THE POISSON-DIRICHLET LAW FOR PRIME PREDECESSORS
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PART I. THE QUESTION AND THE ANSWER
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Chapter 1. A SMALL STORY
Pick a prime number, say p = 101. Now look at p - 1 = 100.
100 factors into primes: 100 = 2 * 2 * 5 * 5.
Those factors tell us something about the prime. Every time you pick
a prime, subtracting 1 gives a number whose prime factors play a
hidden game.
p = 17 p - 1 = 16 = 2 * 2 * 2 * 2 (a power of two)
p = 97 p - 1 = 96 = 2 * 2 * 2 * 2 * 2 * 3 (mostly twos)
p = 101 p - 1 = 100 = 2 * 2 * 5 * 5 (mixed)
p = 257 p - 1 = 256 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 (a big power)
p = 2521 p - 1 = 2520 = 2^3 * 3^2 * 5 * 7 (several pieces)
The question this document answers is:
If you pick a random prime p between 3 and x, and let x grow
large, what does the typical p - 1 look like as a product of
primes?
A more careful way to ask: what is the "typical" list of prime
factors of p - 1? Do they tend to be small? Large? Do they have a
"shape" that is stable as x grows?
We will see that the answer is YES: there is a very specific
stable shape, called the Poisson-Dirichlet distribution with
parameter one. That is the main theorem.
Chapter 2. HOW DO WE MEASURE THE SHAPE OF A FACTORIZATION?
For a number like 2520, the factorization is
2520 = 2 * 2 * 2 * 3 * 3 * 5 * 7,
which contains seven prime factors: three 2s, two 3s, one 5, one 7.
Different primes p give different lists. To say what is "typical"
we need to compare these lists.
The key idea is to look at logarithms. Write the prime factors of
p - 1, with multiplicity, in decreasing order:
q_1 >= q_2 >= q_3 >= ... >= 1.
For p = 2521, we have p - 1 = 2520, so the list is
q_1 = 7, q_2 = 5, q_3 = 3, q_4 = 3, q_5 = 2, q_6 = 2, q_7 = 2.
For p = 97, the list is
q_1 = 3, q_2 = 2, q_3 = 2, q_4 = 2, q_5 = 2, q_6 = 2.
For p = 257, the list is
q_1 = q_2 = ... = q_8 = 2.
Now define, for each index j >= 1,
V_j(p) = log q_j(p) / log(p - 1).
These V_j are real numbers between 0 and 1. They measure how much
of the "logarithmic size" of p - 1 each prime factor accounts for.
Because a product of numbers has logarithm equal to a sum of the
logarithms, the V_j add up:
V_1 + V_2 + V_3 + ... = 1.
So the V's partition the interval [0, 1] of total mass 1 into
pieces of different sizes. The list
V_1 >= V_2 >= V_3 >= ...
is the shape of the factorization of p - 1.
For example, for p = 2521 and p - 1 = 2520:
log q_1 = log 7 ~ 1.946
log q_2 = log 5 ~ 1.609
log q_3 = log 3 ~ 1.099
log q_4 = log 3 ~ 1.099
log q_5 = log 2 ~ 0.693
log q_6 = log 2 ~ 0.693
log q_7 = log 2 ~ 0.693
log q_1 + ... + log q_7 = log 2520 ~ 7.832
So V_1 ~ 1.946 / 7.832 ~ 0.248, meaning the largest prime factor
(7) accounts for roughly a quarter of the total logarithmic size.
The largest piece is quite small, and the factorization is spread
out.
For p = 257, on the other hand, V_1 = ... = V_8 = 1/8 = 0.125, and
the factorization is very concentrated in equally-sized small pieces.
We want to know: as x grows, what is the distribution of the shape
(V_1, V_2, V_3, ...) if p is chosen uniformly among primes 3 <= p <= x?
Chapter 3 tells us what that distribution is.
Chapter 3. THE POISSON-DIRICHLET DISTRIBUTION
We now describe the mystery shape. It was invented by probabilists
in the twentieth century and has a very simple construction that
is usually called "stick-breaking."
Imagine a stick of total length 1. We are going to break it into
pieces, one after another. Here is the algorithm.
Step 1. Pick a random number U_1 uniformly from (0, 1). That is,
U_1 is any real in (0, 1) with equal chance. Snap off a
piece of length B_1 = 1 - U_1. We have used up
length 1 - U_1. The remaining stick has length U_1.
Step 2. Pick another independent random number U_2 uniformly
from (0, 1). Snap off a piece of length
B_2 = U_1 (1 - U_2). We have used up U_1 (1 - U_2) out
of the remaining U_1. The remaining stick has length
U_1 U_2.
Step 3. Pick U_3, snap off B_3 = U_1 U_2 (1 - U_3). The
remaining stick has length U_1 U_2 U_3.
And so on.
General formula: B_j = (U_1 U_2 ... U_{j-1}) (1 - U_j).
The infinite sequence B_1, B_2, B_3, ... has the property that the
pieces sum to one:
B_1 + B_2 + B_3 + ... = 1 (always, with probability 1).
Exercise. Verify the telescoping:
(1 - U_1) + U_1 (1 - U_2) + U_1 U_2 (1 - U_3) + ...
= 1 - U_1 U_2 U_3 U_4 ...
As the product U_1 U_2 U_3 ... shrinks to zero (almost surely), the
sum tends to 1. Good.
The pieces B_j are NOT, in general, in decreasing order. The
largest piece might be B_1, but sometimes B_2 is larger, or B_3,
and so on. The Poisson-Dirichlet distribution with parameter one,
written PD(1), is defined as the distribution of the DECREASING
REARRANGEMENT
L_1 >= L_2 >= L_3 >= ...
of the pieces B_1, B_2, B_3, ... That is: L_1 is the largest B_j,
L_2 is the second largest B_j, and so on. Since the B_j add to 1,
so do the L_j:
L_1 + L_2 + L_3 + ... = 1.
Some facts about PD(1).
- The first piece B_1 is a uniform random variable, so on
average it is 1/2. Actual largest piece L_1 is a bit larger
than that, because it might not be B_1.
- The largest piece L_1 has median roughly 0.606. That is, about
half the time L_1 is bigger than 0.606, and about half the time
it is smaller. So the shape of a typical prime's factorization
is dominated by one big piece that accounts for more than
half of the total logarithmic size.
- The second piece L_2 is also substantial, typically of order a
few tenths.
- There are infinitely many small pieces. For any t > 0 there are
only finitely many pieces larger than t, but the number of
pieces larger than t scales roughly like 1/t as t shrinks. So
small pieces are numerous; big pieces are rare.
- The largest piece L_1 has a very specific distribution called
the "Dickman distribution," and we will meet it again in
Chapter 5 when we discuss ordinary integers.
For us, the important thing is: PD(1) is a very specific random
"shape" for splitting a total of 1 into pieces. It is not uniform:
it has a few big pieces and many small pieces. And it is going to
be the shape of the prime factors of p - 1.
Chapter 4. THE MAIN THEOREM
Now we can state the answer.
THEOREM (Main theorem of the paper).
Fix an integer k >= 1 and a bounded continuous function F on
the cube [0, 1]^k. Then
(1 / (pi(x) - 1)) * sum_{3 <= p <= x, p prime}
F(V_1(p), V_2(p), ..., V_k(p))
converges, as x tends to infinity through all real values, to
the expectation of F(L_1, L_2, ..., L_k),
where L_1, L_2, L_3, ... is the PD(1) rearrangement of the
stick-breaking pieces B_1, B_2, B_3, ....
In plain English: for any finite window of the prime-factor "shape"
of p - 1, the distribution of that window matches the PD(1) shape
as x grows.
Since F was arbitrary, this also tells us that, for every fixed k,
(V_1, V_2, ..., V_k) converges IN DISTRIBUTION to (L_1, ..., L_k).
The normalization uses log(p - 1), not log x, but this makes no
difference in the limit, since p is close to x in the prime average.
The prime average is taken over ordinary counting, not weighted in
any special way.
This theorem strengthens several earlier results and resolves a
conjecture. Chapter 5 tells that story.
PART II. WHAT WAS KNOWN, AND WHY IT IS HARD
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Chapter 5. A SHORT HISTORY
For ordinary integers (not primes), the story starts in the 1930s.
5.1 Dickman (1930) and the largest prime factor of n
Dickman studied: for a uniformly chosen integer n <= x, what is the
distribution of the largest prime factor P+(n)?
Write P+(n) <= x^{1/u} for a parameter u >= 1. Dickman showed:
#{1 <= n <= x : P+(n) <= x^{1/u}} ~ x * rho(u),
where rho is the "Dickman function:"
rho(u) = 1 for 0 <= u <= 1
u rho'(u) + rho(u-1) = 0 for u > 1.
For example, rho(2) = 1 - log 2 ~ 0.307. So roughly 31 percent of
integers up to x have largest prime factor at most sqrt(x).
