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Topic: Heuristic Proofs of the Prime Number Theorem
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Posts: 1
Registered: 1/27/17
Heuristic Proofs of the Prime Number Theorem
Posted: Jan 27, 2017 8:09 AM
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For many of us non-mathematicians (my background is in physics), a real
mathematician's proof of the Prime Number Theorem (PNT) is beyond our
understanding.  As a result, many heuristic proofs can be found on the
internet, meaning that the techniques employ intuitive judgements that
are not necessarily correct.  I developed my heuristic proof for the
PNT in 1977, thinking at the time that it was legitimate.

The heuristic proofs that I have found online are probabilistic
arguments that treat the prime numbers as if they were randomly
distributed.  One type of proof begins with the probability P(x) that x
is a prime number and consider the probability P(x+1) that the next
integer x+1 is a prime [1, 2, 3].  The proof therefore depends on the
possibility of having two consecutive integers that are both prime
numbers, which is not realistic, except for the integers 2 and 3.  The
resulting differential equation has the solution P = 1/ln(cx), where
the constant c is undetermined.

Another heuristic proof considers two large consecutive primes [4].  A
series of heuristic arguments leads to the differential equation  dP/dx
= -P(x) P(sqrt(x))/x  and arrives at the solution P(x) = 0.5 / ln(x).
  (The numerator should be 1.)

I developed this proof in 1977 when I was a mailman, searching for a
job in physics.  At the time, I thought that it was a legitimate proof.
 The proof follows the Sieve of Eratosthenes, but applied to two
consecutive regions between the squares of primes.

Consider two ranges of integers:
    Range A: R_A = (p_n-1)^2 - (p_n)^2
              the number of integers from from (p_n-1)^2   to  (p_n)^2
    Range B: R_B = (p_n)^2 - (p_n+1)^2
              the number of integers from (p_n)^2          to
 (p_n+1)^2 -1

Range A has been sieved by prime divisors up to p_n-1 so that only
primes remain.  The average density of primes in the range is rho_A

          rho_A = (#primes in the range) / R_A

Now use the same sieve process that was applied to Range A and apply it
to Range B, using the same prime divisors up to p_n-1.

ASSUMPTION 1:  Assume that the sieve for Range A, removes the same
proportion of integers from Range B as it did from Range A.  This means
that Range B now has a proportion rho_A of its original R_B integers
remaining.  The difference is that, in Range A, all the remaining
integers are primes but, in Range B, there are still a few remaining
non-primes, which will be removed by the final pass of the sieve, using
the new prime divisor p_n.

ASSUMPTION 2: Assume that the final pass of the sieve, using prime
divisor p_n, leaves a proportion (1- 1/p_n) of the remaining integers
unscathed and they are all primes, with the result that:

    rho_B = (1-1/p_n) rho_A                              [Eq'n 1]

Let y = p_n and rewrite Eq'n 1 so that the change in density (rho_B -
rho_A) is given by:

    delta rho(y^2) ~  - rho(y^2) / y                     [Eq'n 2]

The prime p_n+1 ~ p_n + 1/rho(p_n) and so the range R_n is given by:

    R_n = d(y^2) ~  2p_n / rho(p_n)                      [Eq'n 3]

    delta(y^2) ~  2y / rho(y)                            [Eq'n 4]

Dividing Eq'n 2 by Eq'n 4, we obtain:

    d rho(y^2) / d(y^2)  ~  -rho(y^2) rho(y) / (2y^2)    [Eq'n 5]

or, letting x = y^2,

    d rho(x) / dx  ~   -rho(x) rho(sqrt(x)) / (2x)      [Eq'n 6]

The solution is:

    rho(x) = 1 / ln(x)                      [Eq'n 7]

1 / ln(cx) is not a solution unless c = 1.

My Consecutive-Squares-of-Primes Heuristic appears to correct the
problems found in some other heuristic proofs, and yet it cannot be
considered a legitimate mathematical proof.  What is wrong with it?
 Let's examine the first equation.  Mertens' Third Theorem says that:

     1/ln(p_n) = 1.781 Prod_n        for large n        [Eq'n 8]   

where Prod_n = (1-1/2) (1-1/3) (1-1/5) ....  (1-1/p_n)

Knowing that the the solution of the PNT is 1/ln(x), that tells us that

    rho((p_n)^2) ~ (1.781/2) Prod_n   and     rho((p_n-1)^2) ~
(1.781/2) Prod_n-1

and therefore

    rho((p_n)^2) / rho((p_n-1)^2)  = (1-1/p_n)

This means that Equation 1 is correct.  Equations 2 to 7, even though
not mathematically very rigorous, are also essentially correct.  So,
what is wrong?  The problem is that Equation 1, even though correct,
was rationalized on the basis of two incorrect assumptions.  That is
deadly in mathematics.  Two wrongs do not make a right, even if they
give you the right answer.

