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Multiplicative theory of (additive)partitions
Robert SchneiderUniversity of Georgia
October 27, 2020
1
Additive number theory
Patterns and interconnectionstheory of partitions
beautiful generating functions
surprising bijections
Ramanujan congruences
combinatorics, algebra, analytic num. theory, mod.forms, stat. phys., QT, string theory, chemistry, ...
2
Additive number theory
Patterns and interconnections
theory of partitions
beautiful generating functions
surprising bijections
Ramanujan congruences
combinatorics, algebra, analytic num. theory, mod.forms, stat. phys., QT, string theory, chemistry, ...
3
Additive number theory
Patterns and interconnectionstheory of partitions
beautiful generating functions
surprising bijections
Ramanujan congruences
combinatorics, algebra, analytic num. theory, mod.forms, stat. phys., QT, string theory, chemistry, ...
4
Additive number theory
Patterns and interconnectionstheory of partitions
beautiful generating functions
surprising bijections
Ramanujan congruences
combinatorics, algebra, analytic num. theory, mod.forms, stat. phys., QT, string theory, chemistry, ...
5
Birth of partition theory
Ishango bone (Africa, ca. 20,000 B.C.E.)
6
Birth of partition theory
Ishango bone (Africa, ca. 20,000 B.C.E.)
7
Birth of partition theory
Ahmes papyrus (Egypt, ca. 2,000 B.C.E.)
8
Birth of partition theory
Ahmes papyrus (Egypt, ca. 2,000 B.C.E.)
9
Partition theory
Incan quipu (Peru, 2,000 B.C.E. - 1600s)
10
Partition theory
Incan quipu (Peru, 2,000 B.C.E. - 1600s)
11
Birth of partition theory
G. W. Leibniz (1600s)
12
Birth of partition theory
G. W. Leibniz (1600s)
13
Birth of partition theory
Leibnizwondered about size of p(n) := # of partitions of n
p(n) is called the partition function
14
Birth of partition theory
Leibnizwondered about size of p(n) := # of partitions of n
p(n) is called the partition function
15
Birth of partition theory
Leonhard Euler (1700s)
16
Birth of partition theory
Leonhard Euler (1700s)
17
Birth of partition theory
Partition generating function (Euler)
∞∑n=0
p(n)qn =∞∏
n=1
(1− qn)−1 (q ∈ C, |q| < 1)
Template for partition theoryproduct-sum generating functionscombinatorics encoded in exponents, coefficientsconnected analysis to partitions
18
Birth of partition theory
Partition generating function (Euler)∞∑
n=0
p(n)qn
=∞∏
n=1
(1− qn)−1 (q ∈ C, |q| < 1)
Template for partition theoryproduct-sum generating functionscombinatorics encoded in exponents, coefficientsconnected analysis to partitions
19
Birth of partition theory
Partition generating function (Euler)∞∑
n=0
p(n)qn =∞∏
n=1
(1− qn)−1 (q ∈ C, |q| < 1)
Template for partition theoryproduct-sum generating functionscombinatorics encoded in exponents, coefficientsconnected analysis to partitions
20
Birth of partition theory
Partition generating function (Euler)∞∑
n=0
p(n)qn =∞∏
n=1
(1− qn)−1 (q ∈ C, |q| < 1)
Template for partition theory
product-sum generating functionscombinatorics encoded in exponents, coefficientsconnected analysis to partitions
21
Birth of partition theory
Partition generating function (Euler)∞∑
n=0
p(n)qn =∞∏
n=1
(1− qn)−1 (q ∈ C, |q| < 1)
Template for partition theoryproduct-sum generating functions
combinatorics encoded in exponents, coefficientsconnected analysis to partitions
22
Birth of partition theory
Partition generating function (Euler)∞∑
n=0
p(n)qn =∞∏
n=1
(1− qn)−1 (q ∈ C, |q| < 1)
Template for partition theoryproduct-sum generating functionscombinatorics encoded in exponents, coefficients
connected analysis to partitions
23
Birth of partition theory
Partition generating function (Euler)∞∑
n=0
p(n)qn =∞∏
n=1
(1− qn)−1 (q ∈ C, |q| < 1)
Template for partition theoryproduct-sum generating functionscombinatorics encoded in exponents, coefficientsconnected analysis to partitions
24
Birth of partition theory
Partition generating function (Euler)∞∑
n=0
p(n)qn =∞∏
n=1
(1− qn)−1 (q ∈ C, |q| < 1)
Proof
RHS =∏∞
n=1(1 + qn + q2n + q3n + ...) (geom. series)=∏∞
n=1(1 + qn + qn+n + qn+n+n + ...) (like, second grade)= 1 + q1 + q1+1 + q2 + q1+1+1 + q1+2 + q3 + ...= LHS
25
Birth of partition theory
Partition generating function (Euler)∞∑
n=0
p(n)qn =∞∏
n=1
(1− qn)−1 (q ∈ C, |q| < 1)
ProofRHS
=∏∞
n=1(1 + qn + q2n + q3n + ...) (geom. series)=∏∞
n=1(1 + qn + qn+n + qn+n+n + ...) (like, second grade)= 1 + q1 + q1+1 + q2 + q1+1+1 + q1+2 + q3 + ...= LHS
26
Birth of partition theory
Partition generating function (Euler)∞∑
n=0
p(n)qn =∞∏
n=1
(1− qn)−1 (q ∈ C, |q| < 1)
ProofRHS =
∏∞n=1(1 + qn + q2n + q3n + ...)
(geom. series)=∏∞
n=1(1 + qn + qn+n + qn+n+n + ...) (like, second grade)= 1 + q1 + q1+1 + q2 + q1+1+1 + q1+2 + q3 + ...= LHS
27
Birth of partition theory
Partition generating function (Euler)∞∑
n=0
p(n)qn =∞∏
n=1
(1− qn)−1 (q ∈ C, |q| < 1)
ProofRHS =
∏∞n=1(1 + qn + q2n + q3n + ...) (geom. series)
=∏∞
n=1(1 + qn + qn+n + qn+n+n + ...) (like, second grade)= 1 + q1 + q1+1 + q2 + q1+1+1 + q1+2 + q3 + ...= LHS
28
Birth of partition theory
Partition generating function (Euler)∞∑
n=0
p(n)qn =∞∏
n=1
(1− qn)−1 (q ∈ C, |q| < 1)
ProofRHS =
∏∞n=1(1 + qn + q2n + q3n + ...) (geom. series)
=∏∞
n=1(1 + qn + qn+n + qn+n+n + ...)
(like, second grade)= 1 + q1 + q1+1 + q2 + q1+1+1 + q1+2 + q3 + ...= LHS
29
Birth of partition theory
Partition generating function (Euler)∞∑
n=0
p(n)qn =∞∏
n=1
(1− qn)−1 (q ∈ C, |q| < 1)
ProofRHS =
∏∞n=1(1 + qn + q2n + q3n + ...) (geom. series)
=∏∞
n=1(1 + qn + qn+n + qn+n+n + ...) (like, second grade)
= 1 + q1 + q1+1 + q2 + q1+1+1 + q1+2 + q3 + ...= LHS
30
Birth of partition theory
Partition generating function (Euler)∞∑
n=0
p(n)qn =∞∏
n=1
(1− qn)−1 (q ∈ C, |q| < 1)
ProofRHS =
∏∞n=1(1 + qn + q2n + q3n + ...) (geom. series)
=∏∞
n=1(1 + qn + qn+n + qn+n+n + ...) (like, second grade)= 1 + q1 + q1+1 + q2 + q1+1+1 + q1+2 + q3 + ...
= LHS
31
Birth of partition theory
Partition generating function (Euler)∞∑
n=0
p(n)qn =∞∏
n=1
(1− qn)−1 (q ∈ C, |q| < 1)
ProofRHS =
∏∞n=1(1 + qn + q2n + q3n + ...) (geom. series)
=∏∞
n=1(1 + qn + qn+n + qn+n+n + ...) (like, second grade)= 1 + q1 + q1+1 + q2 + q1+1+1 + q1+2 + q3 + ...= LHS
32
Multiplicative number theory
I.e., most of classical number theoryprimesdivisorsEuler phi function ϕ(n), Möbius function µ(n)arithmetic functions, Dirichlet convolutionzeta functions, Dirichlet series, L-functions
33
Multiplicative number theory
I.e., most of classical number theory
primesdivisorsEuler phi function ϕ(n), Möbius function µ(n)arithmetic functions, Dirichlet convolutionzeta functions, Dirichlet series, L-functions
34
Multiplicative number theory
I.e., most of classical number theoryprimesdivisorsEuler phi function ϕ(n), Möbius function µ(n)arithmetic functions, Dirichlet convolutionzeta functions, Dirichlet series, L-functions
35
Birth of multiplicative number theory
Ishango bone (Africa, ca. 20,000 B.C.E.)
