Multiplicative theory of (additive) partitions...Multiplicative number theory I.e., most of classical number theory primes divisors Euler phi function ’(n), Möbius function (n)

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


Patterns and interconnections

theory of partitions





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4







5

Birth of partition theory

Ishango bone (Africa, ca. 20,000 B.C.E.)

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7


Ahmes papyrus (Egypt, ca. 2,000 B.C.E.)

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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


G. W. Leibniz (1600s)

12


G. W. Leibniz (1600s)

13


Leibnizwondered about size of p(n) := # of partitions of n

p(n) is called the partition function

14


Leibnizwondered about size of p(n) := # of partitions of n

p(n) is called the partition function

15


Leonhard Euler (1700s)

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Leonhard Euler (1700s)

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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


Partition generating function (Euler)∞∑

n=0

p(n)qn

=∞∏

n=1

(1− qn)−1 (q ∈ C, |q| < 1)


19



n=0

p(n)qn =∞∏

n=1

(1− qn)−1 (q ∈ C, |q| < 1)


20



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



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



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



n=0

p(n)qn =∞∏

n=1

(1− qn)−1 (q ∈ C, |q| < 1)


24



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

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n=0

p(n)qn =∞∏

n=1

(1− qn)−1 (q ∈ C, |q| < 1)

ProofRHS

=∏∞

n=1(1 + qn + q2n + q3n + ...) (geom. series)=∏∞


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n=0

p(n)qn =∞∏

n=1

(1− qn)−1 (q ∈ C, |q| < 1)

ProofRHS =

∏∞n=1(1 + qn + q2n + q3n + ...)

(geom. series)=∏∞


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n=0

p(n)qn =∞∏

n=1

(1− qn)−1 (q ∈ C, |q| < 1)

ProofRHS =

∏∞n=1(1 + qn + q2n + q3n + ...) (geom. series)

=∏∞


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n=0

p(n)qn =∞∏

n=1

(1− qn)−1 (q ∈ C, |q| < 1)

ProofRHS =


=∏∞

n=1(1 + qn + qn+n + qn+n+n + ...)

(like, second grade)= 1 + q1 + q1+1 + q2 + q1+1+1 + q1+2 + q3 + ...= LHS

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n=0

p(n)qn =∞∏

n=1

(1− qn)−1 (q ∈ C, |q| < 1)

ProofRHS =


=∏∞

n=1(1 + qn + qn+n + qn+n+n + ...) (like, second grade)

= 1 + q1 + q1+1 + q2 + q1+1+1 + q1+2 + q3 + ...= LHS

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n=0

p(n)qn =∞∏

n=1

(1− qn)−1 (q ∈ C, |q| < 1)

ProofRHS =


=∏∞

n=1(1 + qn + qn+n + qn+n+n + ...) (like, second grade)= 1 + q1 + q1+1 + q2 + q1+1+1 + q1+2 + q3 + ...

= LHS

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n=0

p(n)qn =∞∏

n=1

(1− qn)−1 (q ∈ C, |q| < 1)

ProofRHS =


=∏∞


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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

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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


I.e., most of classical number theoryprimesdivisorsEuler phi function ϕ(n), Möbius function µ(n)arithmetic functions, Dirichlet convolutionzeta functions, Dirichlet series, L-functions

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Birth of multiplicative number theory


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Eratosthenes, Euclid (Alexandria, ca. 300 B.C.E.)

38


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


Eulerfirst to understand (Riemann) zeta function

:

ζ(s) :=∞∑

n=1

1ns (Re(s) > 1)


ζ(2) =π2

6, ζ(4) =

π4

90, ζ(6) =

π6

945, ...

40


Eulerfirst to understand (Riemann) zeta function:

ζ(s) :=∞∑

n=1

1ns (Re(s) > 1)


ζ(2) =π2

6, ζ(4) =

π4

90, ζ(6) =

π6

945, ...

41



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



ζ(s) :=∞∑

n=1

1ns (Re(s) > 1)


ζ(2) =π2

6, ζ(4) =

π4

90, ζ(6) =

π6

945, ...

