Outdated Archival Version

These pages are not updated anymore. They reflect the state of 20 August 2005. 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