# Research: Number Theory

## SERMON 2023

In memory of Kevin James. The PAlmetto Area Number Theory Series (PANTS) in fall 2023 will be dedicated to Kevin

SERMON is a small, friendly, and informal gathering of number theorists and combinatorialists. Faculty members, graduate students, and undergraduate students are all invited to attend, and to give talks if they wish. The goal of SERMON is to provide a welcoming platform to present and share ideas in number theory, combinatorics and related fields. . We also hope this event provides a networking opportunity for students and junior faculty in the region. We ask all participants to help us achieve this goal.

The first SERMON was organized by Theresa Vaughn at UNCG in 1988. SERMON took place every year at various universities in the southeast until 2019. After a COVID-19 pandemic related break this trandition continues. See the home page for the SouthEast Regional Meeting On Numbers for more general information on SERMON and to subscribe to the SERMON e-mail list.

In 2023 SERMON will be held at the University of North Carolina Greensboro (UNCG) on **Saturday May 6** starting at **10 am**.

### Travel Support

SERMON has a limited budget from which to provide support for travel. In order to stretch our budget, we encourage car pooling and students to share lodging as much as possible.

If you have requested travel support fill out the VisitorExpenseFormSERMON2023 and send it to Matt Boylan by email (boylan@math.sc.edu) or U.S Mail (Matt Boylan, Mathematics Department, University of South Carolina, Columbia, SC 29208).

### Practical Information

You can find directions on the For Visitors page of our department.

All talks will take place in room 219 of the UNCG Petty building, see the UNCG Parking Map for its location.

##### Parking

During SERMON parking is free in the McIver Street Parking Deck which is closest to the Petty building It is on the north side of campus and its GPS address is 200 McIver Street.

##### Food

The local hosts recommend the following options located on Tate street which will be within 5 min walking distance from Petty building:

- Boba House: The Vegetarian Restaurant website/menu
- Sushi Republic (opens at 5 on Saturday) website/menu
- Old San Juan Bar & Grill website/menu
- Romeo’s Vegan Burgers website
- Raaz Indian Kitchen website/menu
- Don: website
- Chipotle website

Furthermore on Campus there are

- Old Town Draught House, 1205 Spring Garden St
- Yum Yum Better Ice Cream (and Hot Dogs), 1219 Spring Garden St

We also have a more comprehensive list of restaurants in Greensboro.

### Program

10:00 | Robert Groth (USC) Constructing Generalized Sierpinski Numbers | |
---|---|---|

10:40 | Ricky Farr (SMC): Fractional Stieltjes Constants | |

11:20 | Cuyler Warnock (Wesleyan): Infinite Families of Congruences for p-Core Partition | |

12:00 | Hui Xue (Clemson): Divisibility of an eigenform by an Eisenstein series | |

12:20 | Lunch | |

14:00 | Michael Filaseta (USC): On $n^{\rm th}$ order Euler polynomials of degree $n$ that are Eisenstein | |

14:40 | James Rudzinski (GTCC): Learning Monotone Boolean Functions | |

15:20 | Bailey Heath (USC): Representation Dimensions of Algebraic Tori | |

16:00 | Matt Farmer (UNCG): Indices of Finite Graphs | |

16:20 | Photo and Coffee | |

16:40 | Frank Thorne (USC): Counting Cubic Fields by Other Invariants | |

17:20 | Hannah Powell (UNCC): The over power-partitions | |

17:40 | Tom Wright (Wofford College): Siegel zeroes and primes in arithmetic progressions | |

18:30 | Oden at 802 West Gate City Boulevard, hopefully outside |

### Abstracts

**Robert Groth (USC)**: Constructing Generalized Sierpinski Numbers>

A Sierpinski number is a positive odd integer $k$ such that $k\cdot 2^n+1$ is composite for all $n\in\mathbb{Z}^{+}$. Fix an integer $A$ with $2\le A$. We show there exists a positive odd integer $k$ such that $k\cdot a^n+1$ is composite for all integers $a\in[2, A]$ and all $n\in\mathbb{Z}^{+}$.

**Matt Farmer (UNCG)**: Indices of Finite Graphs

Let G be a connected graph with vertex set [n] := {1, 2, · · · , n} and edge set E ⊆ ([n] choose 2). A topological index is a function from the set of graphs to C which is invariant under graph isomorphism. We study indices such as the Randic index, the radius, and the largest eigenvalues of the adjacency, Laplacian, and signless Laplacian matrices of graphs; i.e., spectral radii. We aim to find the extremal graphs of the ratio of the Randic index against the other indices listed above and present two new theorems as well as our work on an open problem relating the Randic index to the radius of a graph.

