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Prime powers and generalized Benford law

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It is known that the first digits of prime numbers follow a generalized Benford law (GB) with size-dependent exponent that converges asymptotically to the uniform distribution. Based on two different statistics, we show the existence of size-dependent exponents that outperform in precision the optimal size-dependent exponent in Luque and Lacasa (2009) uniformly over some finite ranges. This result also holds for prime squares. Extending the approach to prime powers, different rates of convergence of the size-dependent GB’s to a GB with inverse power exponent are determined and compared. Furthermore, we introduce a criterion of counting compatibility, which indicates whether or not a given size-dependent GB that belongs to the first digits of some integer sequence is compatible with the asymptotic counting function of this sequence if it exists. We show the existence of a one-parametric size-dependent GB for the sequence of prime powers that is counting compatible with the prime number theorem and determine its optimal size-dependence. Finally, based on the theory of distribution functions of integer sequences, it is proved that the first digits of prime powers converge asymptotically to a GB with inverse power exponent. In particular, asymptotically as the power goes to infinity the sequences of prime powers obey Benford’s law.
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PRIME POWERS AND GENERALIZED BENFORD LAW
Werner Hürlimann, Swiss Mathematical Society,
University of Fribourg, CH-1700 Fribourg
E-mail: whurlimann@bluewin.ch
Abstract
It is known that the first digits of prime numbers follow a generalized Benford law (GB) with
size-dependent exponent that converges asymptotically to the uniform distribution. Based on two
different statistics, we show the existence of size-dependent exponents that outperform in
precision the optimal size-dependent exponent in Luque and Lacasa (2009) uniformly over some
finite ranges. This result also holds for prime squares. Extending the approach to prime powers,
different rates of convergence of the size-dependent GB’s to a GB with inverse power exponent
are determined and compared. Furthermore, we introduce a criterion of counting compatibility,
which indicates whether or not a given size-dependent GB that belongs to the first digits of some
integer sequence is compatible with the asymptotic counting function of this sequence if it exists.
We show the existence of a one-parametric size-dependent GB for the sequence of prime powers
that is counting compatible with the prime number theorem and determine its optimal size-
dependence. Finally, based on the theory of distribution functions of integer sequences, it is
proved that the first digits of prime powers converge asymptotically to a GB with inverse power
exponent. In particular, asymptotically as the power goes to infinity the sequences of prime
powers obey Benford’s law.
Keywords: first digit; prime powers; Riemann’s prime counting function; probabilistic number
theory; asymptotic distribution; mean absolute deviation; probability weighted least squares
MSC 2010: 11A41, 11K99, 11N37, 11Y55, 62E20, 62F12
1. Introduction
It is well-known that the first digits of many numerical data sets are not uniformly distributed.
Newcomb (1881) and Benford (1938)) observed that the first digits of many series of real
numbers obey Benford’s law
9,...,2,1),(log)1(log)(
1010
=+= ddddP
B
. (1.1)
The increasing knowledge about Benford’s law and its applications has been collected in two
recent books by Berger and Hill (2015) and Miller (2015). It is also long known that the first
digits of some “artificial” sequences, like integer squares and cubes, do not follow Benford’s law
(e.g. Furlan (1946)). A simple one-parameter extension of (1.1) is the Generalized Benford law
2
(GB) with exponent ]1,0(
α
(e.g. Pietronero et al. (2001), Miller and Nigrini (2007), Luque
and Lacasa (2009), formula (3.1), Hürlimann (2015a), Proposition 3.1) defined by
9,...,2,1,
1
10
)1(
)(
=
+
=
d
dd
dP
GB
α
αα
α
. (1.2)
Clearly, the limiting case
0
α
respectively 1
α
converges weakly to Benford’s law
respectively the uniform distribution. It is remarkable that the asymptotic distribution of the first
digit of
m
-th integer powers is GB with parameter m/1
α
as the number of digits goes to
infinity (e.g. Hürlimann (2004), Theorem 1, Ross (2012), Remarks 5.1 below). Note that
sequences of integer powers have already been studied in the book by Furlan (1946), Section
III.13 for squares, pp. 172-191, Section III.15 for cubes, pp. 207-217, and Section III.16 for
general powers, pp. 219-229. On the other hand, the first digit sequences of prime numbers
follow a GB with size-dependent parameter 1
α
that converges necessarily to the uniform
distribution, as motivated by Luque and Lacasa (2009). Alternative parametric generalizations of
Benford’s law have been considered by various authors including Hürlimann (2003/15a),
Rodriguez (2004), Fu et al. (2007), equation (3), Grendar et al. (2007), etc. Concerning the
present topic, Hürlimann (2009) finds numerical evidence that the sequence of primes might be
Pareto Benford distributed, a first digit two-parameter probability distribution derived from the
double Pareto in Reed (2001).
In view of the promising and simple results by Luque and Lacasa (2009), it appears natural to
refine their approach and extend it to prime powers. In a first step, we ask whether over finite
ranges their explicit specification of the size-dependent GB for primes can be improved. Based
on two appropriate size-dependent goodness-of-fit statistical measures, namely the ETA measure
(derived from the mean absolute deviation) and the WLS measure (weighted least square
measure derived from the chi-square statistics), we show in Section 2 the existence of size-
dependent exponents, which outperform uniformly the asymptotic optimal value. Section 3
extends this result to prime squares. Then, for some sequences of prime powers
4,10 <
mp
mss
,
s
a fixed integer power, we determine numerically different rates of
convergence of the size-dependent GB with minimum ETA and WLS estimators. Besides a
precise measurement of the rather slow rate of convergence, the obtained numerical statistics
motivate the convergence of first digits of prime powers to a GB with exponent
1
s
. Section 4
considers a criterion of counting compatibility, which indicates whether or not a given size-
dependent GB that belongs to the first digits of some integer sequence is compatible with the
asymptotic counting function of this sequence (if it exists). We show the existence of a one-
parametric size-dependent GB for the sequence of prime powers that is counting compatible with
the prime number theorem and determine its optimal size-dependence. Finally, based on the
theory of distribution functions of integer sequences, it is proved in Theorem 5.1 that the first
digits of prime powers converge asymptotically to a GB with inverse power exponent. In
particular, asymptotically as the power goes to infinity the sequences of prime powers obey
Benford’s law. Another more complex derivation of the latter result, which uses the Erdös-Turàn
inequality, is given in Eliahou et al. (2013) (see also Massé and Schneider (2012)).
