A New Approach to the First Digit Phenomena

Scientific Paper
Title	A New Approach to the First Digit Phenomena
Read in full	Link to paper
Author(s)	Dennis P Allen
Keywords	Benford's law, first digit law, prime numbers, Peano axioms, summability, Cesàro summation, prime number theorem
Published	1999

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Abstract

In this paper, we first show by extending the proof of B. J. Flehinger that the integers have the first digit property that the primes represented to the base ten also have the first digit property. We note that R. E. Whitney has also proven this using the logarithmic matrix method of summability. We then abstract from these two proofs in view of the Peano Axioms to obtain a (new) definition of what it means to sum a sequence in the spirit of the Peano axioms for the positive integers (which includes Flehinger's and Whitney's methods as special cases) and conjecture any method of summation of this very general type assigns the limit log₁₀((A+ 1)/A) to the two sequences s_n and t_n where

s_n = 1 if n has first digit equal to A else 0, and
t_n = 1 if the n^th prime has first digit A else 0.

The conjecture, if true, yields a new explanation of the first digit phenomenon by reducing it to its first cause: the basic well ordering of the positive integers.

Overview

This is a short pure-mathematics paper — unusual in this archive — published in The Toth-Maatian Review (vol. 14, no. 3, 1999, pp. 6839–47) under the title "A New Approach to the First Digit Phenomenon", first submitted January 8, 1999. Its subject is the "first digit phenomenon" or logarithmic law, first reported by Frank Benford in 1938: in tables of physical constants and other naturally occurring numerical data, the proportion of entries whose leading significant digit is at most A is close to log₁₀(A+1), so that leading digit 1 occurs about 30.1% of the time and leading digit 9 about 4.6%.

Allen does two things. First he gives a theorem: the primes, written in base ten, obey the same law as the positive integers do, in exactly the same summability sense — a result he obtains by modifying B. J. Flehinger's 1966 proof for the integers and feeding in the prime number theorem with remainder. Second, and this is the part he regards as new, he proposes a philosophical diagnosis of why the law holds. Rather than deriving it from scale invariance (the usual route, following Pinkham), he traces it to the well-ordering of the positive integers themselves — to the successor structure of the Peano axioms. He formulates an abstract condition on a summation method, that it sum a sequence "in the spirit of the Peano axioms", and conjectures that any such method assigns the value log₁₀((A+1)/A) both to the integer indicator sequence and to the prime indicator sequence. The departure from the mainstream treatment is thus not a denial of any result but a reversal of explanatory direction: scale invariance is downgraded from cause to symptom.

The argument

The problem: no ordinary limit exists

Let P¹_n(A) be the proportion of the positive integers up to n whose initial digit is at most A. Flehinger showed that lim_n→∞ P¹_n(A) does not exist — the proportion oscillates forever as n sweeps through successive decades — but that the sequence is Cesàro summable in an iterated sense. Defining repeated Cesàro averages

P^k_n(A) = (1/n) Σ_m=1..n P^k−1_m(A), for k = 2, 3, 4, …

she obtained lim_k→∞ lim inf_n→∞ P^k_n(A) = lim_k→∞ lim sup_n→∞ P^k_n(A) = log₁₀(A+1). Allen stresses in a parenthesis that "this limiting process is very dependent on the ≤ order of the integers which is so central to their definition in terms of the Peano Axioms" — the observation from which the rest of the paper grows.

Extending the result to the primes

Writing P̂¹_N(A) for the proportion of primes up to N with initial digit at most A, Allen follows Flehinger in the change of variable N = α·10^j with 1 ≤ α < 10 and sets

Q¹(α,A) = lim_j→∞ P̂¹_α·10^j(A).

Although P̂¹_N(A) has no limit as N → ∞, this decade-synchronised limit does exist for every α, and he computes it as

Q¹(α,A) = 1 − (9−A)/(9α) for 1 ≤ α < A+1, and 10A/(9α) for A+1 ≤ α < 10.

The computation counts primes decade by decade. In the complete decade [10^k, 10^k+1) the primes with leading digit at most A number π((A+1)·10^k) − π(10^k), and the partial top decade is handled separately. Substituting the prime number theorem with remainder, π(x) = x/ln(x) + O(x/ln²(x)), the whole expression collapses using the elementary limit

lim_j→∞ Σ_k=1..j−1 j/(k·10^j−k) = 1/9.

The error term is dismissed by the estimate [x/ln x] / [x/ln x + O(x/ln²x)] = 1/(1 + O(1/ln x)) → 1, so "the error term will not affect our computation."

Iterating to the logarithmic law

Repeated Cesàro averaging of the primes' proportions gives limits Q^k(α,A), and Allen derives the recursion

Q^k(α,A) = (1/α){ (1/9)∫₁¹⁰ Q^k−1(β,A) dβ + ∫₁^α Q^k−1(β,A) dβ }

"in the same way as it is derived in Flehinger's paper", from which lim_k→∞ Q^k(α,A) = log₁₀(A+1). Hence the probability that a prime has initial digit exactly A is log₁₀((A+1)/A). He notes that R. E. Whitney (1972) had already reached the same conclusion by the logarithmic matrix method of summability, which is stronger than the Cesàro method used here, and that Peter Schatte proved it by uniform distribution theory.

