Some observations on numbers in short intervals

We obtain some results about the primes and the prime factorizations of the positive integers in the interval (x, x+xθ], where 0<θ<1. We also obtain some results about the primes and the prime factorizations of the positive integers in the interval (x, x+f(x)], where f(x) is a function of slow increase. For instance, f(x)=logαx, where α>0.