36
Birth of multiplicative number theory
Ahmes papyrus (Egypt, ca. 2,000 B.C.E.)
37
Birth of multiplicative number theory
Eratosthenes, Euclid (Alexandria, ca. 300 B.C.E.)
38
Birth of multiplicative number theory
Euler
first to understand (Riemann) zeta function:
ζ(s) :=∞∑
n=1
1ns (Re(s) > 1)
explicit zeta values→ compute even powers of π:
ζ(2) =π2
6, ζ(4) =
π4
90, ζ(6) =
π6
945, ...
39
Birth of multiplicative number theory
Eulerfirst to understand (Riemann) zeta function
:
ζ(s) :=∞∑
n=1
1ns (Re(s) > 1)
explicit zeta values→ compute even powers of π:
ζ(2) =π2
6, ζ(4) =
π4
90, ζ(6) =
π6
945, ...
40
Birth of multiplicative number theory
Eulerfirst to understand (Riemann) zeta function:
ζ(s) :=∞∑
n=1
1ns (Re(s) > 1)
explicit zeta values→ compute even powers of π:
ζ(2) =π2
6, ζ(4) =
π4
90, ζ(6) =
π6
945, ...
41
Birth of multiplicative number theory
Eulerfirst to understand (Riemann) zeta function:
ζ(s) :=∞∑
n=1
1ns (Re(s) > 1)
explicit zeta values→ compute even powers of π
:
ζ(2) =π2
6, ζ(4) =
π4
90, ζ(6) =
π6
945, ...
42
Birth of multiplicative number theory
Eulerfirst to understand (Riemann) zeta function:
ζ(s) :=∞∑
n=1
1ns (Re(s) > 1)
explicit zeta values→ compute even powers of π:
ζ(2) =π2
6, ζ(4) =
π4
90, ζ(6) =
π6
945, ...
43
Multiplicative number theory
Product formula for zeta function (Euler)
∞∑n=0
1ns =
∏p∈P
(1− p−s)−1
(Re(s) > 1)
Template for study of L-functions“Euler product” generating functionsconnected analysis to prime numbers
44
Multiplicative number theory
Product formula for zeta function (Euler)∞∑
n=0
1ns
=∏p∈P
(1− p−s)−1
(Re(s) > 1)
Template for study of L-functions“Euler product” generating functionsconnected analysis to prime numbers
45
Multiplicative number theory
Product formula for zeta function (Euler)∞∑
n=0
1ns =
∏p∈P
(1− p−s)−1
(Re(s) > 1)
Template for study of L-functions“Euler product” generating functionsconnected analysis to prime numbers
46
Multiplicative number theory
Product formula for zeta function (Euler)∞∑
n=0
1ns =
∏p∈P
(1− p−s)−1
(Re(s) > 1)
Template for study of L-functions
“Euler product” generating functionsconnected analysis to prime numbers
47
Multiplicative number theory
Product formula for zeta function (Euler)∞∑
n=0
1ns =
∏p∈P
(1− p−s)−1
(Re(s) > 1)
Template for study of L-functions“Euler product” generating functions
connected analysis to prime numbers
48
Multiplicative number theory
Product formula for zeta function (Euler)∞∑
n=0
1ns =
∏p∈P
(1− p−s)−1
(Re(s) > 1)
Template for study of L-functions“Euler product” generating functionsconnected analysis to prime numbers
49
Additive-multiplicative correspondence (Euler)
Partition generating function (encodes addition)∞∏
n=1
(1− qn)−1 =∞∑
n=0
p(n)qn, |q| < 1
Euler product formula (encodes primes / multiplic.)∏p∈P
(1− p−s)−1 =∞∑
k=1
n−s, Re(s) > 1
Proofs feel similar (multiply geometric series)
50
Additive-multiplicative correspondence (Euler)
Partition generating function (encodes addition)∞∏
n=1
(1− qn)−1 =∞∑
n=0
p(n)qn, |q| < 1
Euler product formula (encodes primes / multiplic.)∏p∈P
(1− p−s)−1 =∞∑
k=1
n−s, Re(s) > 1
Proofs feel similar (multiply geometric series)
51
Additive-multiplicative correspondence (Euler)
Partition generating function (encodes addition)∞∏
n=1
(1− qn)−1 =∞∑
n=0
p(n)qn, |q| < 1
Euler product formula (encodes primes / multiplic.)∏p∈P
(1− p−s)−1 =∞∑
k=1
n−s, Re(s) > 1
Proofs feel similar (multiply geometric series)
52
Modern vistas in partition theory
Alladi–Erdos (1970s)Bijection between integer factorizations, prime partitions
study properties of arithmetic functions
QuestionAre other thms. in arithmetic images in prime partitions ofcombinatorial/set-theoretic meta-structures?
53
Modern vistas in partition theory
Alladi–Erdos (1970s)
Bijection between integer factorizations, prime partitionsstudy properties of arithmetic functions
QuestionAre other thms. in arithmetic images in prime partitions ofcombinatorial/set-theoretic meta-structures?
54
Modern vistas in partition theory
Alladi–Erdos (1970s)Bijection between integer factorizations, prime partitions
study properties of arithmetic functions
QuestionAre other thms. in arithmetic images in prime partitions ofcombinatorial/set-theoretic meta-structures?
55
Modern vistas in partition theory
Alladi–Erdos (1970s)Bijection between integer factorizations, prime partitions
study properties of arithmetic functions
QuestionAre other thms. in arithmetic images in prime partitions ofcombinatorial/set-theoretic meta-structures?
56
Modern vistas in partition theory
Alladi–Erdos (1970s)Bijection between integer factorizations, prime partitions
study properties of arithmetic functions
Question
Are other thms. in arithmetic images in prime partitions ofcombinatorial/set-theoretic meta-structures?
57
Modern vistas in partition theory
Alladi–Erdos (1970s)Bijection between integer factorizations, prime partitions
study properties of arithmetic functions
QuestionAre other thms. in arithmetic images in prime partitions
ofcombinatorial/set-theoretic meta-structures?
58
Modern vistas in partition theory
Alladi–Erdos (1970s)Bijection between integer factorizations, prime partitions
study properties of arithmetic functions
QuestionAre other thms. in arithmetic images in prime partitions ofcombinatorial/set-theoretic meta-structures?
59
Modern vistas in partition theory
Andrews (1970s)
Theory of partition idealsinspired by lattice theoryunifies classical results on gen. functions, bijectionssuggestive of a universal algebra of partitions
QuestionIs there an algebra of partitions generalizing arithmetic inintegers (i.e., prime partitions)?
60
Modern vistas in partition theory
Andrews (1970s)Theory of partition ideals
inspired by lattice theoryunifies classical results on gen. functions, bijectionssuggestive of a universal algebra of partitions
QuestionIs there an algebra of partitions generalizing arithmetic inintegers (i.e., prime partitions)?
61
Modern vistas in partition theory
Andrews (1970s)Theory of partition ideals
inspired by lattice theory
unifies classical results on gen. functions, bijectionssuggestive of a universal algebra of partitions
QuestionIs there an algebra of partitions generalizing arithmetic inintegers (i.e., prime partitions)?
62
Modern vistas in partition theory
Andrews (1970s)Theory of partition ideals
inspired by lattice theoryunifies classical results on gen. functions, bijections
suggestive of a universal algebra of partitions
QuestionIs there an algebra of partitions generalizing arithmetic inintegers (i.e., prime partitions)?
63
Modern vistas in partition theory
Andrews (1970s)Theory of partition ideals
inspired by lattice theoryunifies classical results on gen. functions, bijectionssuggestive of a universal algebra of partitions
QuestionIs there an algebra of partitions generalizing arithmetic inintegers (i.e., prime partitions)?
64
Modern vistas in partition theory
Andrews (1970s)Theory of partition ideals
inspired by lattice theoryunifies classical results on gen. functions, bijectionssuggestive of a universal algebra of partitions
Question
Is there an algebra of partitions generalizing arithmetic inintegers (i.e., prime partitions)?