Dickman's work started a large field called "smooth number theory."
N. G. de Bruijn (1951) gave the modern asymptotic form of rho(u)
that we still use today.
5.2 Billingsley (1972): the joint law
Billingsley asked about the joint distribution of the ordered prime
factors of a random integer. He proved: as x -> infinity, the
decreasing sequence of logarithms of large prime divisors of a
random integer n <= x, rearranged and normalized by log x, converges
in distribution to a certain limit. That limit is precisely the
Poisson-Dirichlet distribution.
5.3 Donnelly and Grimmett (1993): the clean version
Donnelly and Grimmett gave a "size-biased" treatment of the
problem, which handles multiplicities cleanly and leads directly to
the stick-breaking construction of PD(1). This is the version we use.
5.4 Arratia, Kochman and Miller (2014): factorial intensities
Arratia, Kochman and Miller recast the result in terms of "joint
intensities" of factor tuples, and gave a factorial-moment form
that is the cleanest analytic version.
Combined, these results for ordinary integers form a complete proof
that for a uniformly chosen n <= x, the shape of the prime
factorization of n converges to PD(1).
5.5 Ford, Konyagin and Luca (2010): the conjecture for primes
The problem for primes is different. Ford, Konyagin and Luca, in
their 2010 paper "Prime chains and Pratt trees," formulated the
conjecture:
For a uniformly chosen prime p <= x, the shape of the prime
factorization of p - 1 also converges to PD(1).
Exactly the same answer as for ordinary integers, but now on
primes. FKL knew this was much harder than the ordinary-integer
case: for ordinary integers, subtracting 1 is not a thing (you
just factor n), but for primes, subtracting 1 destroys the
structure of the number in a way that is hard to control.
5.6 Granville (2008): the smoothness conjecture for primes
Granville, in his survey "Smooth numbers: computational number
theory and beyond," stated the conjectured asymptotic:
#{p <= x : P+(p-1) <= x^{1/u}} ~ pi(x) * rho(u) for u >= 1.
This is the "shifted-prime Dickman conjecture." It is a
consequence of the full PD(1) theorem.
5.7 Partial results before 2026
Some partial results were known.
- Baker and Harman (1998) proved lower bounds of the form
x / (log x)^C for primes p <= x with p - 1 having no large
prime factors. Threshold x^{0.2961}.
- Lichtman (2022) improved this to threshold x^{0.2844} for
primes in (x, 2x].
- Ford (2025) and Gorodetsky (2026) proved strong results about
small prime divisors of shifted primes, using sieve methods.
- Bharadwaj and Rodgers (2026) proved PD(1) for shifted primes
under the Elliott-Halberstam conjecture (a very strong but
unproven distributional hypothesis about primes in arithmetic
progressions). Unconditionally, they got "level 1/2": the
moment formulas hold on the part of the simplex where the
coordinates sum to less than 1/2.
5.8 2026: the full unconditional proof
Chapter 6. WHY IS THIS HARD FOR PRIMES?
For ordinary integers, Dickman, Billingsley, Donnelly-Grimmett and
Arratia-Kochman-Miller all worked. For primes, why should it be any
harder?
The reason is subtle but important. Let's see what happens if we try
to naively adapt the ordinary-integer proof.
Counting trick for integers.
Suppose we want to count integers n <= x whose prime factors include
a specific set of "small" primes q_1, ..., q_r. That is easy: the
answer is just floor(x / (q_1 q_2 ... q_r)), which is x / Q where
Q = q_1 q_2 ... q_r.
Counting trick for primes.
Now suppose we want to count primes p <= x such that p - 1 is divisible
by q_1 q_2 ... q_r = Q. That means p is congruent to 1 modulo Q. To
count such primes, we need the prime number theorem for arithmetic
progressions:
#{p <= x : p ≡ 1 mod Q} = Li(x) / phi(Q) + error.
For FIXED Q, or Q growing slowly like a power of log x, the error
term is smaller than Li(x)/phi(Q) by any fixed power of log x.
That is Siegel-Walfisz.
But for the full PD(1) theorem, we need Q up to x^{1-eps}, and no
unconditional prime-counting tool is sharp enough at that scale.
The best we can do unconditionally is Bombieri-Vinogradov: on
average over Q up to x^{1/2 - eps}, the error is small. Beyond that,
it is conjectural (Elliott-Halberstam).
The gap.
Suppose we try to bound the number of primes p <= x with p - 1
having all prime factors <= x^{1/u}. For u > 1, this is exactly
Granville's conjecture. The naive count from PNT-APs is
sum over Q | m for some m <= x, m has only small prime factors
of (Li(x) / phi(Q)) + (error terms).
If we ignore the errors, the main-term sum equals Li(x) times
rho(u), which is exactly the Dickman/Granville answer. So the naive
calculation WOULD give the right answer if the error terms were
uniformly tiny.
But the error term is x / (log x)^A for each individual modulus Q.
We have to sum over many (potentially exp(some power of log x) many)
moduli Q. The errors then ADD UP and swamp the main term. That is
the whole problem.
So we need a way to control these errors SIMULTANEOUSLY over all
relevant moduli, using cancellation rather than PNT-APs. That is
exactly what the FKL conjecture and Granville's conjecture ask for,
and it is why the problem remained open for so long.
The rest of the document explains the OpenAI proof of this: how,
despite the failure of uniform PNT-APs, there is enough structure
(via "signed sums" and "dilation graphs") to make the error terms
cancel. That cancellation is the big new idea of the paper.
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PART III. THE PROOF
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Chapter 7. THE PROOF IN ONE PAGE
Here is how the proof goes, in one page.
The key realization: trying to directly control the small-prime
divisors of p - 1 via PNT-APs is doomed, as we saw in Chapter 6.
We have to find some structure that lets the errors cancel.
The paper's strategy has four stages. Stages 1 and 2 are the two
analytic inputs. Stage 3 uses them to extract the statistic. Stage
4 is a purely probabilistic argument.
Step 1. One-sided marked correlation.
We prove a "signed-moment" identity for a very special sum: the
sum of a certain "mark" times the indicator of a prime factor of
p - 1 being in a certain range.
The technical heart of Step 1 is called the "determinant estimate."
We construct a weighted bipartite graph between pairs of integers
(m, n) with mn - 1 divisible by a product of small primes, and we
show that a signed sum over the edges of this graph is very small
(like x divided by (log x)^D for any fixed D).
The method is Cauchy-Schwarz plus analytic number theory:
(a) Cauchy-Schwarz turns the one-variable-at-a-time problem into
a two-variable-at-a-time problem.
(b) That becomes a problem about two vectors of small-prime
products (one on each side).
(c) A "determinant" relation t = bmn - ars appears. The
"major" and "minor" arcs around rationals h/k pick out the
major contribution.
(d) The "minor" contribution is bounded by an "exact memory
identity": each prime's "lifespan" (its run of activity) is
tracked, and a signed sum over all lifespans is bounded by
the "probability" of picking its residue line.
The signed-moment approach has precedents in the literature
(Helfgott-Radziwill, Pilatte), but the state in the OpenAI paper
is richer: it keeps track of an "active prime list" AND a primitive
lattice vector, at each prime p. Root averaging gives a uniform
projective line, and the memory preserves that line across gaps
between active runs.
The output of Step 1 is Theorem 6.2 of the paper: the "one-sided
marked Type II estimate."
Step 2. Two-sided marked correlation.
We prove a related estimate where both sides carry marks. The
technical heart here is the "dilation-graph" method from the
companion paper (S), with some new endpoint estimates.
The idea is to represent the correlation statistic as a matrix
element of an operator on L^2 of sequences. We split the operator
into a "raw" piece (no shifts) and two "comparison" pieces (with a
shift). Each comparison is "morally multiplicative" with respect to
the mark structure, and the total sum is bounded via a finite
character expansion and a sparse-frequency analysis.
The output of Step 2 is Theorem 6.1 of the paper: the "two-sided
marked correlation."
Step 3. Extraction.
Now we use the two correlation estimates to prove the interior
prime statistic theorem:
sum over primes p <= x of Phi((p-1)/x) * (f_x(p-1) - C_x)
= o(x / log x),
where f_x counts the number of ways a chosen set of large primes
divides n and C_x is the average.
The proof is a sieve argument with four sub-steps:
(a) Presieve: restrict x to p where p - 1 has no very small
prime factors, using the block sieve.
(b) Apply the one-sided estimate to replace composite values
in the sieve by "rough" values with the right density.
(c) Additional marks let the two-sided estimate bound the
resulting centered sum.
(d) A mean-square average over many disjoint "marking arrays"
(Chapter 14) removes the marks.
The output: the prime average of any suitable factor statistic of
p - 1 matches the expected value.
Step 4. The probabilistic finish.