43


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


Product formula for zeta function (Euler)∞∑

n=0

1ns

=∏p∈P

(1− p−s)−1

(Re(s) > 1)


45



n=0

1ns =

∏p∈P

(1− p−s)−1

(Re(s) > 1)


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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

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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



n=0

1ns =

∏p∈P

(1− p−s)−1

(Re(s) > 1)


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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



n=1

(1− qn)−1 =∞∑

n=0

p(n)qn, |q| < 1


(1− p−s)−1 =∞∑

k=1

n−s, Re(s) > 1


51



n=1

(1− qn)−1 =∞∑

n=0

p(n)qn, |q| < 1


(1− p−s)−1 =∞∑

k=1

n−s, Re(s) > 1


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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?

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Alladi–Erdos (1970s)

Bijection between integer factorizations, prime partitionsstudy properties of arithmetic functions


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Question

Are other thms. in arithmetic images in prime partitions ofcombinatorial/set-theoretic meta-structures?

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QuestionAre other thms. in arithmetic images in prime partitions

ofcombinatorial/set-theoretic meta-structures?

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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


Andrews (1970s)Theory of partition ideals

inspired by lattice theoryunifies classical results on gen. functions, bijectionssuggestive of a universal algebra of partitions


61



inspired by lattice theory

unifies classical results on gen. functions, bijectionssuggestive of a universal algebra of partitions


62



inspired by lattice theoryunifies classical results on gen. functions, bijections

suggestive of a universal algebra of partitions


63





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Question

Is there an algebra of partitions generalizing arithmetic inintegers (i.e., prime partitions)?

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QuestionIs there an algebra of partitions

generalizing arithmetic inintegers (i.e., prime partitions)?

66




QuestionIs there an algebra of partitions generalizing arithmetic inintegers

(i.e., prime partitions)?

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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

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Objects in classical multiplic. number theory

→ special cases of partition-theoretic structures



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Expect arithmetic theorems

→ extend to partitions


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Expect partition properties

→ properties of integers

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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”.

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Partition notations





78

Partition notations





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Partition notations



Let λ = (λ1, λ2, . . . , λr ), λ1 ≥ λ2 ≥ · · · ≥ λr ≥ 1,denote a nonempty partition

, e.g. λ = (3,2,2,1).


80

Partition notations





81

Partition notations




Let PX denote partitions into elements λi ∈ X ⊆ N

,e.g. PP is the “prime partitions”.

82

Partition notations





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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





Define `(∅) = |∅| = mi(∅) = 0, ∅ ` 0.

85

Partition notations





Define `(∅) = |∅| = mi(∅) = 0, ∅ ` 0.

86

Partition notations





Define `(∅) = |∅| = mi(∅) = 0, ∅ ` 0.

87

Partition notations





Define `(∅) = |∅| = mi(∅) = 0, ∅ ` 0.

88

Partition notations





Define `(∅) = |∅| = mi(∅) = 0, ∅ ` 0.

89


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)

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New partition statistic

Define N(λ), the norm of λ, to be the product of the parts:

N(λ) := λ1λ2λ3 · · ·λr


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N(λ) := λ1λ2λ3 · · ·λr


92



N(λ) := λ1λ2λ3 · · ·λr


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N(λ) := λ1λ2λ3 · · ·λr

Define N(∅) := 1 (it is an empty product)

See “The product of parts or ‘norm’ etc.” (S-Sills)

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N(λ) := λ1λ2λ3 · · ·λr


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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



For λ, γ ∈ P let λγ denote multiset union of the parts

,e.g. (3,2)(2,1) = (3,2,2,1).

Identity is ∅


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Identity is ∅


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Identity is ∅


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Identity is ∅


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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



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).



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103




For δ|λ, let λ/δ mean parts of δ deleted from λ

,e.g. (3,2,2,1)/(3,2,1) = (2).