65
Modern vistas in partition theory
Andrews (1970s)Theory of partition ideals
inspired by lattice theoryunifies classical results on gen. functions, bijectionssuggestive of a universal algebra of partitions
QuestionIs there an algebra of partitions
generalizing arithmetic inintegers (i.e., prime partitions)?
66
Modern vistas in partition theory
Andrews (1970s)Theory of partition ideals
inspired by lattice theoryunifies classical results on gen. functions, bijectionssuggestive of a universal algebra of partitions
QuestionIs there an algebra of partitions generalizing arithmetic inintegers
(i.e., prime partitions)?
67
Modern vistas in partition theory
Andrews (1970s)Theory of partition ideals
inspired by lattice theoryunifies classical results on gen. functions, bijectionssuggestive of a universal algebra of partitions
QuestionIs there an algebra of partitions generalizing arithmetic inintegers (i.e., prime partitions)?
68
Multiplicative theory of (additive) partitions
Philosophy of this talk
Exist multipl., division, arith. functions on partitions
Objects in classical multiplic. number theory→ special cases of partition-theoretic structures
Expect arithmetic theorems→ extend to partitions
Expect partition properties→ properties of integers
69
Multiplicative theory of (additive) partitions
Philosophy of this talk
Exist multipl., division, arith. functions on partitions
Objects in classical multiplic. number theory→ special cases of partition-theoretic structures
Expect arithmetic theorems→ extend to partitions
Expect partition properties→ properties of integers
70
Multiplicative theory of (additive) partitions
Philosophy of this talk
Exist multipl., division, arith. functions on partitions
Objects in classical multiplic. number theory
→ special cases of partition-theoretic structures
Expect arithmetic theorems→ extend to partitions
Expect partition properties→ properties of integers
71
Multiplicative theory of (additive) partitions
Philosophy of this talk
Exist multipl., division, arith. functions on partitions
Objects in classical multiplic. number theory→ special cases of partition-theoretic structures
Expect arithmetic theorems→ extend to partitions
Expect partition properties→ properties of integers
72
Multiplicative theory of (additive) partitions
Philosophy of this talk
Exist multipl., division, arith. functions on partitions
Objects in classical multiplic. number theory→ special cases of partition-theoretic structures
Expect arithmetic theorems
→ extend to partitions
Expect partition properties→ properties of integers
73
Multiplicative theory of (additive) partitions
Philosophy of this talk
Exist multipl., division, arith. functions on partitions
Objects in classical multiplic. number theory→ special cases of partition-theoretic structures
Expect arithmetic theorems→ extend to partitions
Expect partition properties→ properties of integers
74
Multiplicative theory of (additive) partitions
Philosophy of this talk
Exist multipl., division, arith. functions on partitions
Objects in classical multiplic. number theory→ special cases of partition-theoretic structures
Expect arithmetic theorems→ extend to partitions
Expect partition properties
→ properties of integers
75
Multiplicative theory of (additive) partitions
Philosophy of this talk
Exist multipl., division, arith. functions on partitions
Objects in classical multiplic. number theory→ special cases of partition-theoretic structures
Expect arithmetic theorems→ extend to partitions
Expect partition properties→ properties of integers
76
Partition notations
Let P denote the set of all integer partitions.
Let ∅ denote the empty partition.
Let λ = (λ1, λ2, . . . , λr ), λ1 ≥ λ2 ≥ · · · ≥ λr ≥ 1,denote a nonempty partition, e.g. λ = (3,2,2,1).
Let PX denote partitions into elements λi ∈ X ⊆ N,e.g. PP is the “prime partitions”.
77
Partition notations
Let P denote the set of all integer partitions.
Let ∅ denote the empty partition.
Let λ = (λ1, λ2, . . . , λr ), λ1 ≥ λ2 ≥ · · · ≥ λr ≥ 1,denote a nonempty partition, e.g. λ = (3,2,2,1).
Let PX denote partitions into elements λi ∈ X ⊆ N,e.g. PP is the “prime partitions”.
78
Partition notations
Let P denote the set of all integer partitions.
Let ∅ denote the empty partition.
Let λ = (λ1, λ2, . . . , λr ), λ1 ≥ λ2 ≥ · · · ≥ λr ≥ 1,denote a nonempty partition, e.g. λ = (3,2,2,1).
Let PX denote partitions into elements λi ∈ X ⊆ N,e.g. PP is the “prime partitions”.
79
Partition notations
Let P denote the set of all integer partitions.
Let ∅ denote the empty partition.
Let λ = (λ1, λ2, . . . , λr ), λ1 ≥ λ2 ≥ · · · ≥ λr ≥ 1,denote a nonempty partition
, e.g. λ = (3,2,2,1).
Let PX denote partitions into elements λi ∈ X ⊆ N,e.g. PP is the “prime partitions”.
80
Partition notations
Let P denote the set of all integer partitions.
Let ∅ denote the empty partition.
Let λ = (λ1, λ2, . . . , λr ), λ1 ≥ λ2 ≥ · · · ≥ λr ≥ 1,denote a nonempty partition, e.g. λ = (3,2,2,1).
Let PX denote partitions into elements λi ∈ X ⊆ N,e.g. PP is the “prime partitions”.
81
Partition notations
Let P denote the set of all integer partitions.
Let ∅ denote the empty partition.
Let λ = (λ1, λ2, . . . , λr ), λ1 ≥ λ2 ≥ · · · ≥ λr ≥ 1,denote a nonempty partition, e.g. λ = (3,2,2,1).
Let PX denote partitions into elements λi ∈ X ⊆ N
,e.g. PP is the “prime partitions”.
82
Partition notations
Let P denote the set of all integer partitions.
Let ∅ denote the empty partition.
Let λ = (λ1, λ2, . . . , λr ), λ1 ≥ λ2 ≥ · · · ≥ λr ≥ 1,denote a nonempty partition, e.g. λ = (3,2,2,1).
Let PX denote partitions into elements λi ∈ X ⊆ N,e.g. PP is the “prime partitions”.
83
Partition notations
`(λ) := r is length (number of parts).
mi = mi(λ) := multiplicity (or “frequency”) of i .
|λ| := λ1 + λ2 + · · ·+ λr is size (sum of parts).
“λ ` n” means λ is a partition of n.
Define `(∅) = |∅| = mi(∅) = 0, ∅ ` 0.
84
Partition notations
`(λ) := r is length (number of parts).
mi = mi(λ) := multiplicity (or “frequency”) of i .
|λ| := λ1 + λ2 + · · ·+ λr is size (sum of parts).
“λ ` n” means λ is a partition of n.
Define `(∅) = |∅| = mi(∅) = 0, ∅ ` 0.
85
Partition notations
`(λ) := r is length (number of parts).
mi = mi(λ) := multiplicity (or “frequency”) of i .
|λ| := λ1 + λ2 + · · ·+ λr is size (sum of parts).
“λ ` n” means λ is a partition of n.
Define `(∅) = |∅| = mi(∅) = 0, ∅ ` 0.
86
Partition notations
`(λ) := r is length (number of parts).
mi = mi(λ) := multiplicity (or “frequency”) of i .
|λ| := λ1 + λ2 + · · ·+ λr is size (sum of parts).
“λ ` n” means λ is a partition of n.
Define `(∅) = |∅| = mi(∅) = 0, ∅ ` 0.
87
Partition notations
`(λ) := r is length (number of parts).
mi = mi(λ) := multiplicity (or “frequency”) of i .
|λ| := λ1 + λ2 + · · ·+ λr is size (sum of parts).
“λ ` n” means λ is a partition of n.
Define `(∅) = |∅| = mi(∅) = 0, ∅ ` 0.
88
Partition notations
`(λ) := r is length (number of parts).
mi = mi(λ) := multiplicity (or “frequency”) of i .
|λ| := λ1 + λ2 + · · ·+ λr is size (sum of parts).
“λ ` n” means λ is a partition of n.
Define `(∅) = |∅| = mi(∅) = 0, ∅ ` 0.