**Ricky Farr (SMC)**: Fractional Stieltjes Constants

We discuss evaluating fractional Stieltjes constants $\gamma_{\alpha}(a)$, arising naturally from the Laurent series expansions of the fractional derivatives of the Hurwitz zeta functions $\zeta^{(\alpha)}(s,a)$. We give an upper bound for the absolute value of $C_\alpha(a)=\gamma_\alpha(a)-\log^\alpha(a)/a$ and an asymptotic formula $\widetilde{C}_{\alpha}(a)$ for $C_{\alpha}(a)$ that yields a good approximation even for most small values of $\alpha$. We also give a bound for $|\widetilde{C}_{\alpha}(a)|$.

**Michael Filaseta (USC)**: On $n^{\rm th}$ order Euler polynomials of degree $n$ that are Eisenstein

For $m$ an even positive integer and $p$ a prime, we show that the generalized Euler polynomial $E_{mp}^{(mp)}(x)$ is in Eisenstein form with respect to $p$ if and only if $p$ does not divide $m (2^m-1)B_m$. As a consequence, we deduce that at least $1/3$ of the generalized Euler polynomials $E_n^{(n)}(x)$ are in Eisenstein form with respect to a prime $p$ dividing $n$ and, hence, irreducible over $\mathbb Q$. This is joint work with Thomas (Tommy) Luckner.

**Bailey Heath (USC)**: Representation Dimensions of Algebraic Tori

Abstract: Algebraic tori over a field k are special examples of affine group schemes over k, such as the multiplicative group of the field or the unit circle. Any algebraic torus can be embedded into the group of n x n invertible matrices with entries in k for some n, and the smallest such n is called the representation dimension of that torus. In this work, I am interested in finding the smallest possible upper bound on the representation dimension of all algebraic tori of a given dimension d. After providing some background, I will discuss how we can rephrase this question in terms of finite groups of invertible integral matrices. Then, I will share some progress that I have made on this question, including exact answers for certain values of d and a conjecture about the remaining values of n.

**Hannah Powell (UNCC):** The over power-partitions

In 2004, Corteel and Lovejoy introduced overpartitions of integers. Later, the concept of overpartitions was studied for other partitions. In this talk we will introduce the overpartition for power partitions of an integer.

We will also give an asymptotic of this over-power-partition function.

**James Rudzinski (GTCC)**: Learning Monotone Boolean Functions

Given a finite set $S$ with power set $\mathcal{P}(S)$, a \emph{Monotone Boolean function} is any function of the form $f:\mathcal{P}(S) \to \{0,1\}$ such that for all subsets $A$ and $B$ of $S$, $A \subseteq B$ implies $f(A) \leq f(B)$. We will discuss two versions of an algorithm for calculating the inverse images of a monotone Boolean function. Many important structures can be defined by monotone Boolean functions such as the \v{C}ech complex from topological data analysis as well as many monotone graph properties including chromatic number, containment of subgraphs, and graph connectivity.. We view a monotone Boolean function $f$ as a function on $B_n = \{0,1\}^n$ and utilize the so-called “Christmas Tree Decomposition” of $B_n$, a well known Symmetric Chain Decomposition of $B_n$. The algorithm organizes the minimal chain elements into a tree preserving monotone relations between them. This algorithm greatly improves the space complexity of the existing Hansel’s algorithm for this problem and unlike Hansel’s algorithm, it is also parallelizable.

**Frank Thorne (USC)**: Counting Cubic Fields by Other Invariants

I will present an asymptotic formula counting cubic fields by any of a one-parameter family of invariants, including the squarefree part of the discriminant. As I will explain, some of these invariance exhibit “independence of probabilities of splitting types” and some don’t.

Our results rely on two counting methods for counting cubic fields. One is a quantitative refinement of the Davenport-Heilbronn theorem, due to Bhargava, Taniguchi, and the speaker. The other is based on class field theory and Kummer theory, based on work of Cohen and Morra. I will explain each of these methods and how they fit together.

This is joint work with Arul Shankar.

**Cuyler Warnock (Wesleyan)**: Infinite Families of Congruences for p-Core Partitions

For each partition $n=n_1+n_2+\dots+n_k$, a Ferrers diagram can be created which is a left-aligned array of vertices with $n_i$ vertices in the $I$-th row. The hook of vertex $(i,j)$ is the total number of vertices directly below the vertex, directly to the right of the vertex, and the vertex itself. The hook number of $(i,j)$, denoted $H(I,j)$, is the total number of vertices on the hook. A partition of $n$ is $t$-core if and only if none of the hook numbers are divisible by $t$. Let $a_t(n)$ denote the number of $t$-core partitions of $n$. In 2003, Garvan showed for $p=5,7,11,13,17,19,23$ and $\delta_p=\frac{p^2-1}{24}$, that each prime $\ell$ that divides $\frac{p-1}{2}$ yields a congruence

$a_p(n-\delta_p)\equiv 0 \bmod \ell$

whenever $\left(\frac{n}{p}\right)=\epsilon_p$ and $n\not\equiv 0\bmod\ell$, where $\epsilon_{11}=\epsilon{13}=1$ and $\epsilon_{5}=\epsilon{7}=\epsilon_{