# New Results on Partitions, Prime Numbers, and Moonshine Open Access

Permanent URL: https://etd.library.emory.edu/concern/etds/nv935404h?locale=en
Published

## Abstract

In this thesis, we prove new results in combinatorics, analytic number theory, and representation theory. In particular, in combinatorics we prove conjectured inequalities regarding the Andrews smallest parts partition function by first establishing effective estimates using new methods from the theory of quadratic forms. In addition, we provide recurrence relations for the coefficients of conjugacy growth series for wreath products of finitary permutation groups, which essentially measure the algebraic complexity of these groups. In analytic number theory, we apply the Chebotarev Density Theorem in order to generalize a theorem of Alladi on the distribution of primes in arithmetic progressions. More precisely, we reproduce the Chebotarev densities of certain subsets of prime numbers through an infinite sum involving the Möbius function, where we sum over only those integers whose smallest prime divisors fall in the specified subsets. Finally, we refine the theory of moonshine so that the modular forms associated to the representation theory of all finite groups uniquely determine those groups up to isomorphism. We obtain this "higher width moonshine" for all finite groups by employing the classical Frobenius r-characters, which we prove satisfy orthogonality relations analogous to Schur's orthogonality relations for ordinary group characters.

1 Introduction 1

1.1 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 The Andrews smallest parts partition function . . . . . . . . . 3

1.1.2 Finitary permutation groups . . . . . . . . . . . . . . . . . . . 6

1.2 Prime numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Moonshine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Modular Forms and Harmonic Maass Forms 18

2.1 Modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Harmonic Maass forms . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Partitions 27

3.1 The theory of quadratic forms . . . . . . . . . . . . . . . . . . . . . . 27

3.2 The smallest parts partition function . . . . . . . . . . . . . . . . . . 29

3.2.1 Eective estimates for spt(n) . . . . . . . . . . . . . . . . . . 32

3.2.2 Inequalities satised by spt(n) . . . . . . . . . . . . . . . . . . 51

3.3 Conjugacy growth series for nitary permutation groups . . . . . . . 65

3.3.1 Recurrence relations . . . . . . . . . . . . . . . . . . . . . . . 65

3.3.2 An application to hook lengths . . . . . . . . . . . . . . . . . 68

4 Prime Numbers 70

4.1 Some algebraic number theory . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Chebotarev densities of subsets of primes . . . . . . . . . . . . . . . . 73

4.2.1 Intermediate estimates . . . . . . . . . . . . . . . . . . . . . . 73

4.2.2 Densities of subsets of smallest prime divisors . . . . . . . . . 80

4.3 Generalization to arbitrary number eld extensions . . . . . . . . . . 83

5 Moonshine 85

5.1 Classical representation theory . . . . . . . . . . . . . . . . . . . . . . 86

5.2 Orthogonality of r-characters . . . . . . . . . . . . . . . . . . . . . . 90

5.3 Higher width moonshine . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.4 An example of higher width moonshine . . . . . . . . . . . . . . . . . 96

Bibliography 99