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105





Replace P with PP

→ mult./div. in Z+

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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



µP(λ) :=



109

Parallel universe


Partition Möbius function

For λ ∈ P, define

µP(λ) :=



110

Parallel universe



µP(λ) :=



111

Parallel universe



µP(λ) :=



112

Parallel universe



µP(λ) :=



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



µP(δ) =


115

Parallel universe


Partition Möbius function

∑δ|λ

µP(δ) =


116

Parallel universe



µP(δ) =


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).

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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).

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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


Partition phi function

For λ ∈ P, define


no repeats

(1− λ−1i ).


121

Parallel universe




no repeats

(1− λ−1i ).


122

Parallel universe




no repeats

(1− λ−1i ).


123

Parallel universe




no repeats

(1− λ−1i ).


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


∑δ|λ


µP(δ)

N(δ)


126

Parallel universe


∑δ|λ


µP(δ)

N(δ)

Replacing P with PP reduces to classical cases.

Many analogs of objects in multiplic. # theory...

127

Parallel universe


∑δ|λ


µP(δ)

N(δ)


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...

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Parallel universe


f (λ)q|λ|)(∑

λ∈P

g(λ)q|λ|)

=∑λ∈P

q|λ|∑δ|λ

f (δ)g(λ/δ)


f (λ)∑δ|λ

g(δ) =∑λ∈P

g(λ)∑γ∈P

f (λγ)


130

Parallel universe


f (λ)q|λ|)(∑

λ∈P

g(λ)q|λ|)

=∑λ∈P

q|λ|∑δ|λ

f (δ)g(λ/δ)


f (λ)∑δ|λ

g(δ) =∑λ∈P

g(λ)∑γ∈P

f (λγ)


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

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n=1 n−s:


ζP ′(s) :=∑λ∈P ′

N(λ)−s


ζPX(s) =∏n∈X

(1− n−s)−1

133



n=1 n−s:

DefinitionFor P ′ ( P , s ∈ C, define a partition zeta function

:

ζP ′(s) :=∑λ∈P ′

N(λ)−s


ζPX(s) =∏n∈X

(1− n−s)−1

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n=1 n−s:


ζP ′(s) :=∑λ∈P ′

N(λ)−s


ζPX(s) =∏n∈X

(1− n−s)−1

135



n=1 n−s:


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



n=1 n−s:


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



n=1 n−s:


ζP ′(s) :=∑λ∈P ′

N(λ)−s


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


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



µP(λ)N(λ)−s =1

ζPX(s)∑λ∈PX


Takeaway from these examples

Different subsets of P induce very diff. zeta valuesClassical zeta theorems→ partition zeta theorems

142



µP(λ)N(λ)−s =1

ζPX(s)∑λ∈PX


Takeaway from these examplesDifferent subsets of P induce very diff. zeta values

Classical zeta theorems→ partition zeta theorems

143



µP(λ)N(λ)−s =1

ζPX(s)∑λ∈PX


Takeaway from these examplesDifferent subsets of P induce very diff. zeta valuesClassical zeta theorems→ partition zeta theorems

144


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


Pretty cool, but it would be a whole lot cooler if we hadfamilies of identities like Euler’s zeta values:




146




Question

Do there exist (non-Riemann) partition zeta functionssuch that, for the “right” choice of P ′ ( P,


147




QuestionDo there exist (non-Riemann) partition zeta functionssuch that, for the “right” choice of P ′ ( P

,


148






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


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


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


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


Theorem (S, 2016)Summing over partitions of fixed length k > 0:

ζP({2}k) =22k−1 − 1





154



ζP({2}k) =22k−1 − 1





155



ζP({2}k) =22k−1 − 1





156



ζP({2}k) =22k−1 − 1





157


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



Natural questions


159



Natural questions


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


Theorem (S.-Sills, 2020)For Re(s) > 1:

ζP({s}k) =∑λ`k


N(λ) m1! m2! m3! · · · mk !



162



ζP({s}k) =∑λ`k


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


163



ζP({s}k) =∑λ`k


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


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ζP({s}k) =∑λ`k


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


165



ζP({s}k) =∑λ`k


N(λ) m1! m2! m3! · · · mk !