At this point we know:
For every fixed k and every bounded continuous F on [0, 1]^k,
as x -> infinity,
(1/pi(x)) * sum over primes p <= x of F(V_1(p), ..., V_k(p))
converges to the "factorial intensity limit" of the
expected joint count of the k primal balls.
The last stage is to identify this limit as PD(1). The argument
goes through "size-biased sampling":
(a) From the average of the factor statistics we can compute
the expected "joint intensity" of factor tuples. That is,
the expected counting measure of ordered k-tuples of factor
balls.
(b) Size-biased sampling turns those joint intensities into the
distribution of the largest, second largest, etc., via the
stick-breaking map.
(c) A total-mass-one argument shows no mass leaks to the boundary.
That is, the pieces really add up to 1.
The output is exactly the PD(1) shape. That is the theorem.
Parts IV, V go into the details of Steps 1, 2, 3, 4, and the tools
they use.
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PART IV. THE DETAILS OF THE PROOF
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Chapter 8. MARKS: A WAY TO PROBE THE FACTORIZATION
Recall from Chapter 6: we cannot directly count primes using the
prime number theorem for arithmetic progressions at the scale we
need. The paper gets around this by using a different strategy
called "marks."
A mark is a small extra piece of information about a factorization
that we attach, which lets us split the problem into tractable
parts. Think of a mark as a "colored sticker" that we put on
specific small primes.
How marks work.
Fix a small set of "bands": intervals of primes that all have size
exp(L^{a_i}) for some fixed 0.1 < a_i < 0.2. For example, if
x = 10^1000, then a band might be all primes between exp(100) and
exp(200), or between exp(200) and exp(400), etc.
For a positive integer n, define
omega_i(n) = the number of prime factors of n (with multiplicity)
that lie in band i.
Define also the "inverse harmonic mass" of band i:
V_i = sum over primes p in band i of 1/p,
which is approximately log 2, by a prime-counting estimate.
The "mark weight" is
W(n) = product over i of [ q^{omega_i(n) - 1} * omega_i(n) / V_i ],
where 0 < q < 1 is a fixed small damping factor.
The factor omega_i(n) / V_i roughly counts the primes in band i.
The factor q^{omega_i - 1} damps cases with very many primes. And
the whole product is ZERO if some band has zero primes, i.e., if n
has no prime factors in some band.
So the weight W(n) "locks onto" the presence of one or more primes
in each band. It is bounded (by a constant depending only on K and
q), and it is what we will use to probe the structure of n.
Why marks help.
The weight W(n) is a bounded function of n that behaves like a
"local indicator" of band structure. It is "almost multiplicative":
W(d*e) approximately factors as W(d)*W(e) up to the damping.
With W in hand we can phrase our target statistic in terms of
marked averages, and marked averages can be attacked by
"correlation estimates" that we introduce next.
Two types of marks in the paper.
The paper uses two closely related kinds of marks.
- "Fixed marking bands," as above, with K of them. These appear
in the one-sided marked Type II estimate (Chapter 9).
- "Active groups," which are up to O(log L) small groups of
primes scattered in ranges like [exp(L^.27), exp(L^.36)] and
[exp(L^.39), exp(L^.46)]. These appear in the two-sided
marked correlation (Chapter 13), where they are used because
they can be "flipped on and off" to give a family of tests
that we can average over.
Both kinds of marks play the same role: they give us a "structured
probe" of the values we're averaging, which cancels what we can't
control directly.
Chapter 9. THE ONE-SIDED MARKED ESTIMATE, IN WORDS
The first analytic input is a "correlation estimate" for marked
sums with a prime condition on one side.
The setup.
Fix an accuracy parameter D_* (say, we want to save a factor of
L^{-D_*} for some fixed D_*). Fix K "marking bands" as in Chapter 8,
with 0.1 < a_1 < ... < a_K < 0.2, and 0 < q < 1.
- alpha_m = m^{iv} * ( 1_{m prime} - 1_{m W-rough} / (V(W) log m) )
with |v| <= L^C for a fixed C. Here "W-rough" means m has no
prime factor <= W = exp(L^{0.24}), and V(W) is the product over
primes p <= W of (1 - 1/p).
In words: alpha_m is a "centered prime indicator." It is the
same as 1_{m prime}, minus a smooth predicted density
(1/(V(W) log m)) that we subtract off.
- beta_n is supported on W-rough n, with |beta_n| <= L^C.
- F : Z -> C is a bounded function invariant under band prime
multiplication: F(ph) = F(h) for every prime p in a marking
band.
- W(h) is the mark weight from Chapter 8.
- X = H_m * H_n ~ x, the total size of the two-variable box.
The theorem (paper Theorem 6.2, one-sided marked Type II).
For every fixed D_* > 0, if K is large enough (depending only on
D_* and the fixed parameters, not on the band exponents a_i), then
| sum over m, n of alpha_m beta_n F(mn - 1) W(mn - 1) |
<= X L^{-D_*}.
Here the sum is over the supports of alpha and beta. The bound is
uniform in the chosen marking bands, in |v| <= L^C, and in the
coefficient bounds on beta. The threshold for x may depend on the
a_i and all fixed parameters, but the required lower bound on K
does not.
What this says, in plain words.
The variables m and n are constrained so that mn - 1 is divisible
by the products that the mark weight W rewards. That is a strong
arithmetic constraint on mn - 1.
The theorem says: after subtracting off the "expected" contribution
(the smooth proxy 1/(V(W) log m)), the m-side has almost no
correlation with the rest of the sum. In other words, the smooth
proxy is a very good approximation to the prime behavior, even
after multiplying by the marked weight on mn - 1.
Why is this hard?
Because one side of the sum carries a genuine prime indicator.
The proposition is saying: the sum over primes m of a specific
function, weighted by the marked statistics of mn - 1, essentially
cancels against the smooth proxy.
The reason this could fail (and would fail with few bands) is that
F and W might correlate with the arithmetic of mn - 1. For example,
if F were very sensitive to whether mn - 1 is prime, we'd get a bias.
The "marks" W are the trick: they're built only from SMALL prime
divisors of mn - 1, so they're "arithmetically cheap" to correlate
with. After enough bands, the correlation with the genuine prime
variable m is negligible.
Chapter 10. THE DETERMINANT METHOD, PART 1: THE SHAPE OF THE ARGUMENT
Here is how the one-sided estimate is proved, at a high level.
Step A. Reduce band structure to "choose one prime per band."
The mark weight W(h) is built from the number of band primes
dividing h. We first reduce W to a sum over choices of one prime
per band, by expanding omega_i(h) as a sum over primes. What
remains is a sum over m, n, and a selected product b_0 of primes.
This is a standard expansion.
Step B. Cauchy-Schwarz.
The m-variable in alpha is a genuine prime indicator. We use
Cauchy-Schwarz to exchange it for a "two-sided" sum over pairs
(m, r) of test variables. The resulting expression is:
| sum_m ... |^2 <= (something) * Q_Y
where Q_Y is a nonnegative expanded sum over (a, b) products of
marking-band primes, m, r, n, s, and a multiplicity h.
Step C. The determinant identity.
The constraint that both (m, n) pairs satisfy the mark condition
turns into a condition on a DETERMINANT. Two products (a, b) from
marking bands and two "first coordinates" (m, r) with their "second
coordinates" (n, s) get coupled through:
t = bmn - ars.
The exact "diagonal" condition for the marked structure to survive is
t = b - a.
This t = b - a relation is the "determinant identity" that gives
the method its name. It says: a specific 2x2 determinant of
variables has a specific value.
Step D. Split major and minor.
The exact determinant identity t = b - a is too sharp to work
with directly. We replace it by a "smeared" version: instead of
imposing "t equals b - a exactly," we impose "t is close to b - a
modulo 1/k for many small denominators k."
This is the classical major/minor arcs decomposition from analytic
number theory. The major arcs (small denominators k) give the MAIN
term, and the minor arcs (large denominators) give the correction.
Step E. Control both terms.
The minor term is small by Fourier analysis alone: the smeared
version has a very small "off-diagonal" contribution by standard
exponential-sum bounds.
The major term is where the interesting arithmetic lives: it is
here that the marking bands interact with the prime condition on
m. The "memory identity" (Chapter 11) gives an exact accounting
of this interaction. The "major-arc evaluation" (Chapter 12) then
evaluates it.
Chapters 11 and 12 spell out these two pieces.
Chapter 11. THE DETERMINANT METHOD, PART 2: THE SIGNED MOMENT AND THE MEMORY
We now explain how the "minor" part (complement of major arcs) of
the determinant estimate is bounded. This is the technical heart
of Step 1.
The main tool is a "signed moment bound" on a certain operator.
We construct a Hilbert space of "physical states" and prove that a
signed moment of the operator is exponentially small. Then a
standard Hilbert-space argument bounds the pairing we care about.
The physical state space.
A "physical state" is a pair P = (P_1, P_2) of primitive integers,
together with an ordered list of M primes from each marking band,
all dividing P_1.
