \tag{2}\label{eq2} \end{equation} uniformly for all $f\in \F $. It was later observed by Tenenbaum \cite{Te} that this result provides some measure of independence of the additive and multiplicative structures of the set of integers, a topic of great interest in number theory. More precisely, he formulated the following question. Writing \begin{equation*}\ff 1x\EF =\left (\ff 1x\sumn f(n)\right )\left (\ff 1x\sumn e(\mya n)\right ) +o(1), \end{equation*} we ask what can be said about the error term. In particular, we would like to characterize those functions $f$ such that for every irrational $\mya $ we have \begin{equation}\ff 1x\EF =o\left (\ff 1x\left |F(x,0)\right |\right ). \tag{3}\label{eq3} \end{equation} (Observe that (\ref{eq2}) implies (\ref{eq3}) only for those functions $f$ for which $F(x,0)\asymp x$.) The question of when (\ref{eq3}) holds was first raised in a paper of Y. Dupain, R. R. Hall and Tenenbaum \cite{DHT}. It was shown there, among other things, that (\ref{eq3}) holds for the special case of the function $f$ given by $n\mapsto y^{\Omega (n)}$, where $\Omega (n)$ denotes the total number of prime factors of $n$ and $00$ be real numbers and set $Q=x/(\log x)^{3}$. Furthermore, let $a$ and $q$ be coprime integers satisfying $q\le Q$ and $|\mya -a/q|\le 1/(qQ)$. Then we have \begin{equation*}\EF \ll _{\ee }\xol +\ff x{\sqrt q(\log x)^{1-\ee }}+\ff x{\sqrt q\log x} e^{S_{q}(x)}\left (\ff q\ffi \right )^{3/2}, \end{equation*} uniformly for all $f\in \F $. \end{theorem} We extend the range of applicability by proving a somewhat weaker bound as follows. \begin{theorem}\label{thm2} Let $x\ge 3$, $\mya $, $R\ge 3$ and $\ee >0$ be real numbers and suppose that $|\mya -s/r|\le 1/r^{2}$ and $R\le r\le x/R$ for some coprime integers $s$ and $r$. Then we have \begin{equation*}\EF \ll _{\ee }\xol +\ff x{\sqrt R(\log x)^{1-\ee }}+ \ff x{\sqrt R\log x}e^{S(x)}\left (\log R\right )^{1/2}\left (\log _{2}R\right )^{3/2}, \end{equation*} uniformly for all $f\in \F $. \end{theorem} Observe that these theorems might be weaker than (\ref{eq10}) in those cases when $S(x)$ is ``small'', e.g., $S(x)\asymp \log _{3}x$. This shortcoming is especially true in view of the fact that we can now replace (\ref{eq10}) by the stronger estimate \begin{equation*}\EF \ll \xol +\ff x{\sqrt R\log x}e^{S(x)}\left (\log R\right )^{1/2}\left (\log _{2}R\right ) \left (\log _{2} x\right )^{1/2}. \end{equation*} Thus by combining these results one obtains a bound superior, in general, to each of them individually. On the other hand, for functions $f$ which are supported on the positive proportion of primes up to $x$, i.e., for $f\in \F _{\la }(x)$ for some $0<\la \le 1$, we obtain the following corollaries of Theorems \ref{thm1} and \ref{thm2} respectively. \begin{corollary}\label{cor1} Let $x\ge 3$ and $\mya $ be real numbers and set $Q=x/(\log x)^{3}$. Furthermore, let $a$ and $q$ be coprime integers satisfying $q\le Q$ and $|\mya -a/q|\le 1/(qQ)$. Then we have \begin{equation*}\EF \ll _{\la }\xol +\ff x{\sqrt q\log x} e^{S_{q}(x)}\left (\ff q\ffi \right )^{3/2}, \end{equation*} uniformly for all $f\in \F _{\la }(x)$. \end{corollary} \begin{corollary}\label{cor2} Let $x\ge 3$, $\mya $ and $R\ge 3$ be real numbers and suppose that $|\mya -s/r|\le 1/r^{2}$ and $R\le r\le x/R$ for some coprime integers $s$ and $r$. Then we have \begin{equation*}\EF \ll _{\la }\xol + \ff x{\sqrt R\log x}e^{S(x)}\left (\log R\right )^{1/2}\left (\log _{2}R\right )^{3/2}, \end{equation*} uniformly for all $f\in \F _{\la }(x)$. \end{corollary} The equivalence of estimates (\ref{eq8}) and \ref{eq8prime} shows that Corollary \ref{cor2} provides a stronger bound than (\ref{eq4}) even in the case when $S(x)$ is maximal. In particular, this shows that our estimates are quite sharp, since we already noted that (\ref{eq4}) was. Furthermore, given $\la $, $0<\la \le 1$, the original examples of Montgomery and Vaughan yielding (\ref{eq5}) can be easily modified to produce a function $f\in \F $ for which the analogue of (\ref{eq5}) \begin{equation*}F(x,\ff sr)\gg \xol + \ff x{\sqrt r\log x}e^{S(x)} \end{equation*} holds with $S(x)\sim \la \log _{2} x$. Thus we summarize these facts somewhat colloquially by saying that our estimates are sharp ``throughout'' the class of functions supported on the positive proportion of primes up to $x$ in the sense that (\ref{eq4}) is sharp only for the subclass $\F _{1}(x)$. On the other hand, our results do not imply either (\ref{eq6}) or \ref{eq3prime}. Of course, we do get a slightly weaker form of (\ref{eq6}) for those functions $f\in \F _{\la }(x)$ for which \begin{equation*}\sumn |f(n)|\asymp \xol e^{S(x)}. \end{equation*} Thus, for example, one readily sees using standard methods that functions $f$ for which Daboussi and Goubin established \ref{eq3prime} (with $0