3
2. Improved size-dependent generalized Benford law for prime numbers
According to Luque and Lacasa (2009) the first digits of prime numbers up to 4,10
m
m
, are
approximately GB distributed with size-dependent exponent of the form
1
))10ln((1),(
=
amam
α
. (2.1)
Based on a least squares approximation they find the constant 1.01.1
a
as a good
approximation (Luque and Lacasa (2009), equation (3.2)). In Section 5 (a), these authors develop
from it an asymptotic prime counting function )(
xL
as alternative to the well-known
logarithmic integral approximation
=
x
dttxLi
2
1
)ln()( of the prime number counting function
)(x
π
. They show that the asymptotic error between )(xL and )(xLi is minimized for the
value 1
a. Since the rate of convergence of the size-dependent GB to the uniform distribution
is rather slow, it is interesting to ask whether over finite ranges mk
k
,...,4],10[1, =, there exists
an optimal size-dependent exponent (2.1), which uniformly outperforms the asymptotic optimal
value 1
a.
To investigate this question, it is necessary to specify a goodness-of-fit (GoF) measure
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 (1946),
Section II.7.1, pp.70-71, who defines the ETA measure, and by rlimann (2009), p. 8, who
applies the probability weighted least squares (WLS) measure used earlier by Leemis et al. (2000)
(chi-square divided by sample size, similar to
2
L
discrepancy in uniform distribution theory).
These two GoF measures are defined as follows. Let 1},{ nx
n
, be an integer sequence, and let
n
d be the (first non-zero) significant digit of
n
x. The number of
n
x’s, Nn ,...,1
=
, with
significant digit dd
n
= is denoted by )(dX
N
. Then Furlan’s ETA measure for the GBL is
defined to be
=
=
=
9
1
)(
)(
9
1
)(),(
2
9
)(
dNdX
GB
NNN
N
dPMADMAD
N
ETA
α
ααα
, (2.2)
where )(
α
N
MAD is the mean absolute deviation measure (similar to the star discrepancy in
uniform distribution theory). The latter measure is also used to assess conformity to Benford’s
law by Nigrini (2000) (see also Nigrini (2012), Table 7.1, p.160). The WLS measure for the GBL
is defined by
=
=
9
1
2
)(
)(
))((
1
)(
dGB NdX
GB
N
dP
dP
N
WLS
N
α
α
α
. (2.3)
In the present paper, we consider the sequence of prime powers
msss
pp
<10},{ , for a fixed
exponent ,...3,2,1
=
s, and arbitrary primes below 4,10 m
m
. Denote by )(dI
s
k
the number of
4
prime powers below 1,10 k
k
, with first digit d. This number is defined recursively by the
relationship
,...2,1),()10()10)1(()(
1
=++=
+
kdIdddI
s
k
sk
sks
k
ππ
. (2.4)
Therefore, with )10(
m
N
π
= one has )()( dIdX
smsN
= in (2.2)-(2.3). A list of the
15,...,4),(
1
=mdI
m
, together with the sample size )10(
m
N
π
=, 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
.9,...,2,1,
)110(
)}ln(10){ln()}1ln(10){ln()1(
)(
,0
)(
)()(
)(
1
)(
21010
1
9
12
2
)(
2
=
++
=
=
=
+
=
d
dddd
dP
dP
dP
dP
N
WLS
dd
GB
dGB NdX
GB
GB
N
N
α
αααα
α
α
α
α
α
α
α
α
(2.5)
For comparison, the ETA and WLS measures for the size-dependent exponent (2.1) with 1
a,
called LL1 estimator, as well as for the size-dependent exponents 1.113546
a, called LL2
estimator, and 1.2737
a, called LL3 estimator, are listed. By construction, the LL2 estimator
matches the minimum ETA for )10(
15
π
=N, while LL3 does the same for the minimum WLS.
We note that both estimators yield optimal size-dependent exponents, which outperform
uniformly the asymptotic optimal value 1
a over the fixed ranges 15,...,6],10[1, =m
m
, for
the ETA respectively WLS measure. This provides an affirmative answer to the question raised
in the first paragraph.