89
Multiplicative theory of (additive) partitions
New partition statisticDefine N(λ), the norm of λ, to be the product of the parts:
N(λ) := λ1λ2λ3 · · ·λr
Define N(∅) := 1 (it is an empty product)See “The product of parts or ‘norm’ etc.” (S-Sills)
90
Multiplicative theory of (additive) partitions
New partition statistic
Define N(λ), the norm of λ, to be the product of the parts:
N(λ) := λ1λ2λ3 · · ·λr
Define N(∅) := 1 (it is an empty product)See “The product of parts or ‘norm’ etc.” (S-Sills)
91
Multiplicative theory of (additive) partitions
New partition statisticDefine N(λ), the norm of λ, to be the product of the parts:
N(λ) := λ1λ2λ3 · · ·λr
Define N(∅) := 1 (it is an empty product)See “The product of parts or ‘norm’ etc.” (S-Sills)
92
Multiplicative theory of (additive) partitions
New partition statisticDefine N(λ), the norm of λ, to be the product of the parts:
N(λ) := λ1λ2λ3 · · ·λr
Define N(∅) := 1 (it is an empty product)See “The product of parts or ‘norm’ etc.” (S-Sills)
93
Multiplicative theory of (additive) partitions
New partition statisticDefine N(λ), the norm of λ, to be the product of the parts:
N(λ) := λ1λ2λ3 · · ·λr
Define N(∅) := 1 (it is an empty product)
See “The product of parts or ‘norm’ etc.” (S-Sills)
94
Multiplicative theory of (additive) partitions
New partition statisticDefine N(λ), the norm of λ, to be the product of the parts:
N(λ) := λ1λ2λ3 · · ·λr
Define N(∅) := 1 (it is an empty product)See “The product of parts or ‘norm’ etc.” (S-Sills)
95
Multiplicative theory of (additive) partitions
Partition multiplication
For λ, γ ∈ P let λγ denote multiset union of the parts,e.g. (3,2)(2,1) = (3,2,2,1).
Identity is ∅
Partitions into one part are like primes, FTA holds
96
Multiplicative theory of (additive) partitions
Partition multiplication
For λ, γ ∈ P let λγ denote multiset union of the parts
,e.g. (3,2)(2,1) = (3,2,2,1).
Identity is ∅
Partitions into one part are like primes, FTA holds
97
Multiplicative theory of (additive) partitions
Partition multiplication
For λ, γ ∈ P let λγ denote multiset union of the parts,e.g. (3,2)(2,1) = (3,2,2,1).
Identity is ∅
Partitions into one part are like primes, FTA holds
98
Multiplicative theory of (additive) partitions
Partition multiplication
For λ, γ ∈ P let λγ denote multiset union of the parts,e.g. (3,2)(2,1) = (3,2,2,1).
Identity is ∅
Partitions into one part are like primes, FTA holds
99
Multiplicative theory of (additive) partitions
Partition multiplication
For λ, γ ∈ P let λγ denote multiset union of the parts,e.g. (3,2)(2,1) = (3,2,2,1).
Identity is ∅
Partitions into one part are like primes, FTA holds
100
Multiplicative theory of (additive) partitions
Partition division (subpartitions)
For λ, δ ∈ P, let δ|λ mean all parts of δ are parts of λ,e.g. (3,2,1)|(3,2,2,1).
For δ|λ, let λ/δ mean parts of δ deleted from λ,e.g. (3,2,2,1)/(3,2,1) = (2).
Replace P with PP → mult./div. in Z+
101
Multiplicative theory of (additive) partitions
Partition division (subpartitions)
For λ, δ ∈ P, let δ|λ mean all parts of δ are parts of λ
,e.g. (3,2,1)|(3,2,2,1).
For δ|λ, let λ/δ mean parts of δ deleted from λ,e.g. (3,2,2,1)/(3,2,1) = (2).
Replace P with PP → mult./div. in Z+
102
Multiplicative theory of (additive) partitions
Partition division (subpartitions)
For λ, δ ∈ P, let δ|λ mean all parts of δ are parts of λ,e.g. (3,2,1)|(3,2,2,1).
For δ|λ, let λ/δ mean parts of δ deleted from λ,e.g. (3,2,2,1)/(3,2,1) = (2).
Replace P with PP → mult./div. in Z+
103
Multiplicative theory of (additive) partitions
Partition division (subpartitions)
For λ, δ ∈ P, let δ|λ mean all parts of δ are parts of λ,e.g. (3,2,1)|(3,2,2,1).
For δ|λ, let λ/δ mean parts of δ deleted from λ
,e.g. (3,2,2,1)/(3,2,1) = (2).
Replace P with PP → mult./div. in Z+
104
Multiplicative theory of (additive) partitions
Partition division (subpartitions)
For λ, δ ∈ P, let δ|λ mean all parts of δ are parts of λ,e.g. (3,2,1)|(3,2,2,1).
For δ|λ, let λ/δ mean parts of δ deleted from λ,e.g. (3,2,2,1)/(3,2,1) = (2).
Replace P with PP → mult./div. in Z+
105
Multiplicative theory of (additive) partitions
Partition division (subpartitions)
For λ, δ ∈ P, let δ|λ mean all parts of δ are parts of λ,e.g. (3,2,1)|(3,2,2,1).
For δ|λ, let λ/δ mean parts of δ deleted from λ,e.g. (3,2,2,1)/(3,2,1) = (2).
Replace P with PP
→ mult./div. in Z+
106
Multiplicative theory of (additive) partitions
Partition division (subpartitions)
For λ, δ ∈ P, let δ|λ mean all parts of δ are parts of λ,e.g. (3,2,1)|(3,2,2,1).
For δ|λ, let λ/δ mean parts of δ deleted from λ,e.g. (3,2,2,1)/(3,2,1) = (2).
Replace P with PP → mult./div. in Z+
107
Parallel universe
Many arithmetic objects have partition counterparts.
Partition Möbius functionFor λ ∈ P, define
µP(λ) :=
{0 if λ has any part repeated,(−1)`(λ) otherwise.
Replacing P with PP reduces to µ(N(λ)), where N(λ)is the norm (product of parts).
108
Parallel universe
Many arithmetic objects have partition counterparts.
Partition Möbius functionFor λ ∈ P, define
µP(λ) :=
{0 if λ has any part repeated,(−1)`(λ) otherwise.
Replacing P with PP reduces to µ(N(λ)), where N(λ)is the norm (product of parts).
109
Parallel universe
Many arithmetic objects have partition counterparts.
Partition Möbius function
For λ ∈ P, define
µP(λ) :=
{0 if λ has any part repeated,(−1)`(λ) otherwise.
Replacing P with PP reduces to µ(N(λ)), where N(λ)is the norm (product of parts).
110
Parallel universe
Many arithmetic objects have partition counterparts.
Partition Möbius functionFor λ ∈ P, define
µP(λ) :=
{0 if λ has any part repeated,(−1)`(λ) otherwise.
Replacing P with PP reduces to µ(N(λ)), where N(λ)is the norm (product of parts).
111
Parallel universe
Many arithmetic objects have partition counterparts.
Partition Möbius functionFor λ ∈ P, define
µP(λ) :=
{0 if λ has any part repeated,(−1)`(λ) otherwise.
Replacing P with PP reduces to µ(N(λ)), where N(λ)is the norm (product of parts).
112
Parallel universe
Many arithmetic objects have partition counterparts.
Partition Möbius functionFor λ ∈ P, define
µP(λ) :=
{0 if λ has any part repeated,(−1)`(λ) otherwise.
Replacing P with PP reduces to µ(N(λ)), where N(λ)is the norm (product of parts).
113
Parallel universe
Just as in classical cases, nice sums over “divisors”...
Partition Möbius function∑δ|λ
µP(δ) =
{1 if λ = ∅,0 otherwise
114
Parallel universe
Just as in classical cases, nice sums over “divisors”...
Partition Möbius function∑δ|λ
µP(δ) =
{1 if λ = ∅,0 otherwise
115
Parallel universe
Just as in classical cases, nice sums over “divisors”...
Partition Möbius function
∑δ|λ
µP(δ) =
{1 if λ = ∅,0 otherwise
116
Parallel universe
Just as in classical cases, nice sums over “divisors”...
Partition Möbius function∑δ|λ
µP(δ) =
{1 if λ = ∅,0 otherwise
117
Parallel universe
Partition Möbius inversionIf we have
f (λ) =∑δ|λ
g(δ)
we also have
g(λ) =∑δ|λ
µP(λ/δ)f (δ).
118
Parallel universe
Partition Möbius inversionIf we have
f (λ) =∑δ|λ
g(δ)
we also have
g(λ) =∑δ|λ
µP(λ/δ)f (δ).
119
Parallel universe
Classically, µ(n) has a close companion in ϕ(n).
Partition phi functionFor λ ∈ P, define
ϕP(λ) := N(λ)∏λi∈λ
no repeats
(1− λ−1i ).
Replacing P with PP reduces to ϕ(N(λ)).
120
Parallel universe
Classically, µ(n) has a close companion in ϕ(n).
