Download ADDING AND COUNTING Definition 0.1. A partition of a natural

ADDING AND COUNTING
NOTES OF THE TALK GIVEN BY KEN ONO
BY JOEY KRAISLER AND CHARLES WALKER
Abstract. This paper is a summary of a talk on partition numbers given by Ken Ono
during the MASS Colloquium on November 7, 2013. We would like to thank Professor Ono
for providing the lecture slides of which this summary is based.
Definition 0.1. A partition of a natural number, n, is a way of writing n as a sum of natural
numbers. Two sums that differ only in the order of their summands are considered to be
the same partition.
Definition 0.2. The sequence of partition numbers is defined by
p(n) = the number of integer partitions of n.
For example, p(5) = 7 as 5 = 4+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1.
Partition numbers play an important role in:
- Pure Mathematics: Algebra, Combinatorics, Number Theory;
- Early development of computers;
- Primality Testing.
The sequence {p(n)} grows very rapidly with n. For example, p(200) ≈ 4000000000000.
So it is very difficult to count the partitions for large n. A theorem due to Euler allows for
much easier calculation of p(n).
Theorem 0.3 (Euler’s Pentagonal Number Theorem (1700s)).
For each n ∈ N,
p(n) = p(n − 1) + p(n − 2) − p(n − 5) − p(n − 7) + p(n − 12) + ...
where 1, 2, 5, 7, 12,... are the pentagonal numbers defined by the formula
3m2 − m
km =
, m = 1, −1, 2, −2, 3, −3, ...
2
Using this formula, the first 200 values of p(n) were calculated in 1915:
1, 2, 3, 5, 7, 11, 15, ......, p(200) = 3972999029388.
Euler’s ideas gave birth to a new branch of mathematics known as “q-series Theory.” Many
people have contributed to this field including Jacobi, Ramanujan, Watson, and Andrews.
1
2
NOTES OF THE TALK GIVEN BY KEN ONO BY JOEY KRAISLER AND CHARLES WALKER
Question: Is there a way we can approximate the partition numbers?
Theorem 0.4 (Hardy-Ramanujan (1918)). For large values of n,
√
1
π 2n
3 .
p(n) ∼ √ e
4n 3
√ 2n
Let Approx(n) = 4n1√3 eπ 3 . The following table shows how the approximate values given
by this formula compare to the exact ones.
Table 1. H-R Approximation Formula
n
p(n)
Approx(n)
10
42
48.104...
20
627
692.384...
30 5604
6080.435...
40 37338 40080.080...
50 204226 217590.501...
.
.
.
.
.
.
.
.
.
∞
∞
∞
Approx(n)
p(n)
1.145...
1.104...
1.085...
1.073...
1.065...
.
.
.
1
To obtain this formula, Hardy and Ramanujan invented the “Circle Method” which is a
mathematical tool now fundamental in Quadratic Forms, Harmonic Analysis, and Diophantine Problems.
Bootstrapping the ideas created by Hardy and Ramanujan, Rademacher was able to write
down an exact formula for p(n).
Theorem 0.5 (Rademacher (1943)). For each n ∈ N we have
√
∞
X
−3
Ak (n)
π 24n − 1
p(n) = 2π(24n − 1) 4
· I3 (
).
2
k
6k
k=1
As p(n) is an integer valued function, one may truncate the infinite sum and round as
long as enough terms have been added so that the difference between the sum and the true
value is < 12 . Although this is simple in theory, it is quite difficult in practice to determine
how many terms must be added for this condition to be met.
The table below shows that if we use this formula to compute p(1) = 1, then contrary to
what one would expect, the tenth approximation is worse than the second.
ADDING AND COUNTING
3
Table 2. The 10th approximation is worse than the 2nd
N
1
2
3
.
.
.
8
9
10
PN (1)
1.13355...
1.00296...
0.97318...
.
.
.
1.00528...
1.00633...
1.00633...
Recently, Bruinier and Ono obtained a finite algebraic formula for the number of partitions.
They found a function P (z) for which
1
· (P (αn,1 ) + · · · + P (αn,hn )).
p(n) =
24n − 1
√
In this formula, the numbers P (αn,m ) are algebraic, and the α’s are roots of hn ∼ n many
quadratic equations.
This provides an efficient and effective method of computing p(n).
Now we turn our attention to the following:
Question: How often is p(n) even?