Summation "in the spirit of the Peano axioms"

The paper's proposed novelty is a general class of summation methods. A regular method of summation — one that returns the ordinary limit on convergent sequences — sums {a_n} in the spirit of the Peano axioms if the transformed sequence {b_n} satisfies: (a) it converges, or at least is closer to convergence, so that repeated application forces convergence "if only in the limit"; (b) b_j depends only on a₁,…,a_j, "just as j is obtained from Peano Axioms by applying the successor function (j−1) times to 1"; and (c) earlier terms are weighted no less heavily than later ones, because with respect to the axioms 2 is no more important than 1, and 3 no more than 2. In matrix language this says b_n = Σ W_n,ma_m with W_n,1 ≥ W_n,2 ≥ … ≥ W_n,n > 0, i.e. a lower-triangular Toeplitz matrix with non-increasing weights (weakened, since no regular limit can depend on finitely many terms, to require the monotonicity only beyond some k).

The conjecture is that every such method assigns log₁₀((A+1)/A) to both the integer and the prime indicator sequences. Flehinger's iterated Cesàro method and Whitney's logarithmic matrix method are then two special cases of one principle.

Semigroups, Benford sets and the philosophical claim

Allen argues that the law looks surprising only because one confuses two different semigroups: the semigroup of finite non-zero-leading digit strings under concatenation, in which the natural probability of initial digit A is 1/9, and the semigroup of positive integers given by the Peano axioms, in which it is log₁₀((A+1)/A). Since primes are defined recursively — multiplication being defined recursively — and are "deeply recursive" in the structure of the integers, "it is, perhaps, not too surprising that the primes also satisfy the logarithm law."

He introduces the notion of a Benford set: a set of integers or decimals closed under addition provided the sum is "important". A table of numbers (his example: the lengths of the rivers of a country, in miles) drawn from such a parent population with equal probability 1/n per member will inherit the parent's digit statistics and so approximately satisfy Benford's law. "Thus we avoid the hypothesis of scale invariance" — though he concedes it "comes in the back door" via the fact that the rivers could equally be measured in yards or feet. He explicitly notes a counterexample regime: seven-digit telephone numbers never begin with 1, and higher algebraic operations "need not preserve Benford's law at all; they can lead far from the original Peano axioms defining their constituents."

Assessment

What checks out. The mathematics in this paper is correct. The two-branch formula for Q¹(α,A) is internally consistent: it is continuous at α = A+1, where both branches give 10A/(9(A+1)), and it matches itself across the decade boundary, giving A/9 at both α = 1 and α → 10, as any decade-periodic quantity must. Reconstructing the derivation independently from the prime number theorem gives A/(9α) from the complete decades plus (α−1)/α from the partial top decade, and A/(9α) + (α−1)/α is algebraically identical to 1 − (9−A)/(9α) — Allen's stated result. The auxiliary limit Σ j/(k·10^j−k) → 1/9 is right (substituting m = j−k turns it into Σ 10^−m). And iterating his integral recursion numerically from his own Q¹ converges, for every A from 1 to 9, to log₁₀(A+1) to six decimal places — 0.301030, 0.477121, 0.602060, …, 1.000000. The recursion also has the right structural property: constants are its fixed points, as substituting Q ≡ c immediately shows. So the paper's central theorem is sound and its numbers reproduce.

The one blemish is typographical: on p. 6842 the second branch is printed as Q¹(α,A) = 10A/α, dropping the 9 that is correctly present in the same formula on p. 6840. The derivation and the final result both require 10A/(9α), so this is a misprint rather than an error of substance — but it is the sort of slip a reader reproducing the calculation should be warned about. There is also a duplicated label: the list of conditions on p. 6844 has two items lettered (c), which p. 6845 then refers to as "(c) and (d)".

What is genuinely attractive. The abstraction is the real contribution, and it is a good idea. Rather than proving the same theorem a third time by a third technique, Allen asks what Flehinger's iterated Cesàro method and Whitney's logarithmic matrix method have in common, and identifies it as a monotonicity condition on the summation weights — earlier integers count for at least as much as later ones — that he reads off the successor structure of the Peano axioms. That is a clean and testable formulation, and the derived matrix condition (lower-triangular Toeplitz, non-increasing rows, positive) is precise enough to be attacked. It also has genuine explanatory content: it locates the logarithmic law in the counting order rather than in a postulated invariance, and it correctly predicts where the law should fail (telephone numbers, digit strings under concatenation, quantities generated by "higher algebraic operations").

The real difficulties. The headline claim is a conjecture, and it is stated but not proved; Allen says so plainly, which is to his credit, but it means the paper's advertised "new explanation" rests on an unverified hypothesis. The conditions are also stated somewhat loosely — condition (a), "closer to convergence than before … if only in the limit", is not given a precise meaning, and the footnoted qualification that earlier terms should perhaps be weighted strictly more heavily ("for many purposes, the integers become less important as they become larger") leaves it unclear which of two inequivalent conditions is intended. A conjecture over a class of methods is only as strong as the definition of the class.

The proof of the prime result, by contrast, is standard in method: the paper says outright that it modifies Flehinger's proof and that "from this point on the proof is the same". Whitney had published the result in 1972 and Schatte proved it again in 1983, both of which Allen acknowledges. So the theorem is a rederivation, not a new discovery, and the paper is honest about that. Finally, the treatment of scale invariance is more a change of emphasis than an elimination of it: Allen's own river example admits that scale invariance "comes in the back door", and the Benford-set construction — a set closed under addition "provided the sum is not too large", with the boundary of "important" left informal — carries much of the weight that the Peano-axiom framing is credited with. On its own terms, though, the paper does what it says: the mathematics is right, the abstraction is interesting, and the outstanding claim is correctly labelled a conjecture.