Partition phi function
For λ ∈ P, define
ϕP(λ) := N(λ)∏λi∈λ
no repeats
(1− λ−1i ).
Replacing P with PP reduces to ϕ(N(λ)).
121
Parallel universe
Classically, µ(n) has a close companion in ϕ(n).
Partition phi functionFor λ ∈ P, define
ϕP(λ) := N(λ)∏λi∈λ
no repeats
(1− λ−1i ).
Replacing P with PP reduces to ϕ(N(λ)).
122
Parallel universe
Classically, µ(n) has a close companion in ϕ(n).
Partition phi functionFor λ ∈ P, define
ϕP(λ) := N(λ)∏λi∈λ
no repeats
(1− λ−1i ).
Replacing P with PP reduces to ϕ(N(λ)).
123
Parallel universe
Classically, µ(n) has a close companion in ϕ(n).
Partition phi functionFor λ ∈ P, define
ϕP(λ) := N(λ)∏λi∈λ
no repeats
(1− λ−1i ).
Replacing P with PP reduces to ϕ(N(λ)).
124
Parallel universe
Phi function identities
∑δ|λ
ϕP(δ) = N(λ), ϕP(λ) = N(λ)∑δ|λ
µP(δ)
N(δ)
Replacing P with PP reduces to classical cases.Many analogs of objects in multiplic. # theory...
125
Parallel universe
Phi function identities
∑δ|λ
ϕP(δ) = N(λ), ϕP(λ) = N(λ)∑δ|λ
µP(δ)
N(δ)
Replacing P with PP reduces to classical cases.Many analogs of objects in multiplic. # theory...
126
Parallel universe
Phi function identities
∑δ|λ
ϕP(δ) = N(λ), ϕP(λ) = N(λ)∑δ|λ
µP(δ)
N(δ)
Replacing P with PP reduces to classical cases.
Many analogs of objects in multiplic. # theory...
127
Parallel universe
Phi function identities
∑δ|λ
ϕP(δ) = N(λ), ϕP(λ) = N(λ)∑δ|λ
µP(δ)
N(δ)
Replacing P with PP reduces to classical cases.Many analogs of objects in multiplic. # theory...
128
Parallel universe
Partition Cauchy product(∑λ∈P
f (λ)q|λ|)(∑
λ∈P
g(λ)q|λ|)
=∑λ∈P
q|λ|∑δ|λ
f (δ)g(λ/δ)
Swapping order of summation∑λ∈P
f (λ)∑δ|λ
g(δ) =∑λ∈P
g(λ)∑γ∈P
f (λγ)
Other multiplicative objects generalize to partition theory...
129
Parallel universe
Partition Cauchy product(∑λ∈P
f (λ)q|λ|)(∑
λ∈P
g(λ)q|λ|)
=∑λ∈P
q|λ|∑δ|λ
f (δ)g(λ/δ)
Swapping order of summation∑λ∈P
f (λ)∑δ|λ
g(δ) =∑λ∈P
g(λ)∑γ∈P
f (λγ)
Other multiplicative objects generalize to partition theory...
130
Parallel universe
Partition Cauchy product(∑λ∈P
f (λ)q|λ|)(∑
λ∈P
g(λ)q|λ|)
=∑λ∈P
q|λ|∑δ|λ
f (δ)g(λ/δ)
Swapping order of summation∑λ∈P
f (λ)∑δ|λ
g(δ) =∑λ∈P
g(λ)∑γ∈P
f (λγ)
Other multiplicative objects generalize to partition theory...
131
Partition zeta functions
In analogy with ζ(s) =∑∞
n=1 n−s:
DefinitionFor P ′ ( P , s ∈ C, define a partition zeta function:
ζP ′(s) :=∑λ∈P ′
N(λ)−s
N(λ) is the norm (product of parts) of λ, N(∅) := 1.For 1 6∈ P ′ = PX (parts in X ⊂ N)→ Euler product:
ζPX(s) =∏n∈X
(1− n−s)−1
132
Partition zeta functions
In analogy with ζ(s) =∑∞
n=1 n−s:
DefinitionFor P ′ ( P , s ∈ C, define a partition zeta function:
ζP ′(s) :=∑λ∈P ′
N(λ)−s
N(λ) is the norm (product of parts) of λ, N(∅) := 1.For 1 6∈ P ′ = PX (parts in X ⊂ N)→ Euler product:
ζPX(s) =∏n∈X
(1− n−s)−1
133
Partition zeta functions
In analogy with ζ(s) =∑∞
n=1 n−s:
DefinitionFor P ′ ( P , s ∈ C, define a partition zeta function
:
ζP ′(s) :=∑λ∈P ′
N(λ)−s
N(λ) is the norm (product of parts) of λ, N(∅) := 1.For 1 6∈ P ′ = PX (parts in X ⊂ N)→ Euler product:
ζPX(s) =∏n∈X
(1− n−s)−1
134
Partition zeta functions
In analogy with ζ(s) =∑∞
n=1 n−s:
DefinitionFor P ′ ( P , s ∈ C, define a partition zeta function:
ζP ′(s) :=∑λ∈P ′
N(λ)−s
N(λ) is the norm (product of parts) of λ, N(∅) := 1.For 1 6∈ P ′ = PX (parts in X ⊂ N)→ Euler product:
ζPX(s) =∏n∈X
(1− n−s)−1
135
Partition zeta functions
In analogy with ζ(s) =∑∞
n=1 n−s:
DefinitionFor P ′ ( P , s ∈ C, define a partition zeta function:
ζP ′(s) :=∑λ∈P ′
N(λ)−s
N(λ) is the norm (product of parts) of λ, N(∅) := 1.
For 1 6∈ P ′ = PX (parts in X ⊂ N)→ Euler product:
ζPX(s) =∏n∈X
(1− n−s)−1
136
Partition zeta functions
In analogy with ζ(s) =∑∞
n=1 n−s:
DefinitionFor P ′ ( P , s ∈ C, define a partition zeta function:
ζP ′(s) :=∑λ∈P ′
N(λ)−s
N(λ) is the norm (product of parts) of λ, N(∅) := 1.For 1 6∈ P ′ = PX (parts in X ⊂ N)
→ Euler product:
ζPX(s) =∏n∈X
(1− n−s)−1
137
Partition zeta functions
In analogy with ζ(s) =∑∞
n=1 n−s:
DefinitionFor P ′ ( P , s ∈ C, define a partition zeta function:
ζP ′(s) :=∑λ∈P ′
N(λ)−s
N(λ) is the norm (product of parts) of λ, N(∅) := 1.For 1 6∈ P ′ = PX (parts in X ⊂ N)→ Euler product:
ζPX(s) =∏n∈X
(1− n−s)−1
138
Partition zeta functions: nice identities
Fix s = 2, vary subset P ′
Summing over partitions into prime parts:
ζPP(2) = ζ(2) =π2
6
Summing over partitions into even parts:
ζPeven(2) =π
2
Summing over partitions into distinct parts:
ζPdistinct(2) =sinh π
π
139
Partition zeta functions: nice identities
Fix s = 2, vary subset P ′
Summing over partitions into prime parts:
ζPP(2) = ζ(2) =π2
6
Summing over partitions into even parts:
ζPeven(2) =π
2
Summing over partitions into distinct parts:
ζPdistinct(2) =sinh π
π
140
Partition zeta functions: nice identites
Partition analogs of classical identities∑λ∈PX
µP(λ)N(λ)−s =1
ζPX(s)∑λ∈PX
ϕP(λ)N(λ)−s =ζPX(s − 1)ζPX(s)
Takeaway from these examplesDifferent subsets of P induce very diff. zeta valuesClassical zeta theorems→ partition zeta theorems
141
Partition zeta functions: nice identites
Partition analogs of classical identities∑λ∈PX
µP(λ)N(λ)−s =1
ζPX(s)∑λ∈PX
ϕP(λ)N(λ)−s =ζPX(s − 1)ζPX(s)
Takeaway from these examples
Different subsets of P induce very diff. zeta valuesClassical zeta theorems→ partition zeta theorems
142
Partition zeta functions: nice identites
Partition analogs of classical identities∑λ∈PX
µP(λ)N(λ)−s =1
ζPX(s)∑λ∈PX
ϕP(λ)N(λ)−s =ζPX(s − 1)ζPX(s)
Takeaway from these examplesDifferent subsets of P induce very diff. zeta values
Classical zeta theorems→ partition zeta theorems
143
Partition zeta functions: nice identites
Partition analogs of classical identities∑λ∈PX
µP(λ)N(λ)−s =1
ζPX(s)∑λ∈PX
ϕP(λ)N(λ)−s =ζPX(s − 1)ζPX(s)
Takeaway from these examplesDifferent subsets of P induce very diff. zeta valuesClassical zeta theorems→ partition zeta theorems
144
Partition zeta functions: nice identities
Pretty cool, but it would be a whole lot cooler if we hadfamilies of identities like Euler’s zeta values
:
ζ(2N) = π2N × rational number
QuestionDo there exist (non-Riemann) partition zeta functionssuch that, for the “right” choice of P ′ ( P,
ζP ′(N) = πM × rational number?
