Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Benford’s law, or: Why the IRS cares aboutnumber theory!

Steven J. Miller, Williams College

sjm1@williams.eduhttp://www.williams.edu/Mathematics/sjmiller/

Hampshire College, July 14, 2011

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Interesting Question

Interesting QuestionFor a nice data set, such as the Fibonacci numbers, whatpercent of the leading digits are 1?

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Interesting Question

Interesting QuestionFor a nice data set, such as the Fibonacci numbers, whatpercent of the leading digits are 1?

Plausible answers:3

Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Interesting Question

Interesting QuestionFor a nice data set, such as the Fibonacci numbers, whatpercent of the leading digits are 1?

Plausible answers:10%4

Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Interesting Question

Interesting QuestionFor a nice data set, such as the Fibonacci numbers, whatpercent of the leading digits are 1?

Plausible answers:10%, 11%5

Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Interesting Question

Interesting QuestionFor a nice data set, such as the Fibonacci numbers, whatpercent of the leading digits are 1?

Plausible answers:10%, 11%, about 30%.6

Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Summary

State Benford’s Law.

Discuss examples and applications.

Sketch proofs.

Describe open problems.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Caveats!

A math test indicating fraud is not proof of fraud:unlikely events, alternate reasons.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Caveats!

A math test indicating fraud is not proof of fraud:unlikely events, alternate reasons.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Benford’s Law: Newcomb (1881), Benford (1938)

StatementFor many data sets, probability of observing a first digit ofd base B is logB

(d+1

d

); base 10 about 30% are 1s.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Benford’s Law: Newcomb (1881), Benford (1938)

StatementFor many data sets, probability of observing a first digit ofd base B is logB

(d+1

d

); base 10 about 30% are 1s.

Not all data sets satisfy Benford’s Law.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Benford’s Law: Newcomb (1881), Benford (1938)

StatementFor many data sets, probability of observing a first digit ofd base B is logB

(d+1

d

); base 10 about 30% are 1s.

Not all data sets satisfy Benford’s Law.⋄ Long street [1, L]: L = 199 versus L = 999.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Benford’s Law: Newcomb (1881), Benford (1938)

StatementFor many data sets, probability of observing a first digit ofd base B is logB

(d+1

d

); base 10 about 30% are 1s.

Not all data sets satisfy Benford’s Law.⋄ Long street [1, L]: L = 199 versus L = 999.⋄ Oscillates between 1/9 and 5/9 with first digit 1.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Benford’s Law: Newcomb (1881), Benford (1938)

StatementFor many data sets, probability of observing a first digit ofd base B is logB

(d+1

d

); base 10 about 30% are 1s.

Not all data sets satisfy Benford’s Law.⋄ Long street [1, L]: L = 199 versus L = 999.⋄ Oscillates between 1/9 and 5/9 with first digit 1.⋄ Many streets of different sizes: close to Benford.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Examples

recurrence relations

special functions (such as n!)

iterates of power, exponential, rational maps

products of random variables

L-functions, characteristic polynomials

iterates of the 3x + 1 map

differences of order statistics

hydrology and financial data

many hierarchical Bayesian models

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Bastille Day, Wikipedia and Google

2 4 6 8 10

0.1

0.2

0.3

0.4

2 4 6 8 10

0.2

0.4

0.6

0.8

1.0

First digit of number of hits Google returns forfirst 80 distinct words and numbers fromWikipedia’s Bastille Day entry on July 10, 2011.Chisquare of 46.2, highly non-Benford (99% oftime should be at most 20).

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Riemann Zeta Function

ζ(s) =∞∑

n=1

1ns

=∏

p prime

(1 − 1

ps

)−1(if Re(s) > 1).

∣∣ζ(

12 + i

k4

)∣∣, k ∈ {0, 1, . . . , 65535}.

2 4 6 8

0.05

0.1

0.15

0.2

0.25

0.3

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Applications

analyzing round-off errors

determining the optimal way to storenumbers

detecting tax and image fraud, and dataintegrity

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

General Theory

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Mantissas

Mantissa: x = M10(x) · 10k , k integer.

M10(x) = M10(x̃) if and only if x and x̃ have thesame leading digits.

Key observation: log10(x) = log10(x̃) mod 1 ifand only if x and x̃ have the same leading digits.Thus often study y = log10 x .

