Distribution of consecutive digits in theq-ary expansions of some subsequences of integers

Abstract

Asymptotic normality is established for consecutive digits in the q-ary expansions of some subsequences of integers.

Normal numbers and the middle prime factor of an integer

“…Final remarks. Instead of considering the middle prime factor of an integer, that is the prime factor whose rank amongst the ω(n) distinct prime factors of an integer n is the 1 2 ω(n) th one, we could have also studied the prime factor whose rank is the αω(n) th one, for any given real number α ∈ (0, 1). In this more general case, say with p (α) (n) in place of p m (n), the same type of results as above would also hold, meaning in particular that log p (α) (n) would be close to log α n instead of √ log n.…”

Construction of Normal Numbers Using the Distribution of the Largest Prime Factor

“…Here, we prove that the same result holds if P(n) is replaced by P k (n), the kth largest prime factor of n. The case of P k (n) relies on the same basic tool we used to study the case of P(n), namely a 1996 result of Bassily and Kátai [1]. However, the P k (n) case raises new technical challenges and therefore needs a special treatment.…”

Distribution of the sum-of-digits function of random integers: A survey

“…In particular, if one applies formally this result to f (n) = e itνq(n) , then the left-hand side corresponds to the characteristic function of X n , while the dominant term on the right-hand side to a multinomial distribution. We cannot however conclude directly from this result that X n is asymptotically multinomially distributed due to lack of uniformity in t. For asymptotic normality and related results for q-additive functions, see [9,10,93,130] and [12, Ch. 9].…”