The measure on this space is counting measure in P, times
prod_i V_i^{-M} (one factor V_i^{-M} per band). This makes the
state space a discrete Hilbert space.
Concretely: P_1 is a "first coordinate," P_2 is a "second
coordinate," and the "prime lists" record which primes of each
marking band divide P_1.
The row operation.
We define an operator T that moves from one physical state to
another, in a specific way:
- Copy the first J slots of each band's list from source to
target (so the "shared" primes are the same on both sides).
- The target's last slot is a fresh prime from the same band.
- The source's last slot product is "b" (product of one
fresh prime per band from source).
- The target's last slot product is "a" (product of one
fresh prime per band from target).
- The shift equation is: b(mn - 1) = a(rs - 1) for source pair
(m, n) and target pair (r, s).
The multiplier is a specific combination of:
- The "det/damping": q^{(omega(P_1) - KM)/2 + (omega(Q_1) - KM)/2}.
- A "sign" from the determinant.
- The "arc integral" or "delta" from the smeared major-arc
version of Chapter 10.
"Good positions" and the G projection.
Two technical conditions ("good" and "separated") are imposed on
positions P. These ensure that the lattice spanned by the "prime
lists" is not too thin, and that the "phase" of the omitted label
is discriminating.
The projection G onto good states is essentially guaranteed by
the geometry of the lattice.
The moment.
The key ingredient in the proof is the moment bound:
(1/UV) * sum over primitive P of < u_P, (A A*)^R u_P >
<= L^{-E_0 * N}
where A = G S T S G is the restricted, symmetrized operator, R is
about L^{0.5}, N = 2R, and u_P is a "delta state" at P. Here U and
V are the first and second coordinate box lengths, respectively.
This says: after R successive applications of A (and its adjoint),
the "return probability" from any state is exponentially small.
Why this is true.
Because each step of A is "highly unlikely to return" in a specific
sense. The determinant condition t = b - a, together with the
good-state conditions, force the trajectory to progress through a
specific geometric structure that quickly escapes from any given
starting state.
The "memory identity" (below) is the technical machinery that lets
us track this quickly-escaping structure exactly. Without it, the
analysis becomes a tangle of estimates for "shared primes."
From the moment to the pairing bound.
Given the moment bound, a standard Hilbert-space argument
(essentially Cauchy-Schwarz with an intermediate spectral bound)
gives the pairing bound:
|<f, A g>| <= (some constant) * UV * L^{-E_0 + O_C(1)}
for suitable test vectors f, g. That is the desired answer, up to a
log-power loss factor.
The memory identity.
Here is the key identity that makes the moment bound work.
A prime p can be active at several "runs" of the trajectory. If
the same p appears in two CONSECUTIVE active runs, the physical
constraint "shared slot" forces the same projective line. But
between runs, the prime is "pending": it is not active, but we
remember its line from before.
The memory object "stores" each pending prime, along with its
projective line, so that when it becomes active again we can
retrieve the same line. The accounting is:
- When a prime becomes "pending" (leaves the active run), we store
it in the memory, paying a factor V_i^{-1}.
- When a prime becomes "active from memory," we retrieve it, also
paying V_i^{-1}.
- For each "pending hit" (the line at a pending prime coincides
with the current projective line), we pay a factor q_j (or
sqrt(q) at the endpoints j = 0, N).
- A "ghost" (memory birth or death that creates or destroys a
pending prime not tied to an active run) has a similar cost.
The upshot: each lifespan of each prime is charged exactly once,
and the total cost is tracked by an adjoint operator on the memory
space.
This is where the q_j and eta_j from the setup come in. The paper
proves the operations are adjoints (in the memory Hilbert space),
and the moment bound reduces to standard operator-norm bounds on
the memory subspace.
For our purposes here, the important takeaway is: the memory
identity is an EXACT accounting of the shared-prime structure, and
it is what makes the moment bound possible.
Figure "lifespan" in the paper gives a picture of a single
prime's trajectory through active and pending states. The picture
is worth a thousand words:
A "lifespan" of a prime p is the segment of the trajectory
between its birth (when it first becomes active) and its death
(when it disappears from the marking lists). During its life,
p can be active, pending, or in "ghost" states at the ends.
Each intermediate visit costs a factor q_j (or hits its
line).
Chapter 12. THE DETERMINANT METHOD, PART 3: THE MAJOR ARCS
We now explain how the "major arcs" are evaluated. This is
essentially paper section 5.
Recall: we need to bound
Q_Y^{maj} = sum over a, b, m, n, r, s of
alpha_m alpha_r-bar beta_n beta_s-bar H_M(bmn - ars; a, b)
where H_M is the "smeared diagonal" (arc integral) from Chapter
10, X = H_m * H_n ~ x, and Y is a dyadic scale with
log Y in [L^{0.1}, L^{0.2}].
We want: |Q_Y^{maj}| <= X Y L^{-D} for any fixed D.
Why naive bounds fail.
The arc integral only gives a bounded multiplier, so the naive
bound on the sum is way too big. We need CANCELLATION in the
alpha factors to make the sum small.
The challenge: alpha has cancellation (prime vs smooth proxy),
but the sum over ars pairs gives the "m, r" variables a lot of
freedom. A naive bound on |alpha| loses all cancellation.
The strategy.
Split the sum into three factors:
- M_alpha(t): the "long coefficient" alpha, twisted by t.
This should be small (like L^{-A} for every A) uniformly in t.
- N_beta(t): the "long coefficient" beta, twisted by t.
This can be as large as L^C, but has controlled mean square.
- B_lambda(t): the "small prime" factor, coming from the
product of marking band primes. This is bounded by 1 in
absolute value but can give strong information about
frequencies.
Then Mellin/Fourier separation writes the arc integral as a
product of these three factors, and chooses an optimal splitting
between "small prime averaged," "M_alpha is small," and
"N_beta has controlled mean-square."
The long coefficient is small.
Paper Lemma 5.1 ("The long coefficient") says:
|M_alpha(t)| <= L^{-A} uniformly for |t| <= 2 X L^B,
for any fixed A, B. This is a "long coefficient" cancellation
bound.
The proof uses:
- The Vinogradov mean-value theorem (Chapter 18).
- The truncated-rough approximation to alpha (replacing the
prime indicator by a Bonferroni-style divisor sum).
- The prime number theorem for arithmetic progressions mod q
for q up to L^C (Siegel-Walfisz).
- The long-prime logarithmic-phase estimate (Chapter 18) for
the qualitative cancellation at high frequencies.
The sparse-frequency analysis.
If B_lambda(t) is LARGE for many t (which it can be), we use a
sparse-frequency argument: divide the frequency axis into unit
intervals; the intervals where |B_lambda(t)| > L^{-A_s} form a
sparse set (of size exp(O(L^{0.91}))); on the sparse set we use
the mean-square bound for N_beta.
Paper Lemma 5.2 ("A small prime factor and sparse frequencies").
For P_s(t) = P^{-1} sum of band-prime contributions, and
N_beta(t) = H_n^{-1} sum of beta coefficients, the "sparse set"
of times where |P_s(t)| > L^{-A_s} has measure exp(O(L^{0.91})),
and on it the supremum of |N_beta(t)|^2 sums to at most L^{2C+1}.
Combining.
Mellin/Fourier separation writes the major-arc sum as an integral
of F_lambda(t/X, s') * Q(t) * Q'(s'-t), where Q = B M_alpha N_beta.
Then:
- Split into |t| <= X L^{B_1} (main) and tail.
- For |t| <= L^{B_2} (low frequencies), use the "constant-term"
cancellation that comes from the alpha-prime-rough identity.
- For L^{B_2} <= |t| <= X L^{B_1} (high frequencies), use the
fact that M_alpha is small by Lemma 5.1.
- On the "sparse" set (where B_lambda is large), use N_beta
mean-square to control it.
- Off the "sparse" set, use M_alpha * B_lambda * N_beta
uniform smallness.
The result: |Q_Y^{maj}| <= X Y L^{-D} for any fixed D. This
finishes the one-sided estimate.
Chapter 13. THE TWO-SIDED MARKED CORRELATION
The second analytic input is a "two-sided correlation" where BOTH
sides of the equation carry marks. This is paper Theorem 6.1.
What it says.
We use a family of "active groups" of primes. These are:
- S, a family of "small" groups in [exp(L^.27), exp(L^.36)].
- B, a family of "big" groups in [exp(L^.39), exp(L^.46)].
- Sizes bounded: c_0 log L <= |S|, |B| <= C_0 log L.
The point of having O(log L) groups is that we can "flip them on
and off" - choose any subset - and still control the total number
of subsets (which is exp(O(log L)) = L^{O(1)}).
For each choice of active groups, we can define a weight
W_t^b(n) that records which primes of the groups divide n, with
damping q_0 and harmonic normalization V_g^{-t_g}.