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Equidistribution and Benford’s Law

Equidistribution{yn}∞n=1 is equidistributed modulo 1 if probabilityyn mod 1 ∈ [a, b] tends to b − a:

#{n ≤ N : yn mod 1 ∈ [a, b]}N

→ b − a.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Equidistribution and Benford’s Law

Equidistribution{yn}∞n=1 is equidistributed modulo 1 if probabilityyn mod 1 ∈ [a, b] tends to b − a:

#{n ≤ N : yn mod 1 ∈ [a, b]}N

→ b − a.

Thm: β 6∈ Q, nβ is equidistributed mod 1.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Equidistribution and Benford’s Law

Equidistribution{yn}∞n=1 is equidistributed modulo 1 if probabilityyn mod 1 ∈ [a, b] tends to b − a:

#{n ≤ N : yn mod 1 ∈ [a, b]}N

→ b − a.

Thm: β 6∈ Q, nβ is equidistributed mod 1.

Examples: log10 2, log10(

1+√

52

)6∈ Q.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Equidistribution and Benford’s Law

Equidistribution{yn}∞n=1 is equidistributed modulo 1 if probabilityyn mod 1 ∈ [a, b] tends to b − a:

#{n ≤ N : yn mod 1 ∈ [a, b]}N

→ b − a.

Thm: β 6∈ Q, nβ is equidistributed mod 1.

Examples: log10 2, log10(

1+√

52

)6∈ Q.

Proof: if rational: 2 = 10p/q.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Equidistribution and Benford’s Law

Equidistribution{yn}∞n=1 is equidistributed modulo 1 if probabilityyn mod 1 ∈ [a, b] tends to b − a:

#{n ≤ N : yn mod 1 ∈ [a, b]}N

→ b − a.

Thm: β 6∈ Q, nβ is equidistributed mod 1.

Examples: log10 2, log10(

1+√

52

)6∈ Q.

Proof: if rational: 2 = 10p/q.Thus 2q = 10p or 2q−p = 5p, impossible.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Example of Equidistribution: n√π mod 1

0.2 0.4 0.6 0.8 1

0.5

1.0

1.5

2.0

n√π mod 1 for n ≤ 10

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Example of Equidistribution: n√π mod 1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1.0

n√π mod 1 for n ≤ 100

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Example of Equidistribution: n√π mod 1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1.0

n√π mod 1 for n ≤ 1000

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Example of Equidistribution: n√π mod 1

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1.0

n√π mod 1 for n ≤ 10, 000

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Logarithms and Benford’s Law

Fundamental EquivalenceData set {xi} is Benford base B if {yi} isequidistributed mod 1, where yi = logB xi .

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Logarithms and Benford’s Law

Fundamental EquivalenceData set {xi} is Benford base B if {yi} isequidistributed mod 1, where yi = logB xi .

0 1log 2 log 10

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Logarithms and Benford’s Law

Fundamental EquivalenceData set {xi} is Benford base B if {yi} isequidistributed mod 1, where yi = logB xi .

0 1

1 102

log 2 log 10

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Examples

2n is Benford base 10 as log10 2 6∈ Q.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Examples

Fibonacci numbers are Benford base 10.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Examples

Fibonacci numbers are Benford base 10.an+1 = an + an−1.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Examples

Fibonacci numbers are Benford base 10.an+1 = an + an−1.Guess an = rn: rn+1 = rn + rn−1 or r2 = r + 1.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Examples

Fibonacci numbers are Benford base 10.an+1 = an + an−1.Guess an = rn: rn+1 = rn + rn−1 or r2 = r + 1.Roots r = (1 ±

√5)/2.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Examples

Fibonacci numbers are Benford base 10.an+1 = an + an−1.Guess an = rn: rn+1 = rn + rn−1 or r2 = r + 1.Roots r = (1 ±

√5)/2.

General solution: an = c1rn1 + c2rn2 .

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Examples

Fibonacci numbers are Benford base 10.an+1 = an + an−1.Guess an = rn: rn+1 = rn + rn−1 or r2 = r + 1.Roots r = (1 ±

√5)/2.

General solution: an = c1rn1 + c2rn2 .

Binet: an = 1√5

(1+

√5

2

)n− 1√

5

(1−

√5

2

)n.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Examples

Fibonacci numbers are Benford base 10.an+1 = an + an−1.Guess an = rn: rn+1 = rn + rn−1 or r2 = r + 1.Roots r = (1 ±

√5)/2.

General solution: an = c1rn1 + c2rn2 .

Binet: an = 1√5

(1+

√5

2

)n− 1√

5

(1−

√5

2

)n.

