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\documentclass{era-l}
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\dateposted{October 29, 1999}
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\PII{S 1079-6762(99)00071-2}
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\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