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A brand new pattern of statistical regularity in
the prime number distribution
Lucas Lacasa Bartolo Luque Dpto.Matemática
Aplicada, ETSI Aeronáuticos, UPM, Spain.
Octavio Miramontes Instituto de Física,
UNAM, México.
3. Generalized Benford's law
1. The Prime Counting Function
An integer is said to be prime if its only
divisors are 1 and itself. Despite the simplicity
of this definition and that the primes work as
the bricks of integers, the prime distribution
shows local randomness but global regularity.
The prime counting function is defined as
As shown by Pietronero et al. Physica A 293
(2001), a list of numbers which comes from a
distribution with density N-1 will also be
Benford
This kind of distributions are easily generated
through multiplicative processes.
Now if the distribution has a general density
N-?, the Benford's law associated will become a
Generalized Benford (GB), of the
shape This law is also extendible to the k
first significant digits.
The prime numbers seem to be random, no one knows
where the next prime is going to appear. The
counting function is a stepped function. However,
it exhibits a surprising 'smooth behavior' when
looked at large scale. This fact has astonished
mathematicians for centuries. Gauss gave a first
statistical approximation to p(x) as x/ln(x),
later improved in terms of logarithm
integral which doesn't have an analytic
expression. In 1896, de la Vallée Poussin and
Hadamard proved the prime number theorem, which
states that the counting function is asymptotic
to x/ln(x)
4. GB in the primes the pattern
Here we represent the first 1, 2, 3 and 4
significant digits distribution of the prime
number sequence, in the interval (1-1011). The
lines represent a fitting to the generalized
Benford's law. Note that the agreement is very
good. What happens if we vary the size of the
set? In the figures below we represent the same
fitting for different intervals from 1 to N10k.

2. The Benford's law
Frank Benford , in 1938, studied a large amount
of data sets (population sizes, river basin
drainage areas, tables of molecular weights, etc)
and realized that the leading digit or first
significant digit (for example, the first
significant digit of 3.14 and 0.0413 is 3 and 4
respectively) tend not to be uniformly
distributed, but follows
This law can be extended with the same functional
form to the k first significant digits.
The proof of Benfords law was done in 1996 by
Theodore Hill (A Statistical Derivation of the
Significant-Digit law). Hill later showed there
was a kind of central limit theorem that applies
to a wide variety of distributions. This one
states that combinations of different
distributions tend towards the distribution
predicted by Benfords law even when the original
distributions don't.
1 30.1 2 17.6 3 12.5
4 9.7 5 7.9 6 6.7
7 5.8 8 5.1 9 4.6
The distribution in each case still fits to GB,
but with different exponent ?(N). When we
increase the size N of the set under study, the
fitting exponent ?(N) decreases, following the
law ?(N)1/(ln(N)-1.2) (up-right figure). In
particular, note that when N?8, ? ?0, and GB
turns into a uniform distribution as expected.
First significant digit distribution of the set
of 201 physical constants. The red line
corresponds to the Benford's law.
4. Deduction of the counting function
Does the Benford law apply to prime numbers?
Using the upper and lower bounds of function
p(n) from the prime number theorem, it can be
shown that the first significant digit
distribution for the prime number sequence
approximates a uniform distribution, that is,
P(d) 1/9 0.11...
We can naturally deduce a new approximation to
the counting function from the pattern, that we
will call Lu(n). In fact, assuming that the
'density' of the prime number distribution goes
like x-?(N) , normalizing, and imposing Lu(x)
x/ln(x), we derive the following counting
function
The figure besides represents the first
significant digit distribution of the prime
number sequence, for the primes between 2 and N
1011. Note that the values are far from those
predicted by Benford law, varying in the range
0.108-0.117, that is, more or less uniformly
distributed.
This new approximation works better than Gauss's
approximation x/ln(x) and Legendre's best
approximation x/(ln(x)-1).