145
Partition zeta functions: nice identities
Pretty cool, but it would be a whole lot cooler if we hadfamilies of identities like Euler’s zeta values:
ζ(2N) = π2N × rational number
QuestionDo there exist (non-Riemann) partition zeta functionssuch that, for the “right” choice of P ′ ( P,
ζP ′(N) = πM × rational number?
146
Partition zeta functions: nice identities
Pretty cool, but it would be a whole lot cooler if we hadfamilies of identities like Euler’s zeta values:
ζ(2N) = π2N × rational number
Question
Do there exist (non-Riemann) partition zeta functionssuch that, for the “right” choice of P ′ ( P,
ζP ′(N) = πM × rational number?
147
Partition zeta functions: nice identities
Pretty cool, but it would be a whole lot cooler if we hadfamilies of identities like Euler’s zeta values:
ζ(2N) = π2N × rational number
QuestionDo there exist (non-Riemann) partition zeta functionssuch that, for the “right” choice of P ′ ( P
,
ζP ′(N) = πM × rational number?
148
Partition zeta functions: nice identities
Pretty cool, but it would be a whole lot cooler if we hadfamilies of identities like Euler’s zeta values:
ζ(2N) = π2N × rational number
QuestionDo there exist (non-Riemann) partition zeta functionssuch that, for the “right” choice of P ′ ( P,
ζP ′(N) = πM × rational number?
149
Partition zeta functions: nice family
Partitions of fixed length k
Define sum over partitions of fixed length `(λ) = k :
ζP({s}k) :=∑λ∈P`(λ)=k
N(λ)−s (Re(s) > 1)
150
Partition zeta functions: nice family
Partitions of fixed length kDefine sum over partitions of fixed length `(λ) = k :
ζP({s}k) :=∑λ∈P`(λ)=k
N(λ)−s (Re(s) > 1)
151
Partition zeta functions: nice family
Partitions of fixed length kDefine sum over partitions of fixed length `(λ) = k :
ζP({s}k) :=∑λ∈P`(λ)=k
N(λ)−s (Re(s) > 1)
152
Partition zeta functions: nice family
Theorem (S, 2016)
Summing over partitions of fixed length k > 0:
ζP({2}k) =22k−1 − 1
22k−2 ζ(2k) = π2k × rational number
Note: k = 0 suggests ζ(0) = −12 (correct value)
Theorem (Ono-Rolen-S, 2017)
For N ≥ 1: ζP({2N}k) = π2Nk × rational number
153
Partition zeta functions: nice family
Theorem (S, 2016)Summing over partitions of fixed length k > 0:
ζP({2}k) =22k−1 − 1
22k−2 ζ(2k) = π2k × rational number
Note: k = 0 suggests ζ(0) = −12 (correct value)
Theorem (Ono-Rolen-S, 2017)
For N ≥ 1: ζP({2N}k) = π2Nk × rational number
154
Partition zeta functions: nice family
Theorem (S, 2016)Summing over partitions of fixed length k > 0:
ζP({2}k) =22k−1 − 1
22k−2 ζ(2k) = π2k × rational number
Note: k = 0 suggests ζ(0) = −12 (correct value)
Theorem (Ono-Rolen-S, 2017)
For N ≥ 1: ζP({2N}k) = π2Nk × rational number
155
Partition zeta functions: nice family
Theorem (S, 2016)Summing over partitions of fixed length k > 0:
ζP({2}k) =22k−1 − 1
22k−2 ζ(2k) = π2k × rational number
Note: k = 0 suggests ζ(0) = −12 (correct value)
Theorem (Ono-Rolen-S, 2017)
For N ≥ 1: ζP({2N}k) = π2Nk × rational number
156
Partition zeta functions: nice family
Theorem (S, 2016)Summing over partitions of fixed length k > 0:
ζP({2}k) =22k−1 − 1
22k−2 ζ(2k) = π2k × rational number
Note: k = 0 suggests ζ(0) = −12 (correct value)
Theorem (Ono-Rolen-S, 2017)
For N ≥ 1: ζP({2N}k) = π2Nk × rational number
157
Partition zeta functions: nice family
Theorem (Ono-Rolen-S., 2017)Some other partition zeta fctns. contin. to right half-plane.
Natural questions
General ζP({s}k)? Analytic continuation? Poles? Roots?
158
Partition zeta functions: nice family
Theorem (Ono-Rolen-S., 2017)Some other partition zeta fctns. contin. to right half-plane.
Natural questions
General ζP({s}k)? Analytic continuation? Poles? Roots?
159
Partition zeta functions: nice family
Theorem (Ono-Rolen-S., 2017)Some other partition zeta fctns. contin. to right half-plane.
Natural questions
General ζP({s}k)? Analytic continuation? Poles? Roots?
160
Partition zeta functions: analytic properties
Theorem (S.-Sills, 2020)
For Re(s) > 1:
ζP({s}k) =∑λ`k
ζ(s)m1ζ(2s)m2ζ(3s)m3 · · · ζ(ks)mk
N(λ) m1! m2! m3! · · · mk !
sum over partitions λ of size k on RHSinherits continuation from ζ(s)poles at s = 1,1/2,1/3,1/4, ...,1/ktrivial roots at s ∈ −2N
Note: ζP({s}k) not zero at roots of ζ(s) for k > 1
161
Partition zeta functions: analytic properties
Theorem (S.-Sills, 2020)For Re(s) > 1:
ζP({s}k) =∑λ`k
ζ(s)m1ζ(2s)m2ζ(3s)m3 · · · ζ(ks)mk
N(λ) m1! m2! m3! · · · mk !
sum over partitions λ of size k on RHSinherits continuation from ζ(s)poles at s = 1,1/2,1/3,1/4, ...,1/ktrivial roots at s ∈ −2N
Note: ζP({s}k) not zero at roots of ζ(s) for k > 1
162
Partition zeta functions: analytic properties
Theorem (S.-Sills, 2020)For Re(s) > 1:
ζP({s}k) =∑λ`k
ζ(s)m1ζ(2s)m2ζ(3s)m3 · · · ζ(ks)mk
N(λ) m1! m2! m3! · · · mk !
sum over partitions λ of size k on RHS
inherits continuation from ζ(s)poles at s = 1,1/2,1/3,1/4, ...,1/ktrivial roots at s ∈ −2N
Note: ζP({s}k) not zero at roots of ζ(s) for k > 1
163
Partition zeta functions: analytic properties
Theorem (S.-Sills, 2020)For Re(s) > 1:
ζP({s}k) =∑λ`k
ζ(s)m1ζ(2s)m2ζ(3s)m3 · · · ζ(ks)mk
N(λ) m1! m2! m3! · · · mk !
sum over partitions λ of size k on RHSinherits continuation from ζ(s)
poles at s = 1,1/2,1/3,1/4, ...,1/ktrivial roots at s ∈ −2N
Note: ζP({s}k) not zero at roots of ζ(s) for k > 1
164
Partition zeta functions: analytic properties
Theorem (S.-Sills, 2020)For Re(s) > 1:
ζP({s}k) =∑λ`k
ζ(s)m1ζ(2s)m2ζ(3s)m3 · · · ζ(ks)mk
N(λ) m1! m2! m3! · · · mk !
sum over partitions λ of size k on RHSinherits continuation from ζ(s)poles at s = 1,1/2,1/3,1/4, ...,1/k
trivial roots at s ∈ −2N
Note: ζP({s}k) not zero at roots of ζ(s) for k > 1
165
Partition zeta functions: analytic properties
Theorem (S.-Sills, 2020)For Re(s) > 1:
ζP({s}k) =∑λ`k
ζ(s)m1ζ(2s)m2ζ(3s)m3 · · · ζ(ks)mk
N(λ) m1! m2! m3! · · · mk !
sum over partitions λ of size k on RHSinherits continuation from ζ(s)poles at s = 1,1/2,1/3,1/4, ...,1/ktrivial roots at s ∈ −2N
Note: ζP({s}k) not zero at roots of ζ(s) for k > 1
166
Partition zeta functions: analytic properties
Theorem (S.-Sills, 2020)For Re(s) > 1:
ζP({s}k) =∑λ`k
ζ(s)m1ζ(2s)m2ζ(3s)m3 · · · ζ(ks)mk
N(λ) m1! m2! m3! · · · mk !
sum over partitions λ of size k on RHSinherits continuation from ζ(s)poles at s = 1,1/2,1/3,1/4, ...,1/ktrivial roots at s ∈ −2N
Note: ζP({s}k) not zero at roots of ζ(s) for k > 1
167
MacMahon-partition zeta correspondence
Proof mimics Sills’ combinatorial proof (2019) ofMacMahon’s partial fraction decomposition...