The "centered core" F_x is:
F_x(n) = W(n) * ( sum_{l in long-prime products} 1_{l | n} - C_x )
where l runs over products of "long primes" from intervals
[x^{a_1}, x^{b_1}], ..., [x^{a_d}, x^{b_d}] with sum b_i <= 1 - eps,
and C_x is the expected density (of order O_eps(1)).
In words: F_x(n) is centered around zero for a "random" n. It picks
out how many "long prime factors" n has, relative to the average.
The theorem.
Let G(n) = G(n_*) where n_* is n with active primes removed, with
|G(n)| <= L^C * tau(n_*)^C. Then for any fixed smooth
compactly-supported Psi and any fixed A > 0:
sum over w >= 1 of Psi(w/x) / w * F_x(w) G(w+1) W_1(w) W_1(w+1)
<= L^{-A}.
The estimate is uniform over the active groups and slot intervals
satisfying the stated bounds.
Why is this hard?
The statistic involves a centered shift indicator (an indicator of
the long-prime-product structure), plus marks on both w and w+1.
The shift from w to w+1 can be correlated with the marks, and we
need that correlation to cancel.
The strategy.
The proof uses the "dilation-graph" method from the companion
paper S, plus some new endpoint estimates specific to this paper.
The idea: represent the correlation statistic as a matrix element
of an operator on L^2 of sequences. Split the operator into:
- The "raw" piece: no shift.
- Two "comparison" pieces: shift by a product of small primes.
Each comparison piece is "morally multiplicative" with respect to
the mark structure. This lets us bound it via a finite character
expansion: expand in characters modulo the small primes, using
the fact that each character is multiplicative.
Then the raw piece is what we want, and the comparison pieces are
bounded by the L^2 operator bound.
What we need from S.
From the companion paper S, we use:
- The "general graph transference" theorem: an operator bound
in an "ideal Hilbert space" transfers to a bound in the
"physical Hilbert space" in which the marks live.
- The "residual ideal family" theorem: the specific signed
combination of patterns we use satisfies the operator bound
in the ideal Hilbert space.
- The character expansion and cell decomposition inputs.
What is new in this paper.
The endpoint estimates: verifying that our specific F_x and G
satisfy the endpoint hypotheses of the S-graphs, including their
growth and centeredness. These are proved in paper section 6 and
are the main new work for Step 2.
Chapter 18 has the specific lemma statements for the analytic
estimates needed to make the endpoint bounds work.
Chapter 14. EXTRACTION, PART 1: SETUP AND PRESIEVE
Now we combine the two correlation estimates to prove the "interior
prime statistic" theorem (paper Theorem 7.1).
The statement.
Fix d >= 1 and eps > 0. Fix intervals I_1, ..., I_d of primes
with lower endpoints at least x^eps and upper endpoints multiplying
to at most x^{1 - eps}.
For a positive integer u, define
f_x(u) = sum over l_i in I_i (i=1..d) with l_1,...,l_d distinct
of 1_{l_1 l_2 ... l_d | u},
and
C_x = sum over the same l-tuples of 1 / (l_1 l_2 ... l_d).
Then C_x is O_eps(1) by the prime number theorem.
The theorem: for every nonnegative smooth Phi : (0, infinity) -> R
with compact support, as x goes to infinity,
sum over primes p <= x of Phi((p-1)/x) * (f_x(p-1) - C_x)
= o(x / log x).
What f_x means.
For a specific tuple of intervals, f_x(u) counts how many "ways" u
is divisible by "one prime from each slot." The C_x is the expected
count.
The theorem says: on average over primes p, the count f_x(p-1)
matches the expected count C_x, up to o(1) in the average weighted
by x/log x.
Why this implies PD(1).
The full PD(1) convergence will follow from a probabilistic
argument that converts "interior average matches PD(1) expected
value" into "distribution matches PD(1)." That argument is Step
4 (Chapter 17).
The proof structure.
We prove the interior theorem via:
1. Presieve: bound the contribution from primes p where p-1 has
a "very small" prime factor. This reduces to a sum over p
where p-1 has NO small prime factors.
2. Replace composite values in the sieve by rough values: this
uses the One-Sided Estimate (Chapter 9).
3. Bound the resulting centered sum: this uses the Two-Sided
Estimate (Chapter 13).
4. Remove the marks by averaging over many arrays: mean-square
argument (Chapter 16).
Presieve details.
We fix a large even integer h (the sieve depth). Let
gamma = c_0 / (4h + 3), z = x^gamma,
where c_0 is a small constant (chosen below eps/2).
Apply the Block Sieve (Chapter 18) to the "bad conditions"
r' | u+1 for primes r' <= z. The sieve density is g(r') = 1/r',
and the required density bounds hold by a prime-counting estimate.
The block sieve bounds the "total weight" of u for which u+1 has
any prime factor <= z. Because the block sieve is a standard
small-sieve argument, this "sieved out" weight is very small -
like a factor of L^{-A} for any fixed A.
The presift formula.
More precisely, applying the Block Sieve to Phi(u/x) W(u) f_x(u)
and to Phi(u/x) W(u) separately and subtracting, one gets:
| sum over u with P^-(u+1) > z of
Phi(u/x) W(u) (f_x(u) - C_x) |
<= (e^{-h}/gamma) * x * m_W / L * (1 + o(1)),
where m_W is the model mean of W (which is bounded above and
below by positive constants). This is the "presift formula." The
factor e^{-h} is the sieve error; it goes to zero as h -> infinity.
What's left.
After presieving, u+1 has no prime factors <= z.
If u+1 is prime, we're directly in the "prime contributions" we
want. If u+1 is COMPOSITE, then u+1 = p_1 p_2 ... p_j with all
p_i > z and j >= 2.
We need to show: composite u+1 contribute only o(x / log x).
Strategy for composite u+1.
For each j >= 2, we want to show:
sum over p_1, ..., p_j > z of
Phi((p_1...p_j - 1)/x) * W(p_1...p_j - 1) *
(f_x(p_1...p_j - 1) - C_x) = o(x / L).
We do this in three stages:
Stage A. Replace each prime variable p_i by a "rough proxy."
Fix the "proxy coefficient":
a(m) = 1_{m > z and m W-rough} / (V(W) log m).
This is a smooth approximation to the "prime indicator" 1_{p_i}.
The One-Sided Estimate (Chapter 9) with the choice F(h) =
f_x(h) - C_x (which is band-prime-invariant because f_x only
depends on divisor structure) and beta_n = the appropriate
coefficient lets us replace p_i by a(m) up to o(x/L). Chaining
through all j variables gives:
Stage B. Apply the Two-Sided Estimate.
After the replacement, we have a sum of the form:
sum over m_1, ..., m_j (all W-rough) of
Phi((m_1...m_j - 1)/x) *
W(m_1...m_j - 1) *
(f_x(m_1...m_j - 1) - C_x) *
prod_i a(m_i).
Now we can choose the active groups of Theorem 6.1 (Chapter 13)
to match the marking weight W that appears in this sum. The
"active one-mark weight" (from the allocation of the primes of v
= m_1...m_j among the j factors) exactly reproduces the "mark
weight at u+1" in the two-sided estimate.
Summing over all j of the two-sided estimate, we get o(x/L).
Summing over j from 2 to 2/gamma gives o(x/L).
Stage C. The allocation identity.
The key combinatorial identity is:
sum over ways to assign the active primes of v to the j slots
= j^{omega_P(v)} * (q_0/j)^{omega_P(v) - s_P}
= j^{s_P} * q_0^{omega_P(v) - s_P},
where omega_P(v) counts active prime hits (all active primes are
> W by definition) and s_P = |P| is the number of active groups.
This identity is what turns the "allocation over factorizations
of v" (which appears naturally in the m_1...m_j decomposition
of u+1) into the "one-mark weight W_1(u+1)" of the two-sided
theorem. That is the central combinatorial step of the extraction
argument.
Chapter 15 fills in some of the technical details of stages A-C,
and Chapter 16 covers the final "averaging" step (removing the
marks W from the interior statistic).
Chapter 15. EXTRACTION, PART 2: THE COIN-PARTITION AND SUCCESSES
Getting from the "proxy-replacement sum" of Chapter 14 to a genuine
application of the Two-Sided Estimate (Theorem 6.1) is more delicate
than it looks. There is a "coin toss" that must succeed often enough.
The candidate groups.
The "active groups" in Theorem 6.1 have to be chosen. They come from:
- Small groups P_g in [exp(L^.27), exp(L^.36)].
- Big groups P_g in [exp(L^.39), exp(L^.46)].
We choose them as ALL primes in consecutive "dyadic bands." That is,
for small groups: [exp(y), exp(2y)] for y = L^.27, 2 L^.27, ...,
2^8 L^.27, giving about 8 or 9 groups. Similarly for big groups.
The coin toss.