Table 2.1:
GB fit for primes up to
m
10 : ETA versus WLS criterion
m=
WLS
ETA
LL1
LL2
LL3
WLS
ETA
LL1
LL2
LL3
WLS
ETA
4
0.875034
0.879530
69.61
70.03
70.65
70.40
69.27
29663
29432
29401
29370
29959
5
0.905076
0.902532
29.96
29.67
29.55
30.01
29.31
6575
6630
6849
6574
6811
6
0.919879
0.918628
15.89
15.47
15.01
14.83
14.24
1187
1079
998.8
992.1
1062
7
0.931875
0.932951
8.286
7.680
7.304
7.425
7.253
377.0
286.5
202.2
172.0
232.7
8
0.941343
0.942066
5.627
4.622
4.391
4.538
4.368
157.8
109.5
70.07
63.19
94.69
9
0.948449
0.948805
4.827
4.060
3.688
3.750
3.669
88.86
60.83
41.08
39.82
48.42
10
0.953965
0.954303
4.025
3.377
3.341
3.418
3.287
58.08
39.69
27.96
27.66
36.32
11
0.958389
0.958671
3.501
2.947
3.018
3.071
2.896
41.19
28.00
19.91
19.78
26.41
12
0.962036
0.962272
3.143
2.663
2.783
2.820
2.635
30.70
20.92
15.13
15.08
20.14
13
0.965095
0.965293
2.850
2.428
2.563
2.585
2.412
23.28
15.87
11.66
11.65
15.51
14
0.967698
0.967865
2.610
2.236
2.375
2.384
2.228
18.06
12.33
9.202
9.199
12.20
15
0.969938
0.970082
2.407
2.069
2.215
2.215
2.069
14.29
9.774
7.396
7.396
9.774
parameters
ETA GoF measures
WLS GoF measures
5
Table 2.2:
GB fit for first digits of primes with Riemann’s approximation
m=
WLS
ETA
LL1
LL2
LL3
WLS
ETA
LL1
LL2
LL3
WLS
ETA
4
0.875508
0.878401
25.83
25.87
25.92
25.89
25.83
5835
5652
5690
5624
5867
5
0.902516
0.903899
9.072
8.384
8.463
8.454
8.381
531.4
390.6
327.3
326.1
396.4
6
0.919999
0.921060
6.835
5.532
5.918
6.095
5.391
311.0
210.2
141.3
138.1
188.2
7
0.932381
0.933214
6.028
4.919
4.897
5.102
4.733
198.9
134.9
88.72
85.35
121.7
8
0.941501
0.942091
5.191
4.270
4.223
4.395
4.116
127.9
86.79
57.60
55.65
76.57
9
0.948468
0.948917
4.472
3.732
3.730
3.858
3.615
84.81
57.55
38.91
37.98
51.69
10
0.953955
0.954308
3.921
3.298
3.345
3.435
3.216
58.39
39.64
27.40
27.00
36.43
11
0.958385
0.958670
3.487
2.949
3.035
3.095
2.896
41.68
28.32
20.01
19.85
26.64
12
0.962035
0.962270
3.136
2.666
2.777
2.815
2.633
30.72
20.91
15.07
15.02
20.05
13
0.965095
0.965292
2.849
2.432
2.561
2.582
2.413
23.28
15.87
11.65
11.63
15.48
14
0.967698
0.967865
2.609
2.235
2.376
2.385
2.228
18.06
12.33
9.200
9.197
12.20
15
0.969938
0.970082
2.407
2.069
2.216
2.216
2.069
14.29
9.774
7.396
7.396
9.774
16
0.971888
0.972012
2.234
1.932
2.076
2.069
1.932
11.51
7.881
6.039
6.037
7.926
17
0.973599
0.973710
2.084
1.813
1.953
1.940
1.811
9.401
6.449
4.996
4.992
6.572
18
0.975114
0.975212
1.953
1.707
1.843
1.827
1.705
7.781
5.345
4.181
4.175
5.479
19
0.976465
0.976551
1.837
1.613
1.745
1.726
1.611
6.513
4.480
3.536
3.527
4.616
20
0.977676
0.977753
1.735
1.529
1.658
1.636
1.526
5.507
3.792
3.017
3.007
3.933
21 0.978768 0.978838 1.643 1.453 1.578 1.555 1.450 4.698 3.239 2.595 2.584 3.370
22
0.979759
0.979822
1.561
1.384
1.506
1.481
1.382
4.040
2.789
2.249
2.237
2.913
23
0.980661
0.980718
1.486
1.322
1.440
1.414
1.319
3.500
2.418
1.962
1.950
2.542
24
0.981486
0.981539
1.418
1.265
1.380
1.353
1.262
3.052
2.111
1.722
1.710
2.227
25 0.982243 0.982292 1.357 1.213 1.324 1.297 1.210 2.677 1.854 1.519 1.507 1.958
26
0.982941
0.982986
1.300
1.164
1.273
1.246
1.161
2.362
1.636
1.348
1.336
1.736
27
0.983586
0.983627
1.248
1.120
1.225
1.198
1.117
2.094
1.452
1.201
1.189
1.545
28
0.984184
0.984222
1.200
1.079
1.181
1.154
1.076
1.865
1.294
1.075
1.063
1.374
29 0.984740 0.984775 1.155 1.041 1.141 1.113 1.038 1.668 1.159 0.966 0.955 1.233
30
0.985258
0.985291
1.114
1.005
1.102
1.075
1.002
1.498
1.041
0.871
0.860
1.112
parameters
ETA GoF measures
WLS GoF measures
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
)3(
10
+
m
(resp.
)7(
10
+
m
).
By trying to extend the results beyond 15
m one encounters at least two difficulties. The
Table 3 in Riesel (1985), which is used to calculate Table A.1, stops at 15
m, and becomes
more and more difficult to evaluate exactly. Though )10(
m
π
is known up to 25
m (e.g.
Wikipedia at http://en.wikipedia.org/wiki/Prime-counting_function) its representation in single
precision is no more feasible. At the cost of a slight loss in accuracy, one can overcome these
difficulties by using appropriate approximation formulas for )(x
π
, for example the logarithmic
integral approximation )(xLi or Riemann’s better approximation )(xR . For both of them there
exist power series expansions in the variable )ln(x, namely the formulas (2.26) and (2.27) in
Riesel (1985), Chapter 2, p.55. In the following, we use the series derived by Gram (1884) of the
Riemann’s prime number formula, which reads
6
+
+=
=
=
=11
)1(! )ln(
1)(
)(
)(
k
k
n
n
kkk x
xLi
nn
xR
ζ
µ
, (2.6)
where )(z
ζ
is Riemann’s zeta function. Based on it we replace )10(
m
N
π
= and formula (2.3)
by Riemann’s approximations )10(
m
RN = and
,...2,1),()10()10)1(()(
1
=++=
+
kdIdRdRdI
s
k
sk
sks
k
(2.7)
In this way the Table 2.1 extends (here in single precision only) to Table 2.2. Again, the ETA
(resp. WLS) measures are given in units of
)3(
10
+m
(resp.