Most linear recurrence relations Benford:

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Examples

Fibonacci numbers are Benford base 10.an+1 = an + an−1.Guess an = rn: rn+1 = rn + rn−1 or r2 = r + 1.Roots r = (1 ±

√5)/2.

General solution: an = c1rn1 + c2rn2 .

Binet: an = 1√5

(1+

√5

2

)n− 1√

5

(1−

√5

2

)n.

Most linear recurrence relations Benford:⋄ an+1 = 2an

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Examples

Fibonacci numbers are Benford base 10.an+1 = an + an−1.Guess an = rn: rn+1 = rn + rn−1 or r2 = r + 1.Roots r = (1 ±

√5)/2.

General solution: an = c1rn1 + c2rn2 .

Binet: an = 1√5

(1+

√5

2

)n− 1√

5

(1−

√5

2

)n.

Most linear recurrence relations Benford:⋄ an+1 = 2an − an−1

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Examples

Fibonacci numbers are Benford base 10.an+1 = an + an−1.Guess an = rn: rn+1 = rn + rn−1 or r2 = r + 1.Roots r = (1 ±

√5)/2.

General solution: an = c1rn1 + c2rn2 .

Binet: an = 1√5

(1+

√5

2

)n− 1√

5

(1−

√5

2

)n.

Most linear recurrence relations Benford:⋄ an+1 = 2an − an−1⋄ take a0 = a1 = 1 or a0 = 0, a1 = 1.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Digits of 2n

First 60 values of 2n (only displaying 30)1 1024 1048576 digit # Obs Prob Benf Prob2 2048 2097152 1 18 .300 .3014 4096 4194304 2 12 .200 .1768 8192 8388608 3 6 .100 .125

16 16384 16777216 4 6 .100 .09732 32768 33554432 5 6 .100 .07964 65536 67108864 6 4 .067 .067

128 131072 134217728 7 2 .033 .058256 262144 268435456 8 5 .083 .051512 524288 536870912 9 1 .017 .046

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Logarithms and Benford’s Law

χ2 values for αn, 1 ≤ n ≤ N (5% 15.5).N χ2(γ) χ2(e) χ2(π)

100 0.72 0.30 46.65200 0.24 0.30 8.58400 0.14 0.10 10.55500 0.08 0.07 2.69700 0.19 0.04 0.05800 0.04 0.03 6.19900 0.09 0.09 1.71

1000 0.02 0.06 2.90

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Logarithms and Benford’s Law: Base 10

log(χ2) vs N for πn (red) and en (blue),n ∈ {1, . . . ,N}. Note π175 ≈ 1.0028 · 1087, (5%,log(χ2) ≈ 2.74).

200 400 600 800 1000

-1.5

-1.0

-0.5

0.5

1.0

1.5

2.0

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Applications

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Applications for the IRS: Detecting Fraud

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Applications for the IRS: Detecting Fraud

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Detecting Fraud

Bank FraudAudit of a bank revealed huge spike ofnumbers starting with

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Detecting Fraud

Bank FraudAudit of a bank revealed huge spike ofnumbers starting with 48 and 49, most dueto one person.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Detecting Fraud

Bank FraudAudit of a bank revealed huge spike ofnumbers starting with 48 and 49, most dueto one person.

Write-off limit of $5,000. Officer had friendsapplying for credit cards, ran up balancesjust under $5,000 then he would write thedebts off.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Data Integrity: Stream Flow Statistics: 130 years, 457,440 records

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Election Fraud: Iran 2009

Numerous protests/complaints over Iran’s 2009elections.Lot of analysis; data moderately suspicious:

First and second leading digits;

Last two digits (should almost be uniform);

Last two digits differing by at least 2.

Warning: enough tests, even if nothing wrongwill find a suspicious result (but when all testsare on the boundary...).

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

The 3x + 1 Problemand

Benford’s Law

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

3x + 1 Problem

Kakutani (conspiracy), Erdös (not ready).

x odd, T (x) = 3x+12k , 2k ||3x + 1.

Conjecture: for some n = n(x), T n(x) = 1.

7

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

3x + 1 Problem

Kakutani (conspiracy), Erdös (not ready).

x odd, T (x) = 3x+12k , 2k ||3x + 1.

Conjecture: for some n = n(x), T n(x) = 1.

7 →1 11

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

3x + 1 Problem

Kakutani (conspiracy), Erdös (not ready).

x odd, T (x) = 3x+12k , 2k ||3x + 1.

