Suppose $f:\mathbb{N}\to [0,1]$ is a multiplicative function (i.e. $f(nm)=f(n)f(m)$ whenever $m$ and $n$ are coprime). Suppose $f$ has non-zero mean, which means
$$
\lim\_{N\to\infty}\frac{1}{N} \sum\_{n=1}^N f(n) >0.
$$
Also, lets assume $f(n)\neq 0$ for all $n$. Then, can we say something about the set of all $n$ such that $f(n)\leq \varepsilon$? What is the density (or upper density) of this set? In particular, does this density go to zero as $\varepsilon\to 0$?


Thank you very much for your help.



A question about $(0,1]$-valued multiplicative functions
Yes, the density goes to zero as $\epsilon \to 0$, but the convergence to zero can be arbitrarily slow (depending on the choice of the function $f$). To see this, first note that the condition that the mean value of $f$ is strictly positive is equivalent to 
$$ 
\sum\_p \frac{1-f(p)}{p}<\infty, 
$$ 
which is also the same as 
$$ 
\sum\_{p^k} \frac{1-f(p^k)}{p^k} <\infty, 
$$ 
where the sum is over all prime powers $p^k$. This is a simple case of results of Wirsing, Halasz, Delange, and follows from the 
estimate (for non-negative multiplicative functions bounded by $1$) 
$$ 
\sum\_{n\le x} f(n) \ll \frac{x}{\log x} \sum\_{n\le x} \frac{f(n)}{n} \ll \frac{x}{\log x} \exp\Big(\sum\_{p\le x} \frac{f(p)}{p}\Big). 
$$ 
(See for example Theorem 2 of [Halberstam and Richert](http://www.sciencedirect.com/science/article/pii/0022314X79900210)). 


Next, for any $1> \delta >0$, put 
$$ 
{\mathcal F}(\delta)= \sum\_{\substack{p^k \\ f(p^k) \le \delta} } \frac{1}{p^k}, 
$$ 
the convergence of the sum for any $\delta <1$ being guaranteed by our previous observation.
We claim that ${\mathcal F}(\delta) \to 0$ as $\delta \to 0$; this step crucially uses the assumption that $f(n)$ is 
strictly positive for all $n$. For each $\delta >0$ let $p(\delta)$ denote the smallest prime power $p^k$ with $f(p^k)<\delta$. Then the assumption that $f(n)>0$ gives that $p(\delta) \to \infty$ as $\delta \to 0$. Therefore, for $\delta \le 1/2$ say,  

$$ 
{\mathcal F}(\delta) \le \sum\_{\substack {p^k \\ p^k \ge p(\delta) \\ f(p)\le 1/2}} \frac{1}{p^{k}}, 
$$ 
 and the RHS is the tail of a convergent series, and therefore goes to zero as $p(\delta)\to \infty$, or in other words as $\delta \to 0$. This proves our claim.


Now suppose $\epsilon >0$ is given (and suitably small), and we want to bound the density of $n$ with $f(n) \le \epsilon$. Put $\delta=\exp(-(\log 1/\epsilon)^{\frac 12})$, which is bigger than $\epsilon$ but still goes to zero with $\epsilon$. Suppose first that $p^k \Vert n$ for some prime power $p^k$ with $f(p^k) <\delta$. The density of such $n$ is clearly at most ${\mathcal F}(\delta)$. Now suppose that if $p^k \Vert n$ then $f(p^k) \ge \delta$. Note here that (using $x \ge \exp(2\log (1/\delta) (x-1))$ for $\delta <x\le 1$)
 $$ 
 \epsilon > f(n) = \prod\_{p^k \Vert n} f(p^k) \ge \exp\Big( 2\log \frac{1}{\delta} \sum\_{p^k \Vert n} (f(p^k)-1) \Big), 
 $$  

 and so, for such $n$, 
 $$ 
 \sum\_{p^k \Vert n} (1-f(p^k)) \ge \frac{1}{2} \Big(\log \frac {1}{\epsilon}\Big)^{\frac 12}. \tag{1}
 $$ 
Since 
$$ 
\sum\_{n\le N} \sum\_{p^k \Vert n} (1-f(p^k)) \ll N \sum\_{p^k}\frac{1-f(p^k)}{p^k} \ll N,
$$ 
we conclude that the density of $n$ satisfying (1) is $\ll (\log \frac {1}{\epsilon})^{-\frac 12}$. Thus our desired 
density is 
$$ 
\ll {\mathcal F}(\delta) + \Big(\log \frac{1}{\epsilon}\Big)^{-\frac 12}, 
$$ 
which goes to $0$ as $\epsilon \to 0$. 


