A First Digit Theorem for Square-Free Integer Powers

For any fixed integer power, it is shown that the first digits of square-free integer powers follow a generalized Benford law (GBL) with size-dependent exponent that converges asymptotically to a GBL with inverse power exponent. In particular, asymptotically as the power goes to infinity the sequences of squarefree integer powers obey Benford’s law. Moreover, we show the existence of a one-parametric size-dependent exponent function that converge to these GBL’s and determine an optimal value that minimizes its deviation to two minimum estimators of the size-dependent exponent over the finite range of square-free integer powers less than 10 ,..., 4 , 10 = ⋅ m m s , where 10 , 5 , 4 , 3 , 2 , 1 = s is a fixed integer power. Mathematics Subject Classification: Primary 11A25, 11K36, 11N37, 11Y55; Secondary 62E20, 62F12


Abstract
For any fixed integer power, it is shown that the first digits of square-free integer powers follow a generalized Benford law (GBL) with size-dependent exponent that converges asymptotically to a GBL with inverse power exponent.In particular, asymptotically as the power goes to infinity the sequences of squarefree integer powers obey Benford's law.Moreover, we show the existence of a one-parametric size-dependent exponent function that converge to these GBL's and determine an optimal value that minimizes its deviation to two minimum estimators of the size-dependent exponent over the finite range of square-free integer powers less than

Introduction
It is well-known that the first digits of many numerical data sets are not uniformly distributed.Newcomb [14] and Benford [3] observed that the first digits of many series of real numbers obey Benford's law (1.1) The increasing knowledge about Benford's law and its applications has been collected in various bibliographies, the most recent being Beebe [2] and Berger and Hill [4].It is also known that for any fixed power exponent 1 ≥ s , the first digits of integer powers, follow asymptotically a Generalized Benford law (GBL) with exponent ) 1 , 0 ( 1 ∈ = − s α such that (see Hürlimann [7]) (1.2) Clearly, the limiting case 0 → α respectively 1 → α of (1.2) converges weakly to Benford's law respectively the uniform distribution.
We study the distribution of first digits of square-free integer powers.The method consists to fit the GBL to samples of first digits using two size-dependent goodness-of-fit measures, namely the ETA measure (derived from the mean absolute deviation) and the WLS measure (weighted least square measure).In Section 2, we determine the minimum ETA and WLS estimators of the GBL over finite ranges of square-free powers up to 4 , 10 a fixed power exponent.Computations illustrate the convergence of the size-dependent GBL with minimum ETA and WLS estimators to the GBL with exponent 1 − s .Moreover, we show the existence of a one-parametric size-dependent exponent function that converge to these GBL's and determine an optimal value that minimizes its deviation to the minimum ETA and WLS estimators.A mathematical proof of the asymptotic convergence of the finite sequences to the GBL with inverse power exponent follows in Section 3.

Size-dependent GBL for square-free integer powers
To investigate the optimal fitting of the GBL to first digit sequences of squarefree integer powers, it is necessary to specify goodness-of-fit (GoF) measures according to which optimality should hold.First of all, a reasonable GoF measure for the fitting of first-digit distributions should be size-dependent.This has been observed by Furlan [5], Section II.7.1, pp.70-71, who defines the ETA measure, and by Hürlimann [8], p.8, who applies the probability weighted least squares (WLS) measure used earlier by Leemis et al. [12] (chi-square divided by sample size).Let , be an integer sequence, and let n d be the (first) . Then, Furlan's ETA measure for the GBL is defined to be A first digit theorem for square-free integer powers 131 where is the mean absolute deviation measure.The latter measure is also used to assess conformity to Benford's law by Nigrini [15] (see also Nigrini [16], Table 7.1, p.160).The WLS measure for the GBL is defined by (e.g.[12]) where the counting function ) (n S is given by (e.g.Pawlewicz [18], Theorem 1)
, together with the sample size , is provided in Table A.1 of the Appendix.Based on this we have calculated the optimal parameters which minimize the ETA (or equivalently MAD) and WLS measures, the so-called minimum ETA (or minimum MAD) and minimum WLS estimators.Together with their GoF measures, these optimal estimators are reported in Table 2.1 below.Note that the minimum WLS is a critical point of the equation For comparison, the ETA and WLS measures for the size-dependent GBL exponent ) , called LL estimator, are listed.This type of estimator is named in honour of Luque and Lacasa [13] who introduced it in their GBL analysis for the prime number sequence.Through calculation one observes that the LL estimator minimizes the absolute deviations between the LL estimator and the ETA (resp.WLS) estimators over the finite ranges of square-free powers Table 2.1 displays exact results obtained on a computer with single precision, i.e. with 15 significant digits.The ETA (resp.WLS) measures are given in units of  ).Taking into account the decreasing units, one observes that the optimal ETA and WLS measures decrease with increasing sample size.

Asymptotic counting function for square-free integer powers
The following is a slight extension of the argument by Luque and Lacasa [13], Section 5(a).It is well-known that a random process with uniform density where the integral pre-factor is chosen to fulfill the asymptotic limiting value for the square-free number counting function, that is (note that In fact, two improved asymptotic expansions of ) (N S are known, namely ( ) The first one is classical and proved in Hardy and Wright [6], p.269, and Jameson [9], Section 2.5, for example.The second improved estimate is due to Jia [11] (see also Pappalardi [17]).However, it suffices to use the simple estimate (3.3), which is obtained as follows.From (3.2) one gets for arbitrary ,...

With (3.1) this transforms to
A first digit theorem for square-free integer powers 135 which is independent of s and simply denoted by reflects the fact that there are as many square-free powers in ? Clearly, the factor ( ) for any fixed c .Its derivative with respect to c satisfies the property which implies the following min-max property of ( The size-dependent exponent (3.1) with 1 = c not only minimizes the absolute deviations between the LL estimator and the ETA (resp.WLS) estimators over the finite ranges of square-free powers 10 , .Moreover, the following limiting asymptotic result has been obtained.
First Digit Square-Free Integer Power Theorem (GBL for square-free integer powers).The asymptotic distribution of the first digit of square-free integer power sequences 4 , 10 , is given by .10 in Section 2, but it turns out to be uniformly best with maximum error less than 3 10 − against the asymptotic estimate, at least if 4 10 ≥ N

Table 2 . 1 :
GBL fit for first digit of square-free powers: ETA vs. WLS criterion

Table 3 . 1 :
Comparison of square-free number counting functions for