Let P r2 (N ) be the proportion of the first N values of p(n) that are even. Then empirical
data suggests that limN →∞ P r2 (N ) = 12 . This leads to the following conjecture:
Conjecture 0.6. Half of all partition numbers are even.
Empirical data also suggests another similar conjecture.
Conjecture 0.7. One-third of all partition numbers are divisible by 3.
One may think it is reasonable to expect that
1
K
of all partition numbers are divisible by
K, but this is not true. If we consider the case of K = 5, the following table suggests that
more than
1
5
of all partition numbers are divisible by 5.
4
NOTES OF THE TALK GIVEN BY KEN ONO BY JOEY KRAISLER AND CHARLES WALKER
Table 3. How often is p(n) a multiple of 5?
N
500
1,000
1,500
2,000
.
.
.
P r5 (N )
0.336...
0.342...
0.348...
0.346...
.
.
.
Some understanding of this phenomena is given by the following theorem due to Ramanujan.
Theorem 0.8 (Ramanujan (1915)). For all n ∈ N,
p(5n + 4) ≡ 0 (mod 5),
p(7n + 5) ≡ 0 (mod 7),
p(11n + 6) ≡ 0 (mod 11).
Now we can explain the proportion of the multiples of 5 as follows: The theorem implies
that starting with 4, every fifth partition number is divisible by 5, and we expect one-fifth
1 4 9
9
of all the remaining partition numbers to be divisible by 5. Thus, + · =
= 0.36 of
5 5 5
25
the partition numbers should be divisible by 5, which is much closer to the values observed
empirically.
Referring to the theorem above, Ramanujan wrote in 1919, “There appear to be corresponding properties in which the moduli are powers of 5, 7, 11 and no simple properties
for any moduli involving primes other than these three.”
Ramanujan, Watson (1938), and Atkin (1967) establishes corresponding properties for the
powers of 5, 7, and 11:
Theorem 0.9 (Ramanujan’s Congruences).
If 1 ≤ δl (m) ≤ lm satisfies 24δl (m) ≡ 1 (mod lm ), then
p(5m n + δ5 (m)) ≡ 0 (mod 5m ),
p(7m n + δ7 (m)) ≡ 0 (mod 7[
m+2
]
2
),
p(11m n + δ11 (m)) ≡ 0 (mod 11m ).
It is a mystery what Ramanujan meant when he wrote that there are no simple properties
for any moduli involving primes other than these three.
ADDING AND COUNTING
5
The theorem below shows that there are no arithmetic progressions for the primes 2 and
3, and the next one tells us that there are no congruences as in Theorem 0.8 for primes other
than 5, 7, and 11.
Theorem 0.10 (Radu 2011). There are no arithmetic progressions An + B such that
p(An + B) ≡ 0 (mod 2) or p(An + B) ≡ 0 (mod 3).
Theorem 0.11 (Ahlgren, Boylan 2005). The only (l, a) such that l is a prime and
p(ln + a) ≡ 0 (mod l) are (5, 4), (7, 5), and (11, 6).
And here is a not so simple property:
Theorem 0.12 (Ono 2000). For primes Q ≥ 5, there exist infinitely many progressions
An + B such that p(An + B) ≡ 0 (mod Q).
For example, here are several “monstrous” congruences:
p(48037937n + 1122838) ≡ 0 (mod 17),
p(1977147619n + 815655) ≡ 0 (mod 19),
p(14375n + 3474) ≡ 0 (mod 23),
p(348104768909n + 43819835) ≡ 0 (mod 29),
p(4063467631n + 30064597) ≡ 0 (mod 31).
Although more congruences have been discovered, the vast majority of partition numbers
fit neither the Ramanujan’s congruences nor the “monsters.” To see what properties these
numbers have, we must consider fractals.
A Fractal is an infinitely complex structure which:
1) is given by a simple inductive/recursive structure,
2) is self similar at arbitrarily small scales.
Theorem 0.13 (Folsom, Kent, Ono). The partition numbers are l-adically fractal for every
2
−1
prime l ≥ 5. Moreover, the Hausdorff dimension is ≤ [ l−1
] − [ l 24l
].
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Corollary 0.14. The dimension is equal to 0 only if l = 5, 7, 11.
This Corollary explains why Ramanujan’s congruences exist for powers of 5, 7, and 11,
and why there are no simple properties for other primes.