Conjecture: for some n = n(x), T n(x) = 1.

7 →1 11 →1 17

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

3x + 1 Problem

Kakutani (conspiracy), Erdös (not ready).

x odd, T (x) = 3x+12k , 2k ||3x + 1.

Conjecture: for some n = n(x), T n(x) = 1.

7 →1 11 →1 17 →2 13

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

3x + 1 Problem

Kakutani (conspiracy), Erdös (not ready).

x odd, T (x) = 3x+12k , 2k ||3x + 1.

Conjecture: for some n = n(x), T n(x) = 1.

7 →1 11 →1 17 →2 13 →3 5

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

3x + 1 Problem

Kakutani (conspiracy), Erdös (not ready).

x odd, T (x) = 3x+12k , 2k ||3x + 1.

Conjecture: for some n = n(x), T n(x) = 1.

7 →1 11 →1 17 →2 13 →3 5 →4 1

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

3x + 1 Problem

Kakutani (conspiracy), Erdös (not ready).

x odd, T (x) = 3x+12k , 2k ||3x + 1.

Conjecture: for some n = n(x), T n(x) = 1.

7 →1 11 →1 17 →2 13 →3 5 →4 1 →2 1,

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

3x + 1 Problem

Kakutani (conspiracy), Erdös (not ready).

x odd, T (x) = 3x+12k , 2k ||3x + 1.

Conjecture: for some n = n(x), T n(x) = 1.

7 →1 11 →1 17 →2 13 →3 5 →4 1 →2 1,2-path (1, 1), 5-path (1, 1, 2, 3, 4).m-path: (k1, . . . , km).

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Heuristic Proof of 3x + 1 Conjecture

an+1 = T (an)

E[log an+1] ≈∞∑

k=1

12k

log(

3an2k

)

= log an + log 3 − log 2∞∑

k=1

k2k

= log an + log(

34

).

Geometric Brownian Motion, drift log(3/4) < 1.64

Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

3x + 1 and Benford

Theorem (Kontorovich and M–, 2005)As m → ∞, xm/(3/4)mx0 is Benford.

Theorem (Lagarias-Soundararajan 2006)

X ≥ 2N , for all but at most c(B)N−1/36X initialseeds the distribution of the first N iterates ofthe 3x + 1 map are within 2N−1/36 of theBenford probabilities.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

3x + 1 Data: random 10,000 digit number, 2k ||3x + 1

80,514 iterations ((4/3)n = a0 predicts 80,319);χ2 = 13.5 (5% 15.5).

Digit Number Observed Benford1 24251 0.301 0.3012 14156 0.176 0.1763 10227 0.127 0.1254 7931 0.099 0.0975 6359 0.079 0.0796 5372 0.067 0.0677 4476 0.056 0.0588 4092 0.051 0.0519 3650 0.045 0.046

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

3x + 1 Data: random 10,000 digit number, 2|3x + 1

241,344 iterations, χ2 = 11.4 (5% 15.5).

Digit Number Observed Benford1 72924 0.302 0.3012 42357 0.176 0.1763 30201 0.125 0.1254 23507 0.097 0.0975 18928 0.078 0.0796 16296 0.068 0.0677 13702 0.057 0.0588 12356 0.051 0.0519 11073 0.046 0.046

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

5x + 1 Data: random 10,000 digit number, 2k ||5x + 1

27,004 iterations, χ2 = 1.8 (5% 15.5).

Digit Number Observed Benford1 8154 0.302 0.3012 4770 0.177 0.1763 3405 0.126 0.1254 2634 0.098 0.0975 2105 0.078 0.0796 1787 0.066 0.0677 1568 0.058 0.0588 1357 0.050 0.0519 1224 0.045 0.046

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

5x + 1 Data: random 10,000 digit number, 2|5x + 1

241,344 iterations, χ2 = 3 · 10−4 (5% 15.5).

Digit Number Observed Benford1 72652 0.301 0.3012 42499 0.176 0.1763 30153 0.125 0.1254 23388 0.097 0.0975 19110 0.079 0.0796 16159 0.067 0.0677 13995 0.058 0.0588 12345 0.051 0.0519 11043 0.046 0.046

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Productsof

Random Variables

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Preliminaries

X1 · · ·Xn ⇔ Y1 + · · ·+ Yn mod 1, Yi = logB Xi

Density Yi is gi , density Yi + Yj is

(gi ∗ gj)(y) =∫ 1

0gi(t)gj(y − t)dt .

hn = g1 ∗ · · · ∗ gn, ĝ(ξ) = ĝ1(ξ) · · · ĝn(ξ).