Fix a tuple (m_1, ..., m_j) with product v and u = v - 1. For each
candidate group g, define a "success probability":
s_g(u, v) = c_1 * q_0^{omega_g(u) - 1} * omega_g(u) / V_g
* (q_0/j)^{omega_g(v) - 1} * omega_g(v) / V_g,
where c_1 is a small enough constant (depending on j, q_0, and the
bounds on V_g).
The probability s_g(u, v) is nonzero ONLY if both u and v have at
least one prime in group g. So a "success" in group g requires:
- u = v - 1 has a prime in band g ("u-hit"),
- v has a prime in band g ("v-hit").
A second-moment argument on successes.
Consider each group g as an independent "coin." Toss it with
probability s_g. Let P be the set of groups where the coin came up
heads (succeeded).
We show: with positive probability (in fact, with 1 - o(1)
probability, uniformly in the tuple), at least c_2 log |S|
successes occur in the S-family and at least c_2 log |B| successes
in the B-family, for some small c_2 > 0.
The key facts needed:
- Each s_g has expectation bounded below by a constant (in the
model where u and v are "random rough tuples").
- Different groups have almost no correlation in their
u-hit / v-hit structure.
This gives the "many successes with high probability" result via
Chebyshev's inequality. The technical input is a uniform
residue-class equidistribution of the u-hit / v-hit structure
across groups.
Why successes are important.
The successes are what make the "allocation identity" of Chapter
14 Stage C work. Because:
- A success in group g means both u and v have a prime in g,
with the "damping" q_0.
- In the ALLOCATION over factorizations of v into m_1 ... m_j,
that prime of v appears in exactly one factor m_i.
So each v-hit is "allocated" to one of the j slots. Grouping
by "which prime goes to which slot" produces the factor
j^{omega_P(v)} that appears in the allocation identity.
- After conditioning on the successes, the mark weight at v
is that j^{omega_P(v)} times the base mark weight W.
The coin-partition expansion.
We expand:
1 = sum over subsets P of C of prod_{g in P} s_g * prod_{g not in P} (1 - s_g).
Restrict to |P cap S| >= c_2 log |S| and |P cap B| >= c_2 log |B|.
This has mass >= 1 - o(1) by the Chebyshev argument.
Then for a fixed P, the sum over (m_1, ..., m_j) with prod s_g
has the form:
(constant) * sum over u of (Phi(u/x) W(u) (f_x(u) - C_x))
* W_1(u) * W_1(u+1) * G_P(u+1),
where:
- W_1 is the "one-mark active weight" (as in Chapter 13).
- G_P(u+1) is a bounded "core" that sums over factorizations of
u+1 into j rough factors with the right mark structure.
This is EXACTLY the form that the Two-Sided Estimate (Theorem 6.1)
handles!
So Chapter 14's "proxy-replacement sum" is bounded as:
<= (sum over P of retained mass) * (L^{-A} from Theorem 6.1)
+ (mass of discarded P) * (bounded coefficient)
= 0 * L^{-A} + o(1) * (bounded)
= o(x/L), as desired.
Exceptions and their bounds.
Two things have to be checked carefully:
- The discarded (few successes) cases have mass o(1) uniformly
over the tuple. The total tuple mass (weighted by prod a(m_i))
is bounded, so the resulting contribution is o(x/L).
- If some candidate band prime p appears SQUARED in v = m_1...m_j,
the "allocation" over factorizations gives an extra factor j
of overcounting. We remove these cases by a bound: they involve
v being divisible by p^2 for p >= exp(L^.27). The number of such
v in the range is at most x * sum(p^{-2}) << x * exp(-c L^.27),
so negligible.
Note on the CORE.
The G_P(u+1) core is bounded by a fixed log-power times a fixed
divisor power. It behaves like a "smooth distribution" of the
marked structure of u+1. That's exactly the "G(n_*) bounded by
L^C tau(n_*)^C" from Theorem 6.1.
Chapter 16 covers the final averaging step: removing the marks W
from the interior statistic.
Chapter 16. EXTRACTION, PART 3: AVERAGING TO REMOVE THE MARKS
We are now almost done with extraction. The previous two chapters
gave us a "marked" version of the interior statistic. We want the
UNMARKED version:
(L/x) * | sum over primes p of Phi((p-1)/x) * (f_x(p-1) - C_x) |
<= small.
To remove the mark W we use a "variance reduction" argument: if we
average over MANY independent marking arrays, the average weight
(W_a/m_a) should be close to 1.
Setup.
Fix b independent marking arrays. Each array a in 1..b gets its
own weight W_a and its own model mean m_a. Define the squared
deviation:
Z_b(u) = ( (1/b) sum_{a=1}^b W_a(u) / m_a - 1 )^2.
Z_b(u) is nonnegative and equals zero when the averaged weight
equals 1.
Model variance.
In the "independent divisor model" (Chapter 7 of the paper), the
mark weights W_a / m_a are independent across a, with mean 1 and
bounded second moment. Hence:
E_model[Z_b] = (1/b^2) * sum_a Var(W_a / m_a) <= C_K / b.
So the model variance of Z_b is O(1/b).
Applying the sieve to Z_b.
Apply the Block Sieve at fixed depth (h' = 2, z' = x^{c_0/20}) to
the weight Phi(u/x) Z_b(u). The sieve keeps only primes u+1 > z'.
The result (a variant of the presieve formula):
limsup_{x -> infinity} (L/x) *
sum over primes p of Phi((p-1)/x) Z_b(p-1) <= C_K / b.
So for large b, the "deviation from the mean" is small on primes.
Cauchy-Schwarz.
We compare the AVERAGED marked sum to the UNMARKED sum:
| avg_a sum_p Phi((p-1)/x) * (W_a(p-1)/m_a) * (f_x(p-1) - C_x)
- sum_p Phi((p-1)/x) * (f_x(p-1) - C_x) |
= | sum_p Phi((p-1)/x) * (avg_a(W_a(p-1)/m_a) - 1)
* (f_x(p-1) - C_x) |
<= sqrt( sum_p Phi((p-1)/x) * Z_b(p-1) )
* sqrt( sum_p Phi((p-1)/x) * |f_x(p-1) - C_x|^2 ),
where the last inequality is Cauchy-Schwarz in p.
The first factor is <= (C_K/b)^{1/2} * (x/L)^{1/2} by the Z_b
bound. The second factor is <= C * (x/L)^{1/2} by the boundedness
of f_x - C_x and the prime number theorem.
So the product is:
<= C_K^{1/2} * C * (x/L) * b^{-1/2} * (1 + o(1)).
Dividing by x/L, we get:
| avg_a (marked sums) - (unmarked sum) | / (x/L)
<= C * C_K^{1/2} * b^{-1/2}.
Finishing the interior theorem.
Combining all the errors:
| sum_p Phi((p-1)/x) * (f_x(p-1) - C_x) | / (x/L)
<= (e^{-h}/gamma) [presieve error]
+ C_1 * (C_K/b)^{1/2} [averaging error]
+ o(1) [all other errors].
Take h -> infinity first (with b, K, marking arrays all fixed),
then b -> infinity. Both errors go to zero, and the o(1) term
also does. This proves Theorem 7.1 (interior prime statistic
theorem).
The order of the limits is important: all fixed parameters (h,
K, b, marking arrays) are chosen BEFORE taking x -> infinity, but
h and b can be taken to infinity after the real x-limit.
Chapter 17. THE PROBABILISTIC FINISH: FROM INTERIOR TO FULL LAW
We now have:
sum over primes p <= x of Phi((p-1)/x) * (f_x(p-1) - C_x)
= o(x / log x)
for any smooth Phi with compact support and any set of intervals
I_1, ..., I_d with lower endpoints >= x^eps and upper endpoints
multiplying to <= x^{1 - eps}.
The remaining task: show that this implies the convergence of the
prime-factor shape to PD(1).
The plan.
We proceed in six steps:
1. Define the joint intensity mu_{x,d} of ordered d-tuples of
factor balls.
2. Convert the interior statistic into convergence of mu_{x,d}
to a reference measure on the open simplex.
3. Size-biased sample: sample factor balls with probability
proportional to their mass. This turns the joint intensity
into the stick-breaking distribution.
4. Show that the resulting probability density has total mass 1
on the open simplex (no mass escapes to the boundary).
5. Parameterize by the "A-coordinates" via a triangular map to
identify the density with the PD(1) stick-breaking law.
6. Sort and pad to get convergence of the DECREASING
rearrangement.
Step 1: Joint intensity.
For a prime p, the prime factors of p - 1, WITH multiplicity, form
a multiset of "labelled balls." A ball corresponding to a prime q
has mass t_q = log q / log(p - 1). The balls add up to 1.
Define the "joint intensity" of ordered d-tuples of distinct balls:
mu_{x,d}(A_1 x ... x A_d) = (1/(pi(x) - 1)) *
sum over primes p <= x of
sum over ordered d-tuples of DISTINCT balls b_1, ..., b_d
of 1_{(t(b_1), ..., t(b_d)) in A_1 x ... x A_d}.
