Eulerian series, zeta functions and the arithmetic of partitions Open Access

Schneider, Robert (Spring 2018)

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In this dissertation we prove theorems at the intersection of the additive and multiplicative branches of number theory, bringing together ideas from partition theory, q-series, algebra, modular forms and analytic number theory. We present a natural multiplicative theory of integer partitions (which are usually considered in terms of addition), and explore new classes of partition-theoretic zeta functions and Dirichlet series --- as well as "Eulerian" q-hypergeometric series --- enjoying many interesting relations. We find a number of theorems of classical number theory and analysis arise as particular cases of extremely general combinatorial structure laws.  

Among our applications, we prove explicit formulas for the coefficients of the q-bracket of Bloch-Okounkov, a partition-theoretic operator from statistical physics related to quasi-modular forms; we prove partition formulas for arithmetic densities of certain subsets of the integers, giving q-series formulas to evaluate the Riemann zeta function; we study q-hypergeometric series related to quantum modular forms and the "strange" function of Kontsevich; and we show how Ramanujan's odd-order mock theta functions (and, more generally, the universal mock theta function of Gordon-McIntosh) arise from the reciprocal of the Jacobi triple product via the q-bracket operator, connecting also to unimodal sequences in combinatorics and quantum modular-like phenomena.

Table of Contents

Chapter 1. Setting the stage: Introduction, background and summary of results

1.1 Visions of Euler and Ramanujan

1.1.1 Zeta functions, partitions and q-series

1.1.2 Mock theta functions and quantum modular forms

1.1.3 Glimpses of an arithmetic of partitions

1.2 The present work

1.2.1 Intersections of additive and multiplicative number theory

1.2.2 Partition zeta functions

1.2.3 Partition formulas for arithmetic densities     

1.2.4 "Strange" functions, quantum modularity, mock theta functions and unimodal sequences


Chapter 2. Combinatorial applications of Moebius inversion

2.1 Introduction and Statement of Results

2.2 Proof of Theorem 2.1.1

2.3 Proof of Theorems 2.1.2 and 2.1.3


Chapter 3. Multiplicative arithmetic of partitions and the q-bracket

3.1 Introduction: the q-bracket operator

3.2 Multiplicative arithmetic of partitions

3.3 Partition-theoretic analogs of classical functions

3.4 Role of the q-bracket

3.5 The q-antibracket and coefficients of power series over integers

3.6 Applications of the q-bracket and q-antibracket

3.6.1 Sum of divisors function

3.6.2 Reciprocal of the Jacobi triple product


Chapter 4. Partition-theoretic zeta functions

4.1 Introduction, notations and central theorem

4.2 Partition-theoretic zeta functions

4.3 Proofs of theorems and corollaries


Chapter 5. Partition zeta functions: further explorations

5.1 Following up on the previous chapter

5.2 Evaluations

5.2.1 Zeta functions for partitions with parts restricted by congruence conditions

5.2.3 Connections to ordinary Riemann zeta values

5.2.6 Zeta functions for partitions of fixed length

5.3 Analytic continuation and p-adic continuity

5.4 Connections to multiple zeta values

5.5 Proofs

5.5.1 Machinery

5.5.3 Proofs of Theorems 5.2.2 and 5.2.4, and their corollaries

5.5.4 Proof of Theorem 5.2.7 and its corollaries

5.5.5 Proofs of results concerning multiple zeta values

5.6 Partition Dirichlet series


Chapter 6. Partition-theoretic formulas for arithmetic densities

6.1 Introduction and statement of results

6.2 The q-Binomial Theorem and its consequences

6.2.1 Nuts and bolts

6.2.2 Case of integers r modulo t

6.3 Proofs of these results

6.3.1 Proof of Theorem 6.1.1

6.3.2 Proof of Theorem 6.1.3

6.3.3 Proofs of Theorem 6.14 and Corollary 6.12

6.4 Examples


Chapter 7. "Strange" functions and a vector-valued quantum modular form

7.1 Introduction and Statement of Results

7.2 Preliminaries

7.2.1 Sums of Tails Identities

7.3 Properties of Eichler Integrals

7.4 Proof of Theorem 7.1.1

7.4.1 Proof of Theorem 7.1.1 (1)

7.4.2 Proof of Theorem 7.1.1 (2)

7.5 Proof of Corollary 7.1.1


Chapter 8. Jacobi's triple product, mock theta functions, unimodal sequences and the q-bracket

8.1 Introduction

8.2 Connecting the triple product to mock theta functions via partitions and unimodal sequences

8.3 Approaching roots of unity radially from within (and without)

8.4 The "feel" of quantum theory


Appendix A. Notes on Chapter 1: Counting partitions

A.1 Elementary considerations

A.2 Easy formula for p(n)


Appendix B. Notes on Chapter 3: Applications and algebraic considerations

B.1 Ramanujan's tau function and k-color partitions

B.2 q-bracket arithmetic

B.3 Group theory and ring theory in the partitions

B.3.1 Antipartitions and group theory

B.3.3 Partitions and diagonal matrices

B.3.4 Partition tensor product and ring theory

B.3.6 Ring theory over rational partitions 


Appendix C. Notes on Chapter 4: Further observations

C.1 Sequentially congruent partitions


Appendix D. Notes on Chapter 5: Faa di Bruno's formula in partition theory

D.1 Faa di Bruno's formula with product version

D.2 Further examples


Appendix E. Notes on Chapter 6: Further relations

E.1 Classical series and arithmetic functions


Appendix F. Notes on Chapter 7: Alternating "strange" functions

F.1 Further "strange" connections to quantum and mock modular forms

F.2 Proofs of results


Appendix G. Notes on Chapter 8: Results from a computational study of f(q)

G.1 Cyclotomic-type structures at certain roots of unity




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