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Modulo 1 Central Limit Theorem

Theorem (M– and Nigrini 2007){Ym} independent continuous random variableson [0, 1) (not necc. i.i.d.), densities {gm}.Y1 + · · ·+ YM mod 1 converges to the uniformdistribution as M → ∞ in L1([0, 1]) if and only iffor all n 6= 0, limM→∞ ĝ1(n) · · · ĝM(n) = 0.

⋄ Gives info on rate of convergence.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Proof under stronger conditions

Use standard CLT to show Y1 + · · ·+ YMtends to a Gaussian.

Use Poisson Summation to show theGaussian tends to the uniform modulo 1.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Proof under stronger conditions

-4 -2 2 4

0.1

0.2

0.3

0.4

Figure: Plot of normal (mean 0, stdev 1).

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Proof under stronger conditions

0.2 0.4 0.6 0.8 1.0

1

2

3

4

Figure: Plot of normal (mean 0, stdev .1) modulo 1.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Proof under stronger conditions

0.2 0.4 0.6 0.8 1.0

0.995

1.000

1.005

1.010

1.015

Figure: Plot of normal (mean 0, stdev .5) modulo 1.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Conclusions

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Current / Future Investigations

Develop more sophisticated tests for fraud.

Study digits of other systems.⋄ Break rod of fixed length a variable numberof times.⋄ Break rods of variable length a variablenumber of times.⋄ Break rods of variable length, each piecethen breaks with given probability.⋄ Break rods of variable integer length, eachpiece breaks until is a prime, or a square, ....

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

Conclusions and Future Investigations

See many different systems exhibit Benfordbehavior.

Ingredients of proofs (logarithms,equidistribution).

Applications to fraud detection / dataintegrity.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

References

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

A. K. Adhikari, Some results on the distribution of the mostsignificant digit, Sankhya: The Indian Journal of Statistics, SeriesB 31 (1969), 413–420.

A. K. Adhikari and B. P. Sarkar, Distribution of most significantdigit in certain functions whose arguments are random variables,Sankhya: The Indian Journal of Statistics, Series B 30 (1968),47–58.

R. N. Bhattacharya, Speed of convergence of the n-foldconvolution of a probability measure ona compact group, Z.Wahrscheinlichkeitstheorie verw. Geb. 25 (1972), 1–10.

F. Benford, The law of anomalous numbers, Proceedings of theAmerican Philosophical Society 78 (1938), 551–572.

A. Berger, Leonid A. Bunimovich and T. Hill, One-dimensional

dynamical systems and Benford’s Law, Trans. Amer. Math. Soc.

357 (2005), no. 1, 197–219.

81

Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

A. Berger and T. Hill, Newton’s method obeys Benford’s law, TheAmer. Math. Monthly 114 (2007), no. 7, 588-601.

J. Boyle, An application of Fourier series to the most significantdigit problem Amer. Math. Monthly 101 (1994), 879–886.

J. Brown and R. Duncan, Modulo one uniform distribution of thesequence of logarithms of certain recursive sequences,Fibonacci Quarterly 8 (1970) 482–486.

P. Diaconis, The distribution of leading digits and uniformdistribution mod 1, Ann. Probab. 5 (1979), 72–81.

W. Feller, An Introduction to Probability Theory and its

Applications, Vol. II, second edition, John Wiley & Sons, Inc.,

1971.

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Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

R. W. Hamming, On the distribution of numbers, Bell Syst. Tech.J. 49 (1970), 1609-1625.

T. Hill, The first-digit phenomenon, American Scientist 86 (1996),358–363.

T. Hill, A statistical derivation of the significant-digit law,Statistical Science 10 (1996), 354–363.

P. J. Holewijn, On the uniform distribuiton of sequences ofrandom variables, Z. Wahrscheinlichkeitstheorie verw. Geb. 14(1969), 89–92.

W. Hurlimann, Benford’s Law from 1881 to 2006: a bibliography,http://arxiv.org/abs/math/0607168.

D. Jang, J. U. Kang, A. Kruckman, J. Kudo and S. J. Miller,

Chains of distributions, hierarchical Bayesian models and

Benford’s Law, Journal of Algebra, Number Theory: Advances

and Applications, volume 1, number 1 (March 2009), 37–60.83

Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

E. Janvresse and T. de la Rue, From uniform distribution toBenford’s law, Journal of Applied Probability 41 (2004) no. 4,1203–1210.