MacMahon’s partial fraction decompositionFor |q| < 1:
k∏n=1
(1− qn)−1 =∑λ`k
(1− q2)−m2 · · · qkmk (1− qk)−mk
N(λ) m1! m2! · · · mk !
Although here we really want to use...
168
MacMahon-partition zeta correspondence
Proof mimics Sills’ combinatorial proof (2019) ofMacMahon’s partial fraction decomposition...
MacMahon’s partial fraction decompositionFor |q| < 1:
k∏n=1
(1− qn)−1 =∑λ`k
(1− q2)−m2 · · · qkmk (1− qk)−mk
N(λ) m1! m2! · · · mk !
Although here we really want to use...
169
MacMahon-partition zeta correspondence
MacMahon’s partial fraction decomposition times qk
qkk∏
n=1
(1− qn)−1 =∑`(λ)=k
q|λ|
=∑λ`k
qm1(1− q)−m1 · q2m2(1− q2)−m2 · · · qkmk (1− qk)−mk
N(λ) m1! m2! · · · mk !
LHS generates all partitions with largest part kConjugation→ also gen. partitions w/ length kRHS = comb. geom. series over partitions of size k
170
MacMahon-partition zeta correspondence
MacMahon’s partial fraction decomposition times qk
qkk∏
n=1
(1− qn)−1 =∑`(λ)=k
q|λ|
=∑λ`k
qm1(1− q)−m1 · q2m2(1− q2)−m2 · · · qkmk (1− qk)−mk
N(λ) m1! m2! · · · mk !
LHS generates all partitions with largest part k
Conjugation→ also gen. partitions w/ length kRHS = comb. geom. series over partitions of size k
171
MacMahon-partition zeta correspondence
MacMahon’s partial fraction decomposition times qk
qkk∏
n=1
(1− qn)−1 =∑`(λ)=k
q|λ|
=∑λ`k
qm1(1− q)−m1 · q2m2(1− q2)−m2 · · · qkmk (1− qk)−mk
N(λ) m1! m2! · · · mk !
LHS generates all partitions with largest part kConjugation→ also gen. partitions w/ length k
RHS = comb. geom. series over partitions of size k
172
MacMahon-partition zeta correspondence
MacMahon’s partial fraction decomposition times qk
qkk∏
n=1
(1− qn)−1 =∑`(λ)=k
q|λ|
=∑λ`k
qm1(1− q)−m1 · q2m2(1− q2)−m2 · · · qkmk (1− qk)−mk
N(λ) m1! m2! · · · mk !
LHS generates all partitions with largest part kConjugation→ also gen. partitions w/ length kRHS = comb. geom. series over partitions of size k
173
MacMahon-partition zeta correspondence
MacMahon’s partial fraction decomposition times qk
qkk∏
n=1
(1− qn)−1 =∑`(λ)=k
q|λ|
=∑λ`k
qm1(1− q)−m1 · q2m2(1− q2)−m2 · · · qkmk (1− qk)−mk
N(λ) m1! m2! · · · mk !
Compare and contrast
ζP({s}k) =∑`(λ)=k
N(λ)−s =∑λ`k
ζ(s)m1ζ(2s)m2 · · · ζ(ks)mk
N(λ) m1! m2! · · · mk !
174
MacMahon-partition zeta correspondence
Gen fctn component Analogous zeta fctn componentq|λ| N(λ)−s
qj
1−qj ζ(js)qk∏k
j=1(1−qj )ζP({s}k)
Multiplication of terms of either shape qn or n−s generatespartitions in exactly the same way:
qλ1qλ2qλ3 · · · qλr = q|λ| ←→ λ−s1 λ−s
2 λ−s3 · · ·λ
−sr = N (λ)−s
175
MacMahon-partition zeta correspondence
Gen fctn component Analogous zeta fctn componentq|λ| N(λ)−s
qj
1−qj ζ(js)qk∏k
j=1(1−qj )ζP({s}k)
Multiplication of terms of either shape qn or n−s generatespartitions in exactly the same way:
qλ1qλ2qλ3 · · · qλr = q|λ| ←→ λ−s1 λ−s
2 λ−s3 · · ·λ
−sr = N (λ)−s
176
MacMahon-partition zeta correspondence
Gen fctn component Analogous zeta fctn componentq|λ| N(λ)−s
qj
1−qj ζ(js)qk∏k
j=1(1−qj )ζP({s}k)
The term q jn in geom series∑∞
n=1 q jn and resp. term n−js
of ζ(js) both encode partition (n)j := (n,n, ...,n) (j times):
q j
1− q j =∞∑
n=1
q|(n)j | ←→ ζ(js) =
∞∑n=1
N((n)j)−s
.
177
MacMahon-partition zeta correspondence
Geometric series-zeta function duality
Correspondence says qj
1−qj and ζ(js) are interchangeableas gen fctns for partitions (n,n, . . . ,n) (j reps / same part).
178
MacMahon-partition zeta correspondence
Geometric series-zeta function duality
Correspondence says qj
1−qj and ζ(js) are interchangeable
as gen fctns for partitions (n,n, . . . ,n) (j reps / same part).
179
MacMahon-partition zeta correspondence
Geometric series-zeta function duality
Correspondence says qj
1−qj and ζ(js) are interchangeableas gen fctns for partitions (n,n, . . . ,n) (j reps / same part).
180
Multiplicative theory of (additive) partitions
Applications
New combinatorial bijectionsComputing coefficients of q-series, mock mod. formsStatistical physicsComputational chemistryComputing arithmetic densitiesComputing π (quite inefficiently, to boot!)
181
Multiplicative theory of (additive) partitions
ApplicationsNew combinatorial bijectionsComputing coefficients of q-series, mock mod. formsStatistical physicsComputational chemistryComputing arithmetic densitiesComputing π
(quite inefficiently, to boot!)
182
Multiplicative theory of (additive) partitions
ApplicationsNew combinatorial bijectionsComputing coefficients of q-series, mock mod. formsStatistical physicsComputational chemistryComputing arithmetic densitiesComputing π (quite inefficiently, to boot!)
183
Application: arithmetic densities
DefinitionThe arithmetic density of a subset S ⊆ Z+ is
limN→∞
#{n ∈ S | n ≤ N}N
,
if the limit exists.
ExamplesIntegers ≡ r (mod t) have density 1/tSquare-free integers have density 6/π2 = 1/ζ(2)
184
Application: arithmetic densities
Definition
The arithmetic density of a subset S ⊆ Z+ is
limN→∞
#{n ∈ S | n ≤ N}N
,
if the limit exists.
ExamplesIntegers ≡ r (mod t) have density 1/tSquare-free integers have density 6/π2 = 1/ζ(2)
185
Application: arithmetic densities
DefinitionThe arithmetic density of a subset S ⊆ Z+ is
limN→∞
#{n ∈ S | n ≤ N}N
,
if the limit exists.
ExamplesIntegers ≡ r (mod t) have density 1/tSquare-free integers have density 6/π2 = 1/ζ(2)
186
Application: arithmetic densities
DefinitionThe arithmetic density of a subset S ⊆ Z+ is
limN→∞
#{n ∈ S | n ≤ N}N
,
if the limit exists.
Examples
Integers ≡ r (mod t) have density 1/tSquare-free integers have density 6/π2 = 1/ζ(2)
187
Application: arithmetic densities
DefinitionThe arithmetic density of a subset S ⊆ Z+ is
limN→∞
#{n ∈ S | n ≤ N}N
,
if the limit exists.