Looking at Equation 1, it seems obvious that the largest prime divisor
p_n, on the final pass of the sieve, is removing a fraction 1/p_n of
the integers that remain after previous passes of the sieve, which used
smaller prime divisors.  This is Assumption 2, one of the two
assumptions used to derive the equation.

Consider the range of 72 integers from 17^2 to 19^2 -1  (289 to 360).
 There are 11 primes in this range.
In the final pass of the Sieve of Eratosthenes, 17 removes 2 integers:
  17x17 & 17x19.  Therefore there must have been 11 + 2 = 13 integers
left in the range before the final pass.  We have assumed that the
final pass removes a proportion 1/17, but the actual proportion is
 2/13 = 2.615/17.  So, in this example, the final pass removes
approximately 2.6 times as many integers as assumed.

I checked the 18 ranges above the squares of p_481 to p_498 in the same
manner as the previous example (p_481 = 3433; p_481^2 = 11,785,489).
 For these 18 cases, p_n removed either 2 or 3 non-primes in every
case, with an average of 2.5.  The proportion removed was between 1.5
and 8.4 times the expected ratio of 1/p_n, with a weighted average of
2.9 times the expected ratio.  The actual number of integers removed
from the range by the final pass is always at least two.  From Eq'ns 3
and 7, for large n, the number of primes in the range is approximately
p_n so, if k integers are removed on the final pass, the proportion is
k/p_n.  k must be 2 or more and, on average, seems to be approximately
2.5 or a little higher.

In the range from (p_n)^2 to (p_n+1)^2 -1, the prime divisor p_n on the
final pass always removes (p_n)^2 and always removes (p_n)(p_n+1).
 Using reasoning like Eq'n 3, it can be shown that (p_n)(p_n+2) is
approximately the same size as (p_n+1)^2, which is the end of the
range.  So, about half the time, (p_n)(p_n+2) is in the range and half
the time it is in the next range.  The average of 2.5 integers that was
seen in the 18 cases that were checked is therefore quite reasonable.
 From this point of view, it now seems unreasonable to expect the final
pass to remove a proportion 1/p_n of the integers that remained after
previous passes of the sieve, which used prime divisors smaller than
p_n and have nothing to do with removing integers like (p_n)^2,
 (p_n)(p_n+1) or (p_n)(p_n+2), which are the integers that p_n removes.
 So, here is the situation: Equation 1 is correct, but 1/p_n is not the
proportion of integers removed on the final pass, which was Assumption
2.  The conclusion is correct, but one of the two premises is
incorrect.  That means the other premise must also be incorrect.  Both
Assumption 1 and Assumption 2 are wrong, but they were used to derive
Equation 1, which is correct.  This demonstrates how dangerous
heuristic reasoning can be, and nowhere more so than in dealing with
prime numbers.

Equation 1 looks deceptively simple and yet, I have no way to
rationalize it.  As demonstrated earlier, if we already know that the
PNT is true and if we use Mertens' Third Theorem, it demonstrates that
Equation 1 is correct.  But, if the PNT is used to prove Equation 1,
then Equation 1 cannot be used to prove the PNT.  Before, I
rationalized Equation 1, based on two assumptions that turned out to be
incorrect.  I would dearly like to know a legitimate rationalization of
Equation 1, but fear that the explanation is buried deep in complicated
number theory that I won't understand.  It is a humorous situation to
have spent considerable effort deriving a fine-looking proof, only to
find out that the satisfying understanding is an illusion, the result
of wrong assumptions conspiring together to tell the truth.  I am not
alone.  Many of you other heuristic-proofers out there are in the same

[1] Heuristic Derivation of the Prime Number Theorem,by Frank Morgan,

[2] God Plays Dice, by Michael Lugo, 2008

[3] Probabilistic Sieve of Eratosthenes, by Joriki, 2012

[4] Differential Equation Modeling Distribution of Primes, by Babak,

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