A. Kontorovich and S. J. Miller, Benford’s Law, Values ofL-functions and the 3x + 1 Problem, Acta Arith. 120 (2005),269–297.

D. Knuth, The Art of Computer Programming, Volume 2:Seminumerical Algorithms, Addison-Wesley, third edition, 1997.

J. Lagarias and K. Soundararajan, Benford’s Law for the 3x + 1Function, J. London Math. Soc. (2) 74 (2006), no. 2, 289–303.

S. Lang, Undergraduate Analysis, 2nd edition, Springer-Verlag,New York, 1997.

84

Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

P. Levy, L’addition des variables aléatoires définies sur unecirconférence, Bull. de la S. M. F. 67 (1939), 1–41.

E. Ley, On the peculiar distribution of the U.S. Stock IndicesDigits, The American Statistician 50 (1996), no. 4, 311–313.

R. M. Loynes, Some results in the probabilistic theory ofasympototic uniform distributions modulo 1, Z.Wahrscheinlichkeitstheorie verw. Geb. 26 (1973), 33–41.

S. J. Miller, When the Cramér-Rao Inequality provides noinformation, Communications in Information and Systems 7(2007), no. 3, 265–272.

S. J. Miller and M. Nigrini, The Modulo 1 Central Limit Theoremand Benford’s Law for Products, International Journal of Algebra2 (2008), no. 3, 119–130.

S. J. Miller and M. Nigrini, Order Statistics and Benford’s law,International Journal of Mathematics and MathematicalSciences, Volume 2008 (2008), Article ID 382948, 19 pages.

85

Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

S. J. Miller and R. Takloo-Bighash, An Invitation to ModernNumber Theory, Princeton University Press, Princeton, NJ, 2006.

S. Newcomb, Note on the frequency of use of the different digitsin natural numbers, Amer. J. Math. 4 (1881), 39-40.

M. Nigrini, Digital Analysis and the Reduction of Auditor LitigationRisk. Pages 69–81 in Proceedings of the 1996 Deloitte & Touche/ University of Kansas Symposium on Auditing Problems, ed. M.Ettredge, University of Kansas, Lawrence, KS, 1996.

M. Nigrini, The Use of Benford’s Law as an Aid in AnalyticalProcedures, Auditing: A Journal of Practice & Theory, 16 (1997),no. 2, 52–67.

M. Nigrini and S. J. Miller, Benford’s Law applied to hydrologydata – results and relevance to other geophysical data,Mathematical Geology 39 (2007), no. 5, 469–490.

86

Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

M. Nigrini and S. J. Miller, Data diagnostics using second ordertests of Benford’s Law, Auditing: A Journal of Practice andTheory 28 (2009), no. 2, 305–324.

R. Pinkham, On the Distribution of First Significant Digits, TheAnnals of Mathematical Statistics 32, no. 4 (1961), 1223-1230.

R. A. Raimi, The first digit problem, Amer. Math. Monthly 83(1976), no. 7, 521–538.

H. Robbins, On the equidistribution of sums of independentrandom variables, Proc. Amer. Math. Soc. 4 (1953), 786–799.

H. Sakamoto, On the distributions of the product and the quotientof the independent and uniformly distributed random variables,Tôhoku Math. J. 49 (1943), 243–260.

P. Schatte, On sums modulo 2π of independent randomvariables, Math. Nachr. 110 (1983), 243–261.

87

Introduction General Theory Applications 3x + 1 Products F Conclusions Refs

P. Schatte, On the asymptotic uniform distribution of sumsreduced mod 1, Math. Nachr. 115 (1984), 275–281.

P. Schatte, On the asymptotic logarithmic distribution of thefloating-point mantissas of sums, Math. Nachr. 127 (1986), 7–20.

E. Stein and R. Shakarchi, Fourier Analysis: An Introduction,Princeton University Press, 2003.

M. D. Springer and W. E. Thompson, The distribution of productsof independent random variables, SIAM J. Appl. Math. 14 (1966)511–526.

K. Stromberg, Probabilities on a compact group, Trans. Amer.Math. Soc. 94 (1960), 295–309.

P. R. Turner, The distribution of leading significant digits, IMA J.Numer. Anal. 2 (1982), no. 4, 407–412.

88

IntroductionGeneral TheoryApplications3x+1Products FConclusionsRefs