ExamplesIntegers ≡ r (mod t) have density 1/t
Square-free integers have density 6/π2 = 1/ζ(2)
188
Application: arithmetic densities
DefinitionThe arithmetic density of a subset S ⊆ Z+ is
limN→∞
#{n ∈ S | n ≤ N}N
,
if the limit exists.
ExamplesIntegers ≡ r (mod t) have density 1/tSquare-free integers have density 6/π2 = 1/ζ(2)
189
Application: arithmetic densities
Classical computation
Well-known relation between arithmetic density andzeta-type sumsIf a subset T ⊆ P has arith. density in P, its density isequal to the Dirichlet density of T
lims→1
∑p∈T p−s∑p∈P p−s
190
Application: arithmetic densities
Classical computationWell-known relation between arithmetic density andzeta-type sums
If a subset T ⊆ P has arith. density in P, its density isequal to the Dirichlet density of T
lims→1
∑p∈T p−s∑p∈P p−s
191
Application: arithmetic densities
Classical computationWell-known relation between arithmetic density andzeta-type sumsIf a subset T ⊆ P has arith. density in P, its density isequal to the Dirichlet density of T
lims→1
∑p∈T p−s∑p∈P p−s
192
Application: arithmetic densities
Theorem (Ono-S-Wagner, 2018)
The arith. density of integers ≡ r mod t is equal to
− limq→1
∑λ∈P
sm(λ)≡r(mod t)
µP(λ)q|λ| =1t.
“sm” is the smallest part of λextends work of Alladi (1977), Locus Dawsey (2017)
193
Application: arithmetic densities
Theorem (Ono-S-Wagner, 2018)The arith. density of integers ≡ r mod t is equal to
− limq→1
∑λ∈P
sm(λ)≡r(mod t)
µP(λ)q|λ| =1t.
“sm” is the smallest part of λextends work of Alladi (1977), Locus Dawsey (2017)
194
Application: arithmetic densities
Theorem (Ono-S-Wagner, 2018)The arith. density of integers ≡ r mod t is equal to
− limq→1
∑λ∈P
sm(λ)≡r(mod t)
µP(λ)q|λ| =1t.
“sm” is the smallest part of λ
extends work of Alladi (1977), Locus Dawsey (2017)
195
Application: arithmetic densities
Theorem (Ono-S-Wagner, 2018)The arith. density of integers ≡ r mod t is equal to
− limq→1
∑λ∈P
sm(λ)≡r(mod t)
µP(λ)q|λ| =1t.
“sm” is the smallest part of λextends work of Alladi (1977), Locus Dawsey (2017)
196
Application: arithmetic densities
Theorem (Ono-S-Wagner, 2018)
The arith. density of k th power-free integers is equal to
− limq→1
∑λ∈P
sm(λ) k th power-free
µP(λ)q|λ| =1
ζ(k).
Proofsq-binomial thm + partition bijection + complex analysis
197
Application: arithmetic densities
Theorem (Ono-S-Wagner, 2018)The arith. density of k th power-free integers is equal to
− limq→1
∑λ∈P
sm(λ) k th power-free
µP(λ)q|λ| =1
ζ(k).
Proofsq-binomial thm + partition bijection + complex analysis
198
Application: arithmetic densities
Theorem (Ono-S-Wagner, 2018)The arith. density of k th power-free integers is equal to
− limq→1
∑λ∈P
sm(λ) k th power-free
µP(λ)q|λ| =1
ζ(k).
Proofsq-binomial thm + partition bijection + complex analysis
199
Application: arithmetic densities
Theorem (Ono-S-Wagner, 2018)The arith. density of k th power-free integers is equal to
− limq→1
∑λ∈P
sm(λ) k th power-free
µP(λ)q|λ| =1
ζ(k).
another partition-zeta connection
200
Application: arithmetic densities
Theorem (Ono-S-Wagner, 2018)The arith. density of k th power-free integers is equal to
− limq→1
∑λ∈P
sm(λ) k th power-free
µP(λ)q|λ| =1
ζ(k).
another partition-zeta connection
201
Application: arithmetic densities
Theorem (Ono-S-Wagner, 2020)For dS arith. density of a q-commensurate subset S ⊆ N:
− limq→1
∑λ∈P
sm(λ)∈S
µP(λ)q|λ| = dS.
More generally, for a(λ) with certain analytic conditions:
− limq→1
∑sm(λ)∈S
(µP ∗ a)(λ)ϕ (sm(λ))
q|λ| = dS.
here ∗ is partition convolution, ϕ(n) classical phisecond formula extends work of Wang (2020)
202
Application: arithmetic densities
Theorem (Ono-S-Wagner, 2020)For dS arith. density of a q-commensurate subset S ⊆ N:
− limq→1
∑λ∈P
sm(λ)∈S
µP(λ)q|λ| = dS.
More generally, for a(λ) with certain analytic conditions:
− limq→1
∑sm(λ)∈S
(µP ∗ a)(λ)ϕ (sm(λ))
q|λ| = dS.
here ∗ is partition convolution, ϕ(n) classical phisecond formula extends work of Wang (2020)
203
Application: arithmetic densities
Theorem (Ono-S-Wagner, 2020)For dS arith. density of a q-commensurate subset S ⊆ N:
− limq→1
∑λ∈P
sm(λ)∈S
µP(λ)q|λ| = dS.
More generally, for a(λ) with certain analytic conditions:
− limq→1
∑sm(λ)∈S
(µP ∗ a)(λ)ϕ (sm(λ))
q|λ| = dS.
here ∗ is partition convolution, ϕ(n) classical phisecond formula extends work of Wang (2020)
204
Application: arithmetic densities
Theorem (Ono-S-Wagner, 2020)For dS arith. density of a q-commensurate subset S ⊆ N:
− limq→1
∑λ∈P
sm(λ)∈S
µP(λ)q|λ| = dS.
More generally, for a(λ) with certain analytic conditions:
− limq→1
∑sm(λ)∈S
(µP ∗ a)(λ)ϕ (sm(λ))
q|λ| = dS.
here ∗ is partition convolution, ϕ(n) classical phisecond formula extends work of Wang (2020)
205
Application: arithmetic densities
Theorem (Ono-S-Wagner, 2020)Proof requires a new theory of q-series densitycomputations based on “q-density” statistic.:
dS(q) :=
∑sm(λ)∈S q|λ|∑
λ q|λ|= (1− q)
∑sm(λ)∈S
q|λ|.
Natural number n Partition λPrime factors of n Parts of λ
Square-free integers Partitions into distinct partsµ(n) µP(λ)ϕ(n) ϕP(λ)
pmin(n) sm(λ)pmax(n) lg(λ)
n−s q|λ|
ζ(s)−1 (q;q)∞s → 1 q → 1
206
Partition-theoretic multiverse
Philosophy of this talk (again)Exist multipl., division, arith. functions on partitions
Objects in classical multiplic. number theory→ special cases of partition-theoretic structures
Expect arithmetic theorems→ extend to partitions
Expect partition properties→ properties of integers
Work in progressWith Akande, Beckwith, Dawsey, Hendon, Jameson, Just,Ono, Rolen, M. Schneider, Sellers, Sills, Wagner, ... you?
207
Partition-theoretic multiverse
Philosophy of this talk (again)Exist multipl., division, arith. functions on partitions
Objects in classical multiplic. number theory→ special cases of partition-theoretic structures
Expect arithmetic theorems→ extend to partitions
Expect partition properties→ properties of integers
Work in progressWith Akande, Beckwith, Dawsey, Hendon, Jameson, Just,Ono, Rolen, M. Schneider, Sellers, Sills, Wagner, ... you?
208
Partition-theoretic multiverse
Philosophy of this talk (again)Exist multipl., division, arith. functions on partitions
Objects in classical multiplic. number theory→ special cases of partition-theoretic structures
Expect arithmetic theorems→ extend to partitions
Expect partition properties→ properties of integers
Work in progressWith Akande, Beckwith, Dawsey, Hendon, Jameson, Just,Ono, Rolen, M. Schneider, Sellers, Sills, Wagner, ...
you?
209
Partition-theoretic multiverse
Philosophy of this talk (again)Exist multipl., division, arith. functions on partitions
Objects in classical multiplic. number theory→ special cases of partition-theoretic structures
Expect arithmetic theorems→ extend to partitions
Expect partition properties→ properties of integers
Work in progressWith Akande, Beckwith, Dawsey, Hendon, Jameson, Just,Ono, Rolen, M. Schneider, Sellers, Sills, Wagner, ... you?
210
Gratitude
Thank you for listening :)
211
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