166



ζP({s}k) =∑λ`k


N(λ) m1! m2! m3! · · · mk !



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


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’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



qkk∏

n=1

(1− qn)−1 =∑`(λ)=k

q|λ|

=∑λ`k


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



qkk∏

n=1

(1− qn)−1 =∑`(λ)=k

q|λ|

=∑λ`k


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



qkk∏

n=1

(1− qn)−1 =∑`(λ)=k

q|λ|

=∑λ`k


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



qkk∏

n=1

(1− qn)−1 =∑`(λ)=k

q|λ|

=∑λ`k


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


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

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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

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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

.

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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




1−qj and ζ(js) are interchangeable

as gen fctns for partitions (n,n, . . . ,n) (j reps / same part).

179




1−qj and ζ(js) are interchangeableas gen fctns for partitions (n,n, . . . ,n) (j reps / same part).

180


Applications

New combinatorial bijectionsComputing coefficients of q-series, mock mod. formsStatistical physicsComputational chemistryComputing arithmetic densitiesComputing π (quite inefficiently, to boot!)

181


ApplicationsNew combinatorial bijectionsComputing coefficients of q-series, mock mod. formsStatistical physicsComputational chemistryComputing arithmetic densitiesComputing π

(quite inefficiently, to boot!)

182


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)

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Definition

The arithmetic density of a subset S ⊆ Z+ is

limN→∞

#{n ∈ S | n ≤ N}N

,



185



limN→∞

#{n ∈ S | n ≤ N}N

,



186



limN→∞

#{n ∈ S | n ≤ N}N

,


Examples

Integers ≡ r (mod t) have density 1/tSquare-free integers have density 6/π2 = 1/ζ(2)

187



limN→∞

#{n ∈ S | n ≤ N}N

,


ExamplesIntegers ≡ r (mod t) have density 1/t

Square-free integers have density 6/π2 = 1/ζ(2)

188



limN→∞

#{n ∈ S | n ≤ N}N

,



189


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


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


191


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


192


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


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.


194



− 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



− limq→1

∑λ∈P

sm(λ)≡r(mod t)

µP(λ)q|λ| =1t.


196


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


Theorem (Ono-S-Wagner, 2018)The arith. density of k th power-free integers is equal to

− limq→1

∑λ∈P


µP(λ)q|λ| =1

ζ(k).


198



− limq→1

∑λ∈P


µP(λ)q|λ| =1

ζ(k).


199



− limq→1

∑λ∈P


µP(λ)q|λ| =1

ζ(k).

another partition-zeta connection

200



− limq→1

∑λ∈P


µP(λ)q|λ| =1

ζ(k).

another partition-zeta connection

201


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



− limq→1

∑λ∈P

sm(λ)∈S

µP(λ)q|λ| = dS.


− limq→1

∑sm(λ)∈S


q|λ| = dS.


203



− limq→1

∑λ∈P

sm(λ)∈S

µP(λ)q|λ| = dS.


− limq→1

∑sm(λ)∈S


q|λ| = dS.


204



− limq→1

∑λ∈P

sm(λ)∈S

µP(λ)q|λ| = dS.


− limq→1

∑sm(λ)∈S


q|λ| = dS.


205


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

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Partition-theoretic multiverse

Philosophy of this talk (again)Exist multipl., division, arith. functions on partitions




Work in progressWith Akande, Beckwith, Dawsey, Hendon, Jameson, Just,Ono, Rolen, M. Schneider, Sellers, Sills, Wagner, ... you?

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Work in progressWith Akande, Beckwith, Dawsey, Hendon, Jameson, Just,Ono, Rolen, M. Schneider, Sellers, Sills, Wagner, ...

you?

209







210

Gratitude

Thank you for listening :)

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Documents

Multiplicative theory of (additive) partitions...Multiplicative number theory I.e., most of classical number theory primes divisors Euler phi function ’(n), Möbius function (n)