Let $0 < \alpha < 1$ be a constant. The expected number of prime factors of a "random" integer near $n$ which are greater than $n^\alpha$ is $-\log \alpha$.


It's my understanding that (properly formulated) this is a well-known fact in analytic number theory but I cannot find a reference for it. Can anybody provide a reference?


**Edited to add (March 28):**: The asymptotic density of positive integers $n$ with $k$th largest factor smaller than $n^{1/\alpha}$ is $\rho\_k(\alpha)$, where we have $L\_0(\alpha) = [\alpha > 0]$ and 
$$ L\_k(\alpha) = [\alpha \ge k] \int\_k^\alpha L\_{k-1}(t-1) \: {dt \over t}, $$
and $1-\rho\_k(\alpha) = \sum\_{n=0}^\infty {-k \choose n} L\_{n+k}(\alpha)$. (See Riesel, p. 162.) The density of positive integers with $k$th largest factor *larger* than $n^{1/\alpha}$ is therefore $1-\rho\_k(\alpha)$, and so the expected number of factors larger than $n^{1/\alpha}$ is $\sum\_{k \ge 1} (1-\rho\_k(\alpha))$. Therefore the expected number of such factors is
$$ \sum\_{k \ge 1} \sum\_{n \ge 0} {-k \choose n} L\_{n+k}(\alpha). $$
Letting $n+k = j$ we can rewrite this sum as
$$ \sum\_{j \ge 1} \sum\_{n=0}^{j-1} {n-j \choose n} L\_j = \sum\_{j \ge 1} L\_j \left( \sum\_{n=-0}^{j-1} (-1)^n {j-1 \choose n} \right) $$
and the inner sum is $0$ except when $j=1$, when it is $1$. So the expected number of factors larger than $n^{1/\alpha}$ is $L\_1(\alpha)$; this is $\log \alpha$.



Reference for the expected number of prime factors of n larger than n^alpha is -log alpha
Theorem 5.4 of Riesel, Prime Numbers and Computer Methods for Factorization, says "the number of prime factors $p$ of integers in the interval $[N-x,N+x]$ such that $a<\log\log p< b$ is proportional to 
$b-a$ if $b-a$ as well as $x$ are sufficiently large as $N\to\infty$." 


