Eulerian series, zeta functions and the arithmetic of partitions By Robert Schneider B. S., University of Kentucky, 2012 M. Sc., Emory University, 2016 Advisor: Ken Ono, Ph.D. A dissertation submitted to the Faculty of the James T. Laney School of Graduate Studies of Emory University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics 2018 Abstract Eulerian series, zeta functions and the arithmetic of partitions By Robert Schneider Inthisdissertationweprovetheoremsattheintersectionoftheadditiveandmultiplicative branchesofnumbertheory, bringingtogetherideasfrompartitiontheory, 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 ofBloch-Okounkov, apartition-theoreticoperatorfromstatisticalphysicsrelatedtoquasi- 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 g of Gordon-McIntosh) arise from 3 the reciprocal of the Jacobi triple product via the q-bracket operator, connecting also to unimodal sequences in combinatorics and quantum modular-like phenomena. “Partitions constitute the sphere in which analysis lives, moves, and has its being... [this] almost neglected (but vast, subtle and universally permeating) element of algebraic thought and expression.” — J. J. Sylvester1 1ThankstoGeorgeAndrewsandJimSmoakforprovidingthisquotation, afootnotein[Syl08], p. 93. For Marci and Max Acknowledgments I am grateful to the following people for many inspiring conversations, not to mention lessons and guidance, which influenced this work: my Ph.D. advisor, Ken Ono, and other dissertation committee members David Borthwick and John Duncan; my collaborators and co-authors Larry Rolen, Marie Jameson, Olivia Beckwith, Ian Wagner, Andrew Sills, Amanda Clemm, James Kindt, Lara Patel and Xiaokun Zhang; Professors George An- drews, KrishnaswamiAlladi, AndrewGranville, DavidLeep, JoeGallian, PennyDunham, William Dunham, Neil Calkin, Colm Mulcahy, Amanda Folsom, Raman Parimala, Suresh Vennapally, Ron Gould, Bree Ettinger, David Zureick-Brown, Dave Goldsman and Shan- shuang Yang; and my colleagues Robert Lemke Oliver, Michael Griffin, Jesse Thorner, Ben Phelan, John Ferguson, Joel Riggs, Maryam Khaqan, Cyrus Hettle, Sarah Trebat- Leder, Lea Beneish, Madeline Locus Dawsey, Victor Manuel Aricheta, Warren Shull, Bill Kay, Anastassia Etropolski, Adele Dewey-Lopez, Alex Rice and Jackson Morrow (whom I also thank for typesetting Chapter 4). I am also grateful to Prof. Vaidy Sunderam, Terry Ingram and Erin Nagle in Emory’s Department of Mathematics and Computer Science; and to Emory’s Laney Graduate School for electing me for the Woodruff Fellowship and Dean’s Teaching Fellowship — in particular, to Dean Lisa Tedesco and Dr. Jay Hughes, and to Prof. Elizabeth Bounds and Rachelle Green in the Candler School of Theology. I am deeply thankful to my wife Marci Schneider and son Maxwell Schneider — to whom I dedicate this work — and my parents and siblings, for their confidence in me and support during my graduate school journey; as well as to Marci for compiling and typesetting the bibliography for this dissertation, and to Max (a talented mathematician and programmer) for a lifetime of discussions, and for checking my ideas on computer. Robert Schneider April 3, 2018 Contents 1 Setting the stage: Introduction, background and summary of results 1 1.1 Visions of Euler and Ramanujan . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Zeta functions, partitions and q-series . . . . . . . . . . . . . . . . . 1 1.1.2 Mock theta functions and quantum modular forms . . . . . . . . . 5 1.1.3 Glimpses of an arithmetic of partitions . . . . . . . . . . . . . . . . 8 1.2 The present work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.1 Intersections of additive and multiplicative number theory . . . . . 10 1.2.2 Partition zeta functions . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.3 Partition formulas for arithmetic densities . . . . . . . . . . . . . . 18 1.2.4 “Strange” functions, quantummodularity, mockthetafunctionsand unimodal sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Combinatorial applications of Möbius inversion 23 2.1 Introduction and Statement of Results . . . . . . . . . . . . . . . . . . . . 23 2.2 Proof of Theorem 2.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3 Proof of Theorems 2.1.2 and 2.1.3 . . . . . . . . . . . . . . . . . . . . . . . 31 3 Multiplicative arithmetic of partitions and the q-bracket 34 3.1 Introduction: the q-bracket operator . . . . . . . . . . . . . . . . . . . . . 34 3.2 Multiplicative arithmetic of partitions . . . . . . . . . . . . . . . . . . . . . 37 3.3 Partition-theoretic analogs of classical functions . . . . . . . . . . . . . . . 39 3.4 Role of the q-bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.5 The q-antibracket and coefficients of power series over Z . . . . . . . . . 49 ≥0 3.6 Applications of the q-bracket and q-antibracket . . . . . . . . . . . . . . . 52 3.6.1 Sum of divisors function . . . . . . . . . . . . . . . . . . . . . . . . 52 3.6.2 Reciprocal of the Jacobi triple product . . . . . . . . . . . . . . . . 53 4 Partition-theoretic zeta functions 56 4.1 Introduction, notations and central theorem . . . . . . . . . . . . . . . . . 56 4.2 Partition-theoretic zeta functions . . . . . . . . . . . . . . . . . . . . . . . 59 4.3 Proofs of theorems and corollaries . . . . . . . . . . . . . . . . . . . . . . . 69 5 Partition zeta functions: further explorations 81 5.1 Following up on the previous chapter . . . . . . . . . . . . . . . . . . . . . 