In words: mu_{x,d} is the expected counting measure of ordered
d-tuples of distinct factor balls, mapped to their masses.
Step 2: Convergence of the intensity.
Suppose we want the "probability" that (t_1, ..., t_d) lies in a
rectangle R = (a_1, b_1] x ... x (a_d, b_d] with 0 < a_i < b_i and
sum b_i < 1. Set I_i = (x^{a_i}, x^{b_i}]. Then:
f_x(p-1) = sum over l_i in I_i of 1_{l_1 ... l_d | p-1}
= number of ordered d-tuples of distinct factor-primes
of p-1 with log-masses in the range (a_i, b_i].
So "mu_{x,d}(R) * (pi(x) - 1)" equals "sum over primes p <= x of
f_x(p-1)" in a small band. That is exactly what the interior
theorem controls.
By the interior theorem:
sum over primes p <= x of (f_x(p-1) - C_x) = o(x/log x).
Where C_x = sum over the same l-tuples of 1/(l_1 ... l_d). By the
prime number theorem and partial summation:
C_x ~ prod_i log(b_i / a_i).
So:
mu_{x,d}(R) * (pi(x) - 1)
= sum over primes p <= x of f_x(p-1)
= pi(x) * prod_i log(b_i/a_i) + o(pi(x)).
Therefore:
mu_{x,d}(R) -> prod_i log(b_i / a_i)
= int_R prod_i (dt_i / t_i).
So the joint intensity converges to the reference measure:
dt_1/t_1 * dt_2/t_2 * ... * dt_d/t_d
on the open simplex D_d = {(t_1, ..., t_d): t_i > 0, sum t_i < 1}.
Step 3: Size-biased sampling.
Given the balls with masses t_1, ..., t_N (adding to 1), sample
them successively WITHOUT replacement: at each step, choose a
remaining ball with probability equal to its mass divided by the
remaining total mass.
Let T_{x,i} be the mass selected at step i. Define T_{x,i} = 0
for i > N.
The probability that the first d samples are the specific ordered
tuple (t_1, ..., t_d) is:
w_d(t_1, ..., t_d) = prod_{i=1}^d t_i / (1 - t_1 - ... - t_{i-1}).
This is the "size-bias weight" of the tuple.
Step 4: The joint probability density.
Combining the intensity limit with the size-bias weight gives:
nu_d(A) = int_A w_d(t) d(mu_{x,d}) -> int_A w_d(t) prod_i (dt_i/t_i).
So the limit measure on D_d has density:
h_d(t_1, ..., t_d) = w_d(t) / (t_1 t_2 ... t_d)
= 1 / prod_i (1 - t_1 - ... - t_{i-1}).
Explicitly: h_d(t_1, ..., t_d) = prod_{i=1}^d
1 / (1 - t_1 - ... - t_{i-1}).
Step 5: Total mass and the A-coordinates.
Direct computation gives:
int_{D_d} h_d(t) dt_1 ... dt_d = 1.
So h_d is a PROBABILITY DENSITY on the open simplex.
The critical computation: under the triangular change of variables
A_i = t_i / (1 - t_1 - ... - t_{i-1}),
t_i = A_i * prod_{r < i} (1 - A_r),
the density h_d transforms to:
dA_1 * dA_2 * ... * dA_d,
which is the UNIFORM measure on (0, 1)^d. And (0, 1)^d has volume 1.
So the total mass is 1, and this confirms "no mass escapes to the
boundary."
Step 6: Stick-breaking and sorting.
Set U_i = 1 - A_i. Then (U_1, ..., U_d) are iid uniform on (0, 1),
and:
t_i = (1 - U_i) * prod_{r < i} U_r for i = 1..d.
Compare to Chapter 3: the stick-breaking construction B_j =
(prod_{r<j} U_r)(1-U_j) is EXACTLY the same formula, applied with
independent uniform U_j.
So the limit measure on D_d (from the "size-biased sampling" step)
equals the law of the first d stick-breaking pieces (B_1, ..., B_d).
From stick-breaking to PD(1).
We want the DECREASING REARRANGEMENT, not the natural ordering
B_1, B_2, B_3, ...
Let R_d = 1 - sum_{i=1}^d B_i = prod_{i=1}^d U_i. Then
E[R_d] = 2^{-d} -> 0. So the remaining mass beyond the first d
sticks is small on average, as d grows.
For a finite or summable sequence with total mass 1, sorting the
first d coordinates gives order statistics S_{d,1} >= S_{d,2} >=
... >= S_{d,d}. The difference |V_i - S_{d,i}| is bounded above
by the unmatched mass R_d (a standard rearrangement inequality).
So:
| Average over primes of F(V_1(p), ..., V_k(p))
- Average over primes of F(sorted first d terms of p-1) |
<= omega_F(eta) + 2 ||F||_inf * E[R_d] / eta,
where omega_F is the modulus of continuity of F.
Take x -> infinity (interior convergence), then d -> infinity (with
the R_d bound above), then eta -> 0 (continuity of F). We get:
(1/pi(x)) * sum over primes p <= x of F(V_1(p), ..., V_k(p))
converges to E[F(L_1, ..., L_k)]
for every fixed bounded continuous F. This is the main theorem.
All real x.
The same argument applies to any real-valued x parameter. Attach
the prime 2 with a small correction (measured by "Pi(x) - 1" vs
"Pi(x)") to fix the exact count. Done.
The theorem holds.
----------------------------------------------------------------------
PART V. THE ANALYTIC TOOLBOX
----------------------------------------------------------------------
Chapter 18. THE ANALYTIC TOOLBOX
The proof uses several analytic estimates as black boxes. We
collect them here for reference. The full statements are in
paper sections 2 and 9.
18.1 Prime estimates (paper Lemma 2.1).
Assume the prime number theorem with error smaller than any fixed
negative power of the logarithm. Also Mertens-type estimates:
sum_{p <= y} 1/p ~ log log y + c_M,
prod_{p <= y} (1 - 1/p) ~ exp(-gamma_E) / log y.
And Siegel-Walfisz: for fixed A, C > 0, uniformly for r <= (log y)^C
and (a, r) = 1,
#{p <= y : p = a mod r} = Li(y)/phi(r) + O(y / (log y)^A).
All standard, from the PNT and its refinements.
18.2 Divisor moments (paper Lemma 2.2).
For every fixed nonnegative integer k, there is C_k > 0 such that:
sum_{n <= Y} tau(n)^k << Y (log(2Y))^{C_k},
sum_{n <= Y} tau(n)^k / n << (log(2Y))^{C_k}.
Also main-value inequalities for Dirichlet polynomials. For
P(t) = sum_{n <= Y} c_n n^{it}:
int_I |P(t)|^2 dt << (T + Y log(2Y)) sum |c_n|^2,
sum_{t in T} |P(t)|^2 << (T + 1 + Y log(2Y)) (log(2Y))^2
sum |c_n|^2,
for I of length T and T subset of I with mutual spacing >= 1.
These are used everywhere.
18.3 Rough counts (paper Lemma 2.3).
Fix a "smoothness cutoff" W = exp(L^{0.24}), where L = log x. For
a W-rough integer H in [x^delta, x^C] with delta > 0:
#{n in I : P^-(n) > W} = V(W) |I| + O(H L^{-A}),
for any A > 0 and any interval I subset of [H, 2H]. There are also
progression and character versions:
#{n in I : n = b mod d, P^-(n) > W} = (V(W)/d) |I| + O(H L^{-A}/d)
for d <= exp(L^a) with all prime divisors > W, and
sum_{n in I, P^-(n) > W} chi(n) = 1_{chi principal} V(W) |I|
+ O(H L^{-A})
for chi mod q <= L^C.
These are proved by Bonferroni approximations to the rough indicator.
18.4 The long prime polynomial (paper Lemma 2.4).
Fix 0 < tau < eta < 1 and C, A > 0. There is B_0 = B_0(tau, eta,
C, A) such that for N in [x^tau/2, x^eta], q <= L^C, and |t| in
[L^{B_0}, x^2]:
|sum_{p in I} chi(p) p^{-1 + it}| <<_{tau,eta,C,A} L^{-A}
for every Dirichlet character chi mod q and every interval
I subset of [N, 2N].
Proved in Appendix 9 (paper section 9). The proof uses:
- A "smooth cutoff" convolution to smooth the interval.
- Mellin inversion to a contour integral.
- A zero-free strip from the logarithmic-phase estimate.
- Standard PNT-AP for low frequencies.
18.5 The logarithmic-phase estimate (paper Lemma 2.5).
Fix C > 0. There is an absolute constant C_3 > 0 such that for
sufficiently large x, for
exp(L/(log L)^2) <= N <= 2 x^5,
q <= L^C,
exp(L/(2(log L)^2)) <= |t| <= 4 x^3,
and for every residue class a mod q and interval I subset of
[N, 2N]:
|sum_{n in I, n = a mod q} n^{it}| << (N/q) * exp(-L/(log L)^{C_3}).
