Distribution of Leading Digits of Numbers

Abstract

Applying the theory of distribution functions of sequences we find the relative densities of the first digits also for sequences xn not satisfying Benford’s law. Especially for sequence xn = nr, n = 1, 2, . . . and $x_n = p_n^r $, n = 1, 2, . . ., where pn is the increasing sequence of all primes and r > 0 is an arbitrary real. We also add rate of convergence to such densities.

Distributional asymptotics mod 1 of $(\log_bn)$

“…mod 1; moreover, the elements of Ω[x n ], have been described in terms of asymptotic distribution functions. Similar results for slowly changing sequences in the literature include logarithms of natural numbers or prime numbers, iterated logarithms, and monotone functions of prime numbers [3,5,7,6,8,9,10,11,12]. As far as the author knows, however, there were virtually no results, in the case of slowly changing sequences, on the rate(s) of convergence for subsequences of (ν N ) to Ω[x n ], not even for very basic sequences such as (log b n) with b ∈ N \ {1}, prior to [13].…”

The Distributional Asymptotics Mod 1 of (log b n)

“…mod 1; moreover, the elements of Ω[x n ], have been described in terms of asymptotic distribution functions. Similar results for slowly changing sequences in the literature include logarithms of natural numbers or prime numbers, iterated logarithms, and monotone functions of prime numbers [3,[5][6][7][8][9][10][11][12]. As far as the author knows, however, there were virtually no results, in the case of slowly changing sequences, on the rate(s) of convergence for subsequences of (ν N ) to Ω[x n ], not even for very basic sequences such as (log b n) with b ∈ N \ {1}, prior to [13].…”

“…As far as the author knows, however, there were virtually no results, in the case of slowly changing sequences, on the rate(s) of convergence for subsequences of (ν N ) to Ω[x n ], not even for very basic sequences such as (log b n) with b ∈ N \ {1}, prior to [13]. Only recently did the author learn that [9] establishes an upper bound (log N/N ) for the latter, as well as their asymptotic distribution functions. Even there, however, the sharpness of the bound (log N/N ) remains obscure.…”

Distribuion of Leading Digits of Numbers II

“…As examples of N i and w, in [8] we gave the following: [8], for rate of convergence of (2) we also proved that…”

“…In [8], we gave some results about the first digit problem in base b ≥ 2 for the sequence (n r ) n≥1 and for the sequence (p r n ) n≥1 , where p n is the nth prime number and r > 0 is a real number. In this paper, we generalize and sharpen them.…”