Random multiplicative functions and typical size of character in short intervals

We examine the conditions under which the sum of random multiplicative functions in short intervals, given by ₗ<₍ ₗ+ₘ f (n), exhibits the phenomenon of better than square-root cancellation. We establish that the point at which the square-root cancellation diminishes significantly is approximately when the ratio (xy) is around x. By modeling characters by random multiplicative functions, we give a sharp bound of 1r-1 \!\!\! ₑ |ₗ<₍ ₗ+ₘ (n) |, where r is a large prime and x+y r. This extends the result of Harper Harpercharac.