The proof uses the Vinogradov mean-value theorem with growing
degree. After Taylor-expanding logarithm to degree k ~ L/(log L)^2,
the phase becomes a polynomial, and the Vinogradov theorem bounds
the moments.
18.6 Vinogradov mean-value theorem (paper Lemma 9.2).
There are absolute constants C_1, C_2 > 0 such that for every
integer k >= 2, some integer s with k <= s <= C_1 k^4 satisfies:
J_{s,k}(M) <= exp(C_1 k^{C_2}) * M^{2s - K + 1/100}
where K = k(k+1)/2 and J_{s,k}(M) counts solutions of the
Vinogradov system:
u_1^j + ... + u_s^j = v_1^j + ... + v_s^j (1 <= j <= k),
1 <= u_i, v_i <= M.
The proof uses the classical Vinogradov mean-value method in
Linnik's p-adic form. Modern proofs using efficient congruencing
(Wooley) give much stronger bounds, but the classical statement
suffices for the logarithmic-phase estimate.
18.7 The block sieve (paper Lemma 2.6, from S's Lemma 2.9).
Given a family of "bad conditions" at primes p <= z with density
g(p) = 1/p, and a weight W, define:
X g(d) + r(d) = the total weight on which all bad conditions
at primes dividing d hold.
Then the "good" weight (avoiding all bad conditions) is:
X prod_{p<=z} (1 - g(p)) * (1 + O(e^{-h}))
+ O( sum_{d <= z^{4h+2}, d squarefree} |r(d)| ).
This is essentially the Erdos-Renyi sieve with two Bonferroni
polynomials (upper and lower) that differ in the "tail" of
degree > h. The h-error gives the O(e^{-h}) factor.
18.8 What we don't cover here.
Two important bookkeeping lemmas from the paper that we do not
fully reproduce:
- The "summed progression remainder" (paper Lemma 7.2): a
progression-sum bound like x * C_x * M_x(H) with error
x L^{-A}.
- The "truncated divisor model" (paper Lemma 8.4): a weight
H_marks can be approximated by a "divisor model" H_0 with
H_0(n) = sum_{d' | n} A_{d'}, d' <= exp(O(L^{.47})),
sum |A_{d'}| / d' <= L^{C_8}.
Both are "bookkeeping" lemmas that essentially say "the smooth
part behaves as expected." Their proofs are in the paper's
sections 7 and 8 and are largely routine, but they are nonempty
work.
Chapter 19. REFERENCES AND FURTHER READING
19.1 About prime factor distributions.
- Karl Dickman, "On the frequency of numbers containing prime
factors of a certain relative magnitude," Arkiv for Matematik,
Astronomi och Fysik 22A (1930), 1-14. The original work on
smooth numbers.
- N. G. de Bruijn, "On the number of positive integers <= x and
free of prime factors > y," Proc. Kon. Ned. Akad. Wetensch. 54
(1951), 50-60. Modern asymptotics for the Dickman function.
- Patrick Billingsley, "On the distribution of large prime
divisors," Periodica Mathematica Hungarica 2 (1972), 283-289.
The joint law for ordinary integers.
- Peter Donnelly and Geoffrey Grimmett, "On the asymptotic
distribution of large prime factors," J. London Math. Soc. 47
(1993), 395-404. The size-biased / stick-breaking approach.
- Richard Arratia, Fred Kochman, Victor S. Miller, "Extensions
of Billingsley's theorem via multi-intensities,"
arxiv:1401.1555 (2014). The factorial-intensity form.
19.2 About shifted primes and the FKL conjecture.
- Kevin Ford, Sergei V. Konyagin, Florian Luca, "Prime chains
and Pratt trees," GAFA 20 (2010), 1231-1258. Contains the
original conjecture about shifted primes.
- Andrew Granville, "Smooth numbers: computational number theory
and beyond," in Algorithmic Number Theory (MSRI Publ. 44),
2008, 267-323. Contains the shifted-Dickman conjecture.
- R. C. Baker, Glyn Harman, "Shifted primes without large prime
factors," Acta Arith. 83 (1998), 331-361. Early partial
results.
- Jared Duker Lichtman, "Primes in arithmetic progressions to
large moduli, and shifted primes without large prime factors,"
arxiv:2211.09641 (2022). Improved partial results.
- Kevin Ford, "Poisson approximation of prime divisors of
shifted primes," IMRN 2025, rnaf079. Small-prime divisors
of shifted primes.
- Ofir Gorodetsky, "A Kubilius model for sieve-theoretic
sequences," Analysis Math. (2026). Small-prime models.
- Abhishek Bharadwaj, Brad Rodgers, "Large prime factors of
well-distributed sequences," Canadian Math. Bull. (2026).
PD(1) for shifted primes under Elliott-Halberstam; level 1/2
unconditionally.
19.3 About the analytic tools.
- H. L. Montgomery and R.C. Vaughan, "Hilbert's inequality,"
J. London Math. Soc. 8 (1974), 73-82. Mean-value theorems.
- Trevor D. Wooley, "Vinogradov's mean value theorem via
efficient congruencing," Annals 175 (2012), 1575-1627.
- Kannan Soundararajan, "Moments of the Riemann zeta function,"
Annals 170 (2009), 981-993. Factorial prime polynomial
coefficient bounds.
- Kevin Ford, "Vinogradov's integral and bounds for the Riemann
zeta function," Proc. London Math. Soc. 85 (2002), 565-633.
- Harald Andres Helfgott and Maksym Radziwill, "Expansion,
divisibility and parity," arxiv:2103.06853 (2021). Precursor
of the signed-moment method.
- Cedric Pilatte, "Improved bounds for the two-point logarithmic
Chowla conjecture," arxiv:2310.19357 (2026). More signed-moment
development.
- Kaisa Matomaki and Maksym Radziwill, "Multiplicative functions
in short intervals," arxiv:1501.04585 (2017). Technique for
sparse frequencies.
19.4 The companion paper.
- OpenAI, "Weighted dilation graphs, smooth shifted primes and
totient fibers," September 24, 2026. The companion paper S,
which supplies the graph-theoretic apparatus for Step 2.
19.5 About the present paper.
- OpenAI, "The Poisson-Dirichlet law for prime predecessors,"
September 24, 2026. The primary source.
19.6 Other related reading.
- W. R. Alford, Andrew Granville, Carl Pomerance, "There are
infinitely many Carmichael numbers," Annals of Math. 139
(1994), 703-722.
- Paul Erdos, "On the normal number of prime factors of p-1 and
some related problems concerning Euler's phi-function,"
Quart. J. Math. 6 (1935), 205-213.
- Carl Pomerance, "Popular values of Euler's function,"
Mathematika 27 (1980), 84-89.
----------------------------------------------------------------------
A CLOSING NOTE
You have just read a plain-English guide to a research paper that
is itself about 100 dense pages of mathematics. The guide is not
the paper; it omits many epsilons, constants, and "routine"
arguments. But the LOGICAL structure is complete: every step in
the proof is at least named, and the main ideas are described.
If you want to read the actual paper, the LaTeX source is
available in the same directory as this file. The paper is
structured in nine sections that roughly correspond to this
guide's chapters, as follows.
Paper section This guide
-------------------------------- ----------------------------------
1. Introduction Chapters 1-6
2. Conventions / analytic inputs Chapter 18
3. Determinant estimate Chapters 9-12 (setup / proof)
4. Memory Chapter 11 (subsection)
5. Major arcs Chapter 12 (subsection)
6. Two-sided correlations Chapter 13
7. Extraction Chapters 14-16
8. Poisson-Dirichlet limit Chapter 17
9. Long prime appendix Chapter 18
Two figures in the paper are worth looking at side by side with
this guide:
- figures/lifespan.tex shows a schematic "lifespan" of a prime
through active, pending, and ghost states. Chapter 11 explains
the physical meaning.
- figures/proof-map.tex shows the dependency graph of the whole
proof. Chapter 7 tries to convey the same picture in words.
A final thought.
The theorem says: the prime factors of a random prime's
predecessor follow the same universal shape as the prime factors
of a random integer, but with a very different proof. The same
answer, reached by entirely different methods. In the end, the
"shape" of a factorization doesn't care about whether the number
came from an ordinary integer or from a predecessor of a prime.
That is the beautiful fact at the heart of this paper.
If the reader remembers nothing else from this document, they
should remember: p - 1 factors most of the time into ONE big
piece (log-size > half of the total) and many tiny pieces, and
the shape of that split, when you sort it from big to small, is
ALWAYS the same distribution, called Poisson-Dirichlet with
parameter 1.
End of document.
----------------------------------------------------------------------
----------------------------------------------------------------------
(End of file.)