81 5.2 Evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.2.1 Zeta functions for partitions with parts restricted by congruence conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.2.3 Connections to ordinary Riemann zeta values . . . . . . . . . . . . 83 5.2.6 Zeta functions for partitions of fixed length . . . . . . . . . . . . . . 85 5.3 Analytic continuation and p-adic continuity . . . . . . . . . . . . . . . . . 86 5.4 Connections to multiple zeta values . . . . . . . . . . . . . . . . . . . . . . 88 5.5 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.5.1 Machinery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.5.3 Proofs of Theorems 5.2.2 and 5.2.4, and their corollaries . . . . . . 93 5.5.4 Proof of Theorem 5.2.7 and its corollaries . . . . . . . . . . . . . . . 96 5.5.5 Proofs of results concerning multiple zeta values . . . . . . . . . . . 99 5.6 Partition Dirichlet series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6 Partition-theoretic formulas for arithmetic densities 103 6.1 Introduction and statement of results . . . . . . . . . . . . . . . . . . . . . 103 6.2 The q-Binomial Theorem and its consequences . . . . . . . . . . . . . . . . 110 6.2.1 Nuts and bolts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.2.2 Case of F (q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 S r,t 6.3 Proofs of these results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.3.1 Proof of Theorem 6.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.3.2 Proof of Theorem 6.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.3.3 Proofs of Theorem 6.1.4 and Corollary 6.1.2 . . . . . . . . . . . . . 116 6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 7 “Strange” functions and a vector-valued quantum modular form 120 7.1 Introduction and Statement of Results . . . . . . . . . . . . . . . . . . . . 120 7.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.2.1 Sums of Tails Identities . . . . . . . . . . . . . . . . . . . . . . . . . 125 7.3 Properties of Eichler Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 126 7.4 Proof of Theorem 7.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.4.1 Proof of Theorem 7.1.1 (1) . . . . . . . . . . . . . . . . . . . . . . . 129 7.4.2 Proof of Theorem 7.1.1 (2) . . . . . . . . . . . . . . . . . . . . . . . 130 7.5 Proof of Corollary 7.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 8 Jacobi’s triple product, mock theta functions, unimodal sequences and the q-bracket 133 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 8.2 Connecting the triple product to mock theta functions via partitions and unimodal sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 8.3 Approaching roots of unity radially from within (and without) . . . . . . . 144 8.4 The “feel” of quantum theory . . . . . . . . . . . . . . . . . . . . . . . . . 154 A Notes on Chapter 1: Counting partitions 156 A.1 Elementary considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 A.2 Easy formula for p(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 B Notes on Chapter 3: Applications and algebraic considerations 162 B.1 Ramanujan’s tau function and k-color partitions . . . . . . . . . . . . . . . 162 B.2 q-bracket arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 B.3 Group theory and ring theory in P . . . . . . . . . . . . . . . . . . . . . . 166 B.3.1 Antipartitions and group theory . . . . . . . . . . . . . . . . . . . . 166 B.3.3 Partitions and diagonal matrices . . . . . . . . . . . . . . . . . . . . 168 B.3.4 Partition tensor product and ring theory . . . . . . . . . . . . . . . 171 B.3.6 Ring theory in Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 C Notes on Chapter 4: Further observations 181 C.1 Sequentially congruent partitions . . . . . . . . . . . . . . . . . . . . . . . 181 D Notes on Chapter 5: Faà di Bruno’s formula in partition theory 189 D.1 Faà di Bruno’s formula with product version . . . . . . . . . . . . . . . . . 189 D.2 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 E Notes on Chapter 6: Further relations involving F 196 S r,t E.1 Classical series and arithmetic functions . . . . . . . . . . . . . . . . . . . 196 F Notes on Chapter 7: Alternating “strange” functions 199 F.1 Further “strange” connections to quantum and mock modular forms . . . . 199 F.2 Proofs of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 G Notes on Chapter 8: Results from a computational study of f(q) 206 G.1 Cyclotomic-type structures at certain roots of unity . . . . . . . . . . . . . 206 Bibliography 210 1 Chapter 1 Setting the stage: Introduction, background and summary of results 1.1 Visions of Euler and Ramanujan In antiquity, storytellers began their narratives by invoking the Muses, whose influence would guide the unfolding imagery. It is fitting, then, that we begin this work by praising Euler and Ramanujan, whose imaginations ranged playfully across much of the landscape of modern mathematical thought. 1.1.1 Zeta functions, partitions and q-series One marvels at the degree to which our contemporary understanding of q-series, integer partitions, and what is now known as the Riemann zeta function all emerged nearly fully-formed from Euler’s pioneering work [And98,Dun99]. Euler made spectacular use of product-sum relations, often arrived at by unexpected avenues, thereby inventing one of the principle archetypes of modern number theory. Among his many profound identities is the product formula for ζ(s), the Riemann zeta