nt.number theory - Reference for the expected number of prime factors of n larger than n^alpha is -log alpha

Do you mean that $log(1/alpha)$ is the expected number of prime factors in $(x^alpha,x]$ when $alphato0$? For fixed $alphain(0,1)$ what you are claiming is not true. For example,

$|{nle x:exists p|n;{rm with};sqrt{x}lt ple x}|sim xlog2$

and

$|{nle x:plesqrt{x};{rm for all};p|n}|sim(1-log2)x$.

When $alphato0$ it is not hard to prove what you need, but I am not sure where you can find a precise reference. For example, setting $omega(n;y,z)=|{p|n:ylt ple z}|$ and following the proof of Theorem 6 in page 311 in Tenenbaum's book "Introduction to Analytic and Probabilistic Number Theory" gives that

$|{nle x:|omega(n;x^alpha,x)-log(1/alpha)|ge(1+delta)log(1/alpha)}|ll xalpha^{Q(1+delta)},$

where $Q(1+delta)=int_1^{1+delta}log tdt$.

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