)7(
10
+m
). For Riemann’s
approximation to the first digits of primes, one notes that the LL2 and LL3 estimators yield
optimal size-dependent exponents, which outperform uniformly the asymptotic optimal value
1
a over the fixed ranges 30,...,5],10[1, =m
m
.
3. Size-dependent generalized Benford law for prime squares and higher powers
The results of the preceding Section are extended to prime power sequences
msss
pp
<10},{ , for
a fixed power ,...3,2,1
=
s, and arbitrary primes below 4,10 m
m
. For this purpose, it is natural
to replace the size-dependent exponent of the form (2.1) by the more general form
}))10ln((1{),,(
11
= amsams
α
, (3.1)
where
a
is some real parameter. Since the rate of convergence of the size-dependent GB with
exponent (3.1) to the GB with exponent
1
s is rather slow, it is interesting to ask whether over
finite ranges mk
ks
,...,4],10[1, =
, there exists an optimal exponent (3.1), which uniformly
outperforms the asymptotic optimal value 1
a that will be derived in Corollary 4.1.
This question is investigated along the line of Section 2 using again the WLS and ETA GoF
measures. The method is first illustrated at prime squares with fixed 2
s. A Table of prime
squares count in form 15,...,4),(
2
2
=
mdI
m
, which extends Table A.1 for primes, does not seem
to be readily available. Table A.2 is a small sample of it for 6,5,4
=
m. It suggests that
Riemann’s approximation (2.7) suffices for the present purpose. Table 3.1 below is similar to
Table 2.2 and holds in single precision only. Here, we compare the size-dependent exponent (3.1)
with 1
a, called LL1 estimator, with the size-dependent exponent 1.107206
a, called LL2
estimator, that by construction matches the minimum ETA for prime squares over the range
]10[1,
60
. The ETA (resp. WLS) measures are given in the somewhat changed units of
)4(
10
+m
(resp.
)8(
10
+m
). One observes that the LL2 estimator yields optimal size-dependent exponents,
which outperform uniformly the asymptotic optimal value 1
a over the fixed ranges
30,...,6],10[1,
2
=
m
m
.
7
Table 3.1:
GB fit for first digit of prime squares with Riemann’s approximation
m=
WLS
ETA
LL1
LL2
WLS
ETA
LL1
LL2
WLS
ETA
4
0.445254
0.441754
279.0
288.3
284.7
249.46
85085
88604
72474
76554
5
0.452761
0.452039
60.50
56.41
64.39
55.655
2717
2953
2674
2895
6
0.460557
0.460417
23.41
17.91
16.27
15.132
236.2
146.7
141.7
151.8
7 0.466537 0.466509 18.79 14.34 13.49 13.425 153.29 74.89 60.43 60.91
8
0.471007
0.470974
15.85
12.25
11.77
11.691
97.53
46.79
37.63
38.39
9
0.474434
0.474408
13.36
10.59
10.38
10.304
62.57
30.34
25.65
26.20
10
0.477138
0.477116
11.495
9.418
9.270
9.206
41.42
20.54
18.30
18.71
11
0.479324
0.479306
10.114
8.476
8.373
8.316
28.51
14.55
13.51
13.82
12 0.481128 0.481113 9.012 7.705 7.633 7.581 20.36 10.73 10.25 10.50
13
0.482641
0.482628
8.119
7.062
7.014
6.967
15.03
8.178
7.965
8.158
14
0.483929
0.483918
7.383
6.520
6.488
6.445
11.4
6.401
6.312
6.468
15
0.485039
0.485029
6.768
6.056
6.035
5.996
8.852
5.120
5.087
5.215
16
0.486005
0.485997
6.256
5.654
5.643
5.606
7.013
4.170
4.160
4.268
17 0.486854 0.486846 5.839 5.302 5.298 5.264 5.652 3.447 3.446 3.533
18
0.487605
0.487599
5.474
4.992
4.993
4.961
4.623
2.886
2.886
2.960
19
0.488275
0.488269
5.152
4.716
4.721
4.691
3.83
2.444
2.442
2.505
20
0.488877
0.488871
4.866
4.470
4.478
4.449
3.209
2.089
2.084
2.140
21
0.489419
0.489414
4.611
4.248
4.258
4.231
2.716
1.800
1.793
1.842
22
0.489911
0.489907
4.381
4.047
4.059
4.034
2.32
1.564
1.554
1.595
23
0.49036
0.490355
4.172
3.864
3.878
3.854
1.997
1.367
1.355
1.392
24
0.49077
0.490766
3.983
3.698
3.713
3.689
1.731
1.203
1.189
1.221
25
0.491146
0.491143
3.810
3.545
3.561
3.540
1.511
1.064
1.049
1.074
26
0.491493
0.49149
3.652
3.404
3.421
3.399
1.327
0.946
0.931
0.956
27
0.491814
0.491811
3.506
3.274
3.291
3.271
1.172
0.845
0.829
0.852
28
0.492112
0.492109
3.372
3.153
3.171
3.151
1.04
0.758
0.742
0.763
29
0.492388
0.492386
3.247
3.042
3.060
3.042
0.927
0.683
0.666
0.682
30 0.492646 0.492644 3.131 2.938 2.956 2.938 0.83 0.617 0.601 0.617
parameters
ETA GoF measures
WLS GoF measures
Table 3.2:
GB fit for first digit of higher prime powers with Riemann’s approximation
s=
3
4
5
10
3
4
5
10
m =
10
0.318113
0.23859
0.190874
0.095438
0.318110
0.238566
0.190859
0.095435
15
0.323369
0.242529
0.194024
0.097012
0.323367
0.242518
0.194017
0.097011
20
0.325923
0.244444
0.195555
0.097778
0.325922
0.244438
0.195552
0.097777
25 0.327434 0.245576 0.196461 0.098231 0.327434 0.245573 0.196459 0.098230
30
0.328433
0.246325
0.197061
0.098530
0.328433
0.246323
0.197059
0.098530
m =
10
364.3512
115.4094
47.1891
2.9554
4.1069
2.2914
1.4426
0.3508
15
101.6183
32.2024
13.1851
0.8210
2.6768
1.4963
0.9424
0.2277
20 41.6841 13.2147 5.4115 0.3370 1.9870 1.1117 0.6999 0.1694
25
21.0058
6.6609
2.7279
0.1699
1.5805
0.8849
0.5575
0.1347
30
12.0323
3.8160
1.5629
0.0973
1.3128
0.7350
0.4631
0.1118
minimum WLS exponents minimum ETA exponents
WLS GoF measures ETA GoF measures
8
For higher prime powers the convergence of the size-dependent GB with minimum ETA and
WLS estimators to the GB with exponent
1
s is illustrated in Table 3.2 below. Here, the ETA
(resp. WLS) GoF measures are given in units of
)4(
10
+m
(resp.
)10(
10
+m
). Over the finite ranges
10,5,4,3,30,25,20,15,10],10[1, ==
sm
ms
, the size-dependent minimum WLS and ETA
exponents increase to the expected limiting GB exponent
1
s, and the fit in the WLS and ETA
GoF measures becomes better as
s
increases.
4. Counting compatibility of the size-dependent GB with the prime number theorem
The preceding Section shows that there is a strong numerical evidence for the approximation
)(
)10(
)(
),,(
dP
dI
GB ams
m
sms
α
π
, whose precision increases by growing value of
m
. The present Section
considers a criterion, called counting compatibility, which indicates whether or not a given size-
dependent GB that belongs to the first digits of some integer sequence is compatible with the
asymptotic counting function of this sequence (if it exists). Applied to prime powers, it is shown
in Theorem 4.1 below that for the size-dependent exponent (3.1) this criterion is satisfied.
Moreover, the optimal value of the parameter
a
that is counting compatible with the prime
number theorem is the single value 1
a, as shown in Corollary 4.1.
To do so, let 1},{ nx
n
, be an arbitrary integer sequence, and suppose that the asymptotic
counting function )(NQ as
N of this sequence exists. Further, let ]1,0[)(
N
α
be a
size-dependent exponent such that the sequence of numbers generated by the power-law density
)(N
x
α
, has a GB first-digit distribution
)(
)(1
dP
GB N
α
with exponent )(1 N
α
.
Definition 4.1.
The generalized Benford law
)(
)(1
dP
GB N
α
is counting compatible with the
counting function )(NQ if there exists a constant )(Nc such that the generalized Benford
counting function defined by
NN
dxxNc
2
)(
)(
α
is asymptotically equivalent to )(NQ .
Let us apply this criterion to the sequence of prime powers. Starting point is the prime number
theorem (e.g. Overholt (2014), Chapter 6), which states that the number of primes
p
in the
inteval ][1,
N
, usually denoted by )(
N
π
, follows the asymptotic distribution
)(ln/~)(
NNNN
π
. (4.1)
Similarly, for any fixed positive integer 1
s
, the number of prime powers
s
p
in the interval
][1,
s
N
, denoted by )(
s
s
N
π
, follows the same asymptotic distribution
)(ln/~)(
NNNN
s
s
π
. (4.2)
9
This follows from the fact that
ss
Np
< if, and only if, one has
Np
<
. The relation (4.2),
which is equivalent to (4.1), can be called prime power number theorem. In the above notation,
consider the prime power relation over an interval ][1,
s
N
that belongs to the GB size-
dependent exponent (3.1), namely
aN
aN
saNs
aNs
=
+
=)ln( 1
),(
~
,
),(
1
),,(
~
α
α
α
, (4.3)
Theorem 4.1
(
Counting compatibility of the GB for prime powers
). For any fixed positive
integer 1
s
and any 1
m
, set
.
)10ln(1
1
1
),10,(
~
1),,(
==
ams
asams
m
αα
(4.4)
Then, the generalized Benford law
9,...,1),(
),,(
=ddP
GB ams
α
, is counting compatible with the prime
power number theorem (4.2). More precisely, the choice of the constant
)(ln1
),,(
aNs
asNc
= (4.5)
implies that the generalized Benford counting function
=
s
NaNss
s
dxxasNcNL
2
),,(
~
),,()(
α
is
asymptotically equivalent to )(ln/~)()( =
NNNNN
s
s
ππ
.
Proof.
Counting compatibility holds provided the following limiting condition holds:
1
)ln(/ )(
lim =
NN NL
s
s
N
. (4.6)
Using (4.4) one obtains
+
=
=
=
aN a
aN N
N
aN aNe
N
aNss aNe
NL
aNaNsss
s
)ln(
exp
)1()ln(),(
~
1),(
)),,(
~
1( ),(
)(
),(
~
1)),,(
~
1(
αα
α
α
α
α
.
This coincides with the prime counting function in Luque and Lacasa (2009), formula (5.4), and
reflects the fact that there are as many prime powers in ][1,
s
N
than there are primes in ][1,
N
.
One has the asymptotic expansion
+
+
++= )(ln 1
)(ln
1
)ln(
1
1
)ln(
)(
32
2
2
1
N
O
N
aa
NN
N
NL
s
s
, (4.7)
10
which implies (4.6). The form (4.4) of the GB exponent in Definition 4.1 follows by setting
s
N10= in Equation (4.3).
Corollary 4.1. The optimal value of the parameter
a
, which minimize the error between
)(
s
s
NL and the logarithmic integral approximation
=
N
dttNLi
2
1
)ln()( to )(N
π
, is 1
a.
Proof. The asymptotic expansion of the logarithmic integral reads
+++= )(ln 1
)(ln 2
)ln(
1
1
)ln(
)(
32
N
O
N
NN
N
NLi .
The error between this and (4.7), namely
+
+
== )(ln 1
)(ln
1
)(ln 2
)ln(
)()()(
32
2
2
1
2
N
O
N
aa
N
N
N
NLNLiNE
s
ss
is minimized at 1
a.
Remarks 4.1. The counting function )(
s
s
NL generalizes the prime counting function (5.1) in
Luque and Lacasa (2009). Counting compatibility has been established recently for many other
important integer sequences. This property holds for square-free integer powers, powers of
perfect powers and powerful integer powers, as shown in Hürlimann (2014a/b/2015b).
5. Asymptotic distribution of the first digits of prime powers
The main difficulty encountered with the first digit problem is the fact that integer sequences like
the integer powers }{
s
n and the prime powers }{
s
n
p, with
n
p the
n
-th prime, ,...2,1
=
n, has
no solution with respect to the natural density (e.g. Diaconis (1977) for the sequence of primes).
Applying the theory of distributions of integer sequences (e.g. Strauch and Porubský (2013)) a
new perspective emerges and it is sometimes possible to determine the relative density, or at
least its asymptotic limit (e.g. Baláž, Nagasaka and Strauch (2010)). Section 4 shows that
counting compatibility is another mathematical evidence besides numerical evidence for the
following exact asymptotic convergence of the relative density )()(
)10(
)(
1
mdP
dI
GB
s
m
sms
π
.
Theorem 5.1 (GB for prime powers). The asymptotic distribution of the first digit of prime
power sequences 1,10 <
mp
mss
, for fixed 1
s, as
m
, is given by
11
.
)10ln(1
1
1
),,(,9,...,1),()(lim
)10(
)(
lim
1
),,(
====
ams
amsddPdP
dI
GB
s
GB ams
m
m
sms
m
α
π
α
(5.1)
Proof. Since )(),,(
1
msams
α
it suffices to prove that )(
)10(
)(
lim
1
dP
dI
GB
s
m
sms
m
=
π
. The
derivation follows a private communication of O. Strauch dated September 30, 2015. First, let
1},{ nx
n
, be an arbitrary sequence of real numbers, and let 0
x be a real number. Then, the
first (non-zero) digit of
x
is equal to d if, and only if, the following inequality
)1(log)(log
1010
dxd +< (5.2)
holds. The step distribution of }1mod{
n
x is given by
N
xxNncard
xF
n
N
]},0[1mod;{
)(
=,
and the limit (if it exists) )()(lim xgxF
m
N
m
=
is called distribution function of }{
n
x with
respect to the sequence
...
21
<< NN
of indices. As an application, suppose that )(xg is a
distribution function of the sequence }1mod{log
10
n
x with respect to
...
21
<< NN
, then the
first digit problem for the sequence }{
n
x has, in virtue of the inequality (5.2), the solution
))((log))1((log
};{
lim
1010
dgdg
NdxofdigitfirstNncard
m
nm
Nm
+=
=
. (5.3)
Now, it follows from Strauch and Blažeková (2006) and Okhubo (2011) that for fixed 1
s the
sequences }1mod{log
10
s
n and }1mod{log
10
s
n
p have the same type of distribution function,
namely
su
susx
s
sx
su
u
xg
/
//
/1
/
/
10
1)10,10min(
1
10
110
10
1
)(
+
= ,
where the parameter ]1,0[
u depends on the sequence of indices
...
21
<< NN
as follows.
For the sequence }{
s
n one has
}1mod{loglim
10 s
m
m
Nu
=
while for the sequence }{
s
n
p the
parameter is
}1mod{loglim
10 s
N
m
m
pu
=
. In the second case, if one replaces
m
N by )(
m
N
π
then one has also
}1mod{loglim
10 s
m
m
Nu
=
. It remains to select the sequence
...
21
<< NN
appropriately such that the limit ]1,0[
u exists. Setting
m
m
N10= for the sequence }{
s
n
respectively )10(
m
m
N
π
= for the sequence }{
s
n
p and using that 01mod10log
10
=
sm
one
sees that necessarily 0
u. One concludes by inserting )110/()110()(
/1/
0
=
ssx
xg into (5.3)
above to see that for the sequence }{
s
n
p of primes one gets
12
=
+
=+=
=
=
).(
1
10
)1(
))((log))1((log
)10(
};10{
lim
)10(
)(
lim
1
/1
/1/1
100100
dP
dd
dgdg
dpofdigitfirstpcarddI
GB
s
s
ss
m
s
n
sms
n
m
m
sms
m
ππ
Remarks 5.1. The exact asymptotic relative density of the first digits of the integer power
sequence }{
s
n is part of the proof. One obtains similarly (see Hürlimann (2004) for numerical
analysis)
)(
1
10
)1(
10
};10{
lim
1
/1
/1/1
dP
dd
dnofdigitfirstncard
GB
s
s
ss
m
ssms
m
=
+
=
=
. (5.4)
Theorem 5.1 and (5.4) hold in arbitrary base b. It suffices to use in the proof the distribution
function of }1mod{log
s
b
n and }1mod{log
s
nb
p given by
]1,0[,
1),min(
1
11
)(
/
//
/1
/
/
+
= u
b
bb
b
b
b
xg
su
susx
s
sx
su
u
. (5.5)
It is remarkable that in the special case eb
of the natural logarithm the distribution function
(5.5) is actually the content of Exercise 179 in Part 2 of Pólya and Szegö (1970), whose first
edition dates back to 1924. Note that the whole set of distribution functions for the sequence
}1mod{lnn has been first determined by Wintner (1935). This author also observes that the
sequence }1mod{ln
n
p behaves like the sequence }1mod{ln n, a precursor of the result used
in the proof of Theorem 5.1. Several questions remain open. Clearly, counting compatibility
yields more information than the exact asymptotic relative density because it reveals the
analytical form of the size-dependent GB exponent. But, what is the logical relationship between
counting compatibility and exact asymptotic relative density? Does counting compatibility
always predict the correct asymptotic relative density of the generalized Benford law?
Appendix: Tables of first digit distributions for primes and prime squares
Based on the recursive relation (2.4), the calculation of 15,...,4),(
1
=mdI
m
, is straightforward by
using Table 3 in Riesel (1985), p.374-376. These numbers are listed in Table A.1. The exact
calculation of ,...3,2,15,...,4),( == sssmdI
s
m
, using (2.4) requires some algorithm to compute
)(x
π
exactly. A number of methods for this are exposed in Riesel (1985), Chapter 1. Special
mention should be given to the analytical method by Lagarias and Odlyzko (1987) and its recent
rigorous implementation by Platt (2013), which has been used up to )10(
24
π
. For the sake of
illustration we present in Table A.2 a comparison between the exact count of first digits of prime
squares and its Riemann approximation over a rather small range. Since only Riemann’s
approximation is used in Section 3, this is not of relevance here.
13
Table A.1: First digit distribution of primes up to 15,...,4,10 =k
k
k
4
5
6
7
8
9
10
sample size
1'229
9'592
78'498
664'579
5'761'455
50'847'534
455'052'511
/ first digit
1
160
1'193
9'585
80'020
686'048
6'003'530
53'378'283
2
146
1'129
9'142
77'025
664'277
5'837'665
52'064'915
3
139
1'097
8'960
75'290
651'085
5'735'086
51'247'361
4
139
1'069
8'747
74'114
641'594
5'661'135
50'653'546
5
131
1'055
8'615
72'951
633'932
5'602'768
50'193'913
6
135
1'013
8'458
72'257
628'206
5'556'434
49'815'418
7
125
1'027
8'435
71'564
622'882
5'516'130
49'495'432
8
127
1'003
8'326
71'038
618'610
5'481'646
49'221'187
9
127
1'006
8'230
70'320
614'821
5'453'140
48'982'456
k
11
12
13
14
15
sample size
4'118'054'813
37'607'912'018
346'065'536'839
3'204'941'750'802
29'844'570'422'669
/ first digit
1
480'532'488
4'369'582'734
40'063'566'855
369'893'939'287
3'435'376'839'800
2
469'864'125
4'281'198'201
39'319'600'765
363'545'360'347
3'380'562'309'312
3
463'196'868
4'225'763'390
38'851'672'813
359'541'975'662
3'345'924'530'873
4
458'352'691
4'185'483'176
38'510'936'699
356'622'729'564
3'320'632'228'693
5
454'577'490
4'153'943'134
38'243'708'524
354'330'372'215
3'300'752'009'165
6
451'476'802
4'128'049'326
38'024'311'091
352'446'754'137
3'284'400'373'590
7
448'855'139
4'106'164'356
37'838'546'363
350'849'788'546
3'270'531'245'684
8
446'590'932
4'087'194'991
37'677'478'288
349'465'615'584
3'258'501'713'644
9
444'608'278
4'070'532'710
37'535'715'441
348'245'215'460
3'247'889'171'908
Table A.2: First digit distribution of prime squares (exact versus Riemann approximation)
2
·
m
8
10
12
8
10
12
sample size
1'229
9'592
78'498
1'227
9'587
78'527
/ first digit
1
251
1'932
15'678
249
1'926
15'649
2
187
1'437
11'722
184
1'439
11'750
3
157
1'195
8'789
153
1'193
9'771
4
129
1'036
8'520
134
1'041
8'525
5
125
944
7'655
118
930
7'645
6
97
844
6'981
107
849
6'985
7
107
775
6'483
99
784
6'465
8
93
745
6'004
93
732
6'042
9
83
684
5'666
90
693
5'695
exact count
Riemann approximation
14
Acknowledgements. I am indebted to O. Strauch for pointing out to me the proof of Theorem
5.1 and sending me related information about it.
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... Two recent books are Berger and Hill [5], and Miller [15]. In Number Theory, it is known that for any fixed power exponent 1  s , the first digits of some integer sequences, like integer powers and prime powers, follow asymptotically a Generalized Benford law (GB) with exponent ] 1 , 0 ( 1    s  (see Hürlimann [9,11]) such that 9 ..., , ...
... As a follow-up to Hürlimann [11,12], we study the first digits of powers of the first prime in twin prime pairs using a numerical and an analytical method. Based on the numerical method we fit the GB to appropriate 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 derived from the chi-square statistics). ...
... Moreover, we show the existence of a one-parametric size-dependent exponent function that converges to these GB's and determine some approximate value that is close enough to the minimum ETA and WLS estimators to support the suggested convergence. Section 4 uses the analytical criterion of first digit counting compatibility introduced in [11,12]. In general, this criterion permits to decide whether or not a given sizedependent GB that belongs to the first digits of some integer sequence is compatible with the asymptotic counting function of this sequence, if it exists. ...
Article
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The first digits of twin primes follow a generalized Benford law with size-dependent exponent and tend to be uniformly distributed, at least over the finite range of twin primes up to 10^m, m=5,...,16. The extension to twin prime powers for a fixed power exponent is considered. Assuming the Hardy-Littlewood conjecture on the asymptotic distribution of twin primes, it is claimed that the first digits of twin prime powers associated to any fixed power exponent converge asymptotically to a generalized Benford law with inverse power exponent. In particular, the sequences of twin prime power first digits presumably converge asymptotically to Benford’s law as the power exponent goes to infinity. Numerical calculations and the analytical first digit counting compatibility criterion support these conjectured statements.
... In number theory, it is known that for any fixed power exponent 1  s , the first digits of some integer sequences, like integer powers and prime powers, follow asymptotically a Generalized Benford law (GB) with exponent ] 1 , 0 ( 1    s  (e.g. Hürlimann [10]) such that 9 ..., , ...
... In Section 2, we determine the minimum MAD and WLS estimators of the GB Besides a precise measurement of the rate of convergence, the obtained numerical statistics motivate the asymptotic convergence of first digits of Niven integer powers to a GB with exponent 1  s . Section 3 relies on the criterion of first digit counting compatibility recently introduced in [10]. In general, this criterion permits to decide whether or not a given size-dependent GB that belongs to the first digits of some integer sequence is compatible with the asymptotic counting function of this sequence, if it exists. ...
... It is known that the first digits of integer powers and prime powers follow asymptotically a GB with inverse power exponent (e.g. [10], Theorem 5.1 and Remarks 5.1). For other integer sequences (with an asymptotic counting function) like square-free integer powers, powers of perfect powers, powerful integer powers, Niven integer powers, etc., the observed discrepancies can be explained in a non-trivial way. ...
Article
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It is claimed that the first digits of Niven integer powers follow a generalized Benford law with a specific parameter-free size-dependent exponent that converges asymptotically to the inverse power exponent. Numerical and other mathematical evidence, called first digit counting compatibility, is provided for this statement.
... It is important to note that the size-dependent GBL parameter (3.9) is proportional to half of the inverse power. This contrasts with [5], [6], [17], where the size-dependent GBL parameters are proportional to the inverse power. Finally, the next Table 2 compares the new counting function Q(N) = Q s (N s ), for all s = 1, 2, . . ...
... Departures from Benford's law occur quite frequently. For the sequences of integer powers, square-free integer powers, powers of perfect powers, and prime numbers (see [17]), the observed discrepancies can be explained in a non-trivial way. More precisely, the first significant digits of these sequences obey a generalized Benford law with size dependent parameter proportional to the inverse of a multiple of the power exponent. ...
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For any fixed power exponent, it is shown that the first digits of powers from perfect power numbers follow a generalized Benford law (GBL) with size-dependent parameter that converges asymptotically to a GBL with half of the inverse power exponent. In particular, asymptotically as the power goes to infinity these first digit sequences obey Benford’s law. Moreover, we show the existence of a one-parameter size-dependent function that converges to the parameter of these GBL’s and determine an optimal value that minimizes its deviation to two minimum estimators of the size-dependent parameter over the finite range of powers from perfect power numbers less than 10^5ms, m=2,...,6, where s=1,2,3,4,5 is a fixed power exponent.
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Benford's law states that the leading digits of many data sets are not uniformly distributed from one through nine, but rather exhibit a profound bias. This bias is evident in everything from electricity bills and street addresses to stock prices, population numbers, mortality rates, and the lengths of rivers. Here, Steven Miller brings together many of the world's leading experts on Benford's law to demonstrate the many useful techniques that arise from the law, show how truly multidisciplinary it is, and encourage collaboration. Beginning with the general theory, the contributors explain the prevalence of the bias, highlighting explanations for when systems should and should not follow Benford's law and how quickly such behavior sets in. They go on to discuss important applications in disciplines ranging from accounting and economics to psychology and the natural sciences. The contributors describe how Benford's law has been successfully used to expose fraud in elections, medical tests, tax filings, and financial reports. Additionally, numerous problems, background materials, and technical details are available online to help instructors create courses around the book. Emphasizing common challenges and techniques across the disciplines, this accessible book shows how Benford's law can serve as a productive meeting ground for researchers and practitioners in diverse fields.
Article
Full-text available
For any fixed power exponent, it is shown that the first digits of powers from perfect power numbers follow a generalized Benford law (GBL) with size-dependent parameter that converges asymptotically to a GBL with half of the inverse power exponent. In particular, asymptotically as the power goes to infinity these first digit sequences obey Benford’s law. Moreover, we show the existence of a one-parameter size-dependent function that converges to the parameter of these GBL’s and determine an optimal value that minimizes its deviation to two minimum estimators of the size-dependent parameter over the finite range of powers from perfect power numbers less than 10^5ms, m=2,...,6, where s=1,2,3,4,5 is a fixed power exponent.
Article
We describe a rigorous implementation of the Lagarias and Odlyzko Analytic Method to evaluate the prime counting function and its use to compute unconditionally the number of primes less than 1024.
Article
The first significant digit patterns arising from a mixture of uniform distributions with a random upper bound are revisited. A closed-form formula for its first significant digit distribution (FSD) is obtained. The one-parameter model of Rodriguez is recovered for an extended truncated Pareto mixing distribution. Considering additionally the truncated Erlang, gamma and Burr mixing distributions, and the generalized Benford law, for which another probabilistic derivation is offered, we study the fitting capabilities of the FSD’s for various Benford like data sets from scientific research. Based on the results, we propose the general use of a fine structure index for Benford’s law in case the data is well fitted by the truncated Erlang member of the uniform random upper bound family of FSD’s.