We propose a new necessary and sufficient condition to test whether a sequence is Benford (base-b) or not and apply this characterization to some kinds of sequences (re)obtaining some well known results, as the fact that the sequence of powers of 2 is Benford (base-10).Benford's law, equidistributed sequences, ergodic endomorphisms

Alberto Zorzi

Andrea Ellero

Marco Corazza

English

Research Papers in Economics

Department of Applied Mathematics,  University of Venice  
WORKING PAPER SERIES 
Marco Corazza, Andrea Ellero and  
Alberto Zorzi 
What Sequences obey  
Benford's Law ? 
Working Paper n. 185/2008 
November 2008 
ISSN: 1828-6887 This Working Paper is published under the auspices of the Department of Applied 
Mathematics of the Ca’ Foscari University of Venice. Opinions expressed herein are 
those of the authors and not those of the Department. The Working Paper series is 
designed to divulge preliminary or incomplete work, circulated to favour discussion 
and comments. Citation of this paper should consider its provisional nature. What Sequences obey Benford’s Law?
Marco Corazza Andrea Ellero
<corazza@unive.it> <ellero@unive.it>
Alberto Zorzi
<albzorzi@unive.it>
Dipartimento di Matematica Applicata
Universit` a Ca’ Foscari di Venezia
Abstract. We propose a new necessary and suﬃcient condition to test whether a se-
quence is Benford (base-b) or not and apply this characterization to some kinds of sequences
(re)obtaining some well known results, as the fact that the sequence of powers of 2 is Benford
(base-10).
Keywords: Benford’s law, equidistributed sequences, ergodic endomorphisms
JEL Classiﬁcation Numbers: C02, C83
MathSci Classiﬁcation Numbers: 28D99, 60F99
Correspondence to:
Marco Corazza Dept. of Applied Mathematics, University of Venice
Dorsoduro 3825/e
30123 Venezia, Italy
Phone: [++39] (041)-234-6921
Fax: [++39] (041)-522-1756
E-mail: corazza@unive.it1 The importance of being one
If we consider the most signiﬁcant digit of the powers of two, 21,22,23...2n it turns out
that the frequencies are not the same for all the ﬁgures: for example, among the ﬁrst 1000
powers of two the ones which start with digit 1 appear more often (30.1 %), the powers
which have 2 as ﬁrst digit follow (17.6 %), then the ones with 3 (12.5 %), and so on the
frequencies decrease until 4.5 % for digit 9. In fact, it is possible to prove (see Section 3)
that the probability for a generic term of the sequence to display d as the most signiﬁcant
digit is
log10
µ
1 +
1
d
¶
,
it means that the sequence of powers of 2 obeys Benford’s Law [5].
In general a sequence of real numbers represented in base b is said to be Benford (base-b)
if the probability to observe digit d as the ﬁrst digit of a term of the sequence is
logb
µ
1 +
1
d
¶
,
for each integer d such that 1 ≤ d < b [8]. For an overview of Benford’s Law and a discussion
of its possible applications see e.g. [5]. 1
The main result we propose in this paper is a new necessary and suﬃcient condition
to test whether a sequence is Benford (base-b) or not: we then apply this characterization
to some kinds of sequences (re)obtaining, for example, that the sequence of powers of 2 is
Benford (base-10) but not Benford (base-4).
We also show how the proposed characterization is related in a natural way to Birkhoﬀ’s
ergodic theorem.
2 Benford (base-b) sequences: a necessary and suﬃcient con-
dition
We ﬁrst show how to deﬁne the most signiﬁcant digit of a number by means of elementary
functions. Given a real number x, we use the ﬂoor and ceiling functions deﬁned, respectively,
by ⌊x⌋ = max{n ∈ Z : n ≤ x} and ⌈x⌉ = min{n ∈ Z : n ≥ x}. this way the fractional part
of x is x mod 1 = x − ⌊x⌋.
Lemma 1 If x is a positive real number, then its ﬁrst digit in base b is ⌊b(logb x)mod1⌋.
Proof. Since ⌊logb x⌋ ≤ logb x < ⌊logb x⌋ + 1 we have b⌊logb x⌋ ≤ x < b⌊logb x⌋+1 and 1 ≤
x
b⌊logb x⌋ < b. Therefore ⌊ x
b⌊logb x⌋⌋ is the most signiﬁcant digit of x. To complete the proof
it is suﬃcient to observe that x = blogb x = b⌊logb x⌋b(logb x)mod1, i.e., each real number is the
sum of its integer and fractional parts; this way x
b⌊logb x⌋ = b(logb x)mod1. ⋄
1As observed in [8] (Remark 9.2.5), the deﬁnition of Benford (base-b) sequence we use here is less
restrictive than the deﬁnition of b-Benford sequence given for example in [1, 7].
1The next result provides a possible way to count the number of terms of a sequence
which display a ﬁrst signiﬁcant digit which is lower than d: it will be used later on to prove
a general criterion to test whether a sequence is Benford (base-b) or not.
Lemma 2 If x1,x2,...xn are positive real numbers and if 1 ≤ d < b, then
♯{xk : 1 ≤ k ≤ n, ﬁrst digit of xk ≤ d} =
n X
k=1
»
1
b(logb xk)mod1 −
1
d + 1
¼
.
Proof. According to Lemma 1, the ﬁrst digit of xk is ≤ d if and only if b(logb xk)mod1 < d+1,
that is, if and only if (logb xk)mod1 < logb(d + 1). To complete the proof we just need to
prove that the function
fd+1(y) =
»
1
by −
1
d + 1
¼
can be rewritten as
fd+1(y) =
½
1 if y ∈ [0,logb(d + 1))
0 if y ∈ [logb(d + 1),1] .
In fact, function g(y) = 1
by − 1
d+1 is strictly decreasing on [0,1] since g′(y) = − 1
by logb < 0.
Moreover 0 < g(0) = d
d+1 < 1, −1 < g(1) = 1
b − 1
d+1 ≤ 0 and g(logb(d + 1)) = 0. ⋄
Now it is possible to prove the following theorem, which characterizes a Benford (base-b)
sequence.
Theorem 1 The sequence xk of positive real numbers is Benford (base-b) if and only if
lim
n→+∞
1
n
n X
k=1
»
1
b(logb xk)mod1 −
1
d + 1
¼
= logb(d + 1)
for each integer d such that 1 ≤ d < b.
Proof. Consider a generic term xk of the sequence. Using Lemma 2, we have that the
probability that the ﬁrst digit of xk is not greater than d is given by
Prob( ﬁrst digit of xk is ≤ d) = lim
n→+∞
1
n
n X
k=1
»
1
b(logb xk)mod1 −
1
d + 1
¼
.
Observe now that the deﬁnition of Benford (base-b) sequence given in Section 1, since
Prob( ﬁrst digit of xk ≤ d) =
d X
i=1
Prob( ﬁrst digit of xk = i)
and
Prob( ﬁrst digit of xk = i) = Prob( ﬁrst digit of xk ≤ i)−Prob( ﬁrst digit of xk ≤ i−1) ,
2allows to claim that xk is b-Benford if and only if
Prob( ﬁrst digit of xk ≤ d) = logb(d + 1)
for each integer d such that 1 ≤ d < b. This completes the proof. ⋄
The theorem above, suggests a way to verify whether a sequence is Benford or not, com-
puting a rather complicated limit, it seems to be a diﬃcult task. The following Lemma
suggests, as a way to deal with the limit, to compute a suitably deﬁned integral: we will
show how to use this idea in the next Section.
Lemma 3 Let be d an integer such that 1 ≤ d < b and fd+1(y) =
l
1
by − 1
d+1
m
. If the
sequence yk is contained in the interval [0,1] and if
lim
n→+∞
1
n
n X
k=1
fd+1(yk) =
Z 1
0
fd+1(y)dy
then
lim
n→+∞
1
n
n X
k=1
»
1
byk −
1
d + 1
¼
= logb(d + 1) .
Proof. It is suﬃcient to observe that
fd+1(y) =
»
1
by −
1
d + 1
¼
=
½
1 if y ∈ [0,logb(d + 1))
0 if y ∈ [logb(d + 1),1]
,
thus
R 1
0 fd+1(y)dy = logb(d + 1) . ⋄
3 Some Benford (base-b) sequences
As a possible application of Theorem 1, we give a new kind of proof of a well known
theorem, due to Diaconis [3]: a sequence of positive real numbers xk is Benford (base-b) if
the sequence of their logarithms logb xk is uniformly distributed modulo 1. We recall that
a sequence yk is uniformly distributed modulo 1, or equidistributed in [0,1], if
lim
n→+∞
♯{yk : 1 ≤ k ≤ n, a ≤ yk ≤ b}
n
= b − a
for each interval [a,b] ⊆ [0,1]. Another, and equivalent way to deﬁne an equidistributed se-
quence is the following (see for example [9, Problem 162]): the sequence yk is equidistributed
if and only if
lim
n→+∞
1
n
n X
k=1
f(yk) =
Z 1
0
f(y)dy (1)
for every function f which is Riemann integrable on [0,1]. It is now an easy matter to prove
the following well known result, stating that the exponentials of equidistributed sequences
are Benford.
3Theorem 2 ([3, 8]) A sequence xk of positive real numbers is Benford (base-b) if the
sequence yk = logb xk is uniformly distributed modulo 1.
Proof. For a ﬁxed integer d consider function fd+1(y) =
l
1
by − 1
d+1
m
, which is Riemann
integrable on [0,1], and the sequence yk. Applying (1) we obtain
lim
n→+∞
1
n
n X
k=1
fd+1(yk) =
Z 1
0
fd+1(y)dy .
By Lemma 3 we have therefore
lim
n→+∞
1
n
n X
k=1
»
1
byk −
1
d + 1
¼
= logb(d + 1) .
Thus the sequence xk is Benford (base-b) due to Theorem 1. ⋄
The previous theorem allows to easily use known results on equidistributed sequences to
state that related geometric sequences are Benford (base-b).
For example, the sequence yk = (kα) mod 1 is equidistributed if α is an irrational
number, as independently proved by Bohl, Sierpinski and Weyl (see e.g. [8], Theorem
12.3.2); hence, the sequence xk = rk is Benford (base-b) if logb r is irrational ([8], Theorem
9.2.6). Incidentally, this property allows to prove the claim made at the beginning of Section
1: the sequence xk = 2k is Benford (base-10), since log10 2 is irrational. The same sequence
is clearly not Benford (base-4), instead, as one can easily observe 2.
A more general context where Lemma 3 applies concerns ergodic theory. Consider an
ergodic endomorphism T deﬁned on a probability space with domain [0,1], and function
fd+1 deﬁned as in Lemma 3. Birkhoﬀ’s ergodic theorem (see e.g. [2], Appendix 3) claims
that for almost every y1 ∈ [0,1],
lim
n→+∞
1
n
n X
k=1
fd+1(Tk(y1)) =
Z 1
0
fd+1(y)dy
while Lemma 3 tells us that
Z 1
0
fd+1(y)dy = logb(d + 1) .
Thus, by Theorem 1, we obtain that a sequence xk of positive real is Benford (base-b)
if yk = logb xk is generated via an ergodic endomorphism. More precisely, xk is Benford
(base-b) as soon as there exists an ergodic endomorphism T such that Tky1 = yk ∀k and
the equality of Birkhoﬀ’s ergodic theorem holds for the initial term y1 and for each integer
d such that 1 ≤ d < b.
2The sequence xk = 2
k written using base 4 reads
1,2,10,20,100,200,...
4References
[1] A. Berger, L.A. Bunimovich, T.P. Hill (2005), “One-dimensional dynamical systems
and Benford’s law”, Transactions of the American Math. Soc., 357 (1), 197–219.
[2] I.P. Cornfeld, S.V. Fomin and Ya.G. Sinai (1982), Ergodic Theory, Springer Verlag,
New York.
[3] P. Diaconis (1977), The Annals of Probabilty, 5 (1), 72–81.
[4] G.H. Hardy, E.M. Wright (1979), An Introduction to the Theory of Numbers, ﬁfth ed.,
Oxford Univ. Press, New York.
[5] T.P. Hill (1995), “The Signiﬁcant-Digit Phenomenon”, The Amer. Mathematical
Monthly, 102 (4), 322–327.
[6] T.P. Hill (1995), “A Statistical Derivation of the Signiﬁcant-Digit Law”, Statistical Sc.,
10 (4), 354–363.
[7] A.V. Kontorovich, S.J. Miller (2005), “Benford’s law, values of L-ﬁnctions and the
3x+1 problem”, Acta Arithmetica, 120, 269–297.
[8] S.J. Miller, R. Takloo-Bighash (2006), An Invitation to Modern Number Theory,
Princeton Univ. Press, Princeton and Oxford.
[9] G. Polya, G. Szeg¨ o,(1972), Problems and Theorems in Analysis vol.I, Springer-Verlag,
Heidelberg.
5

What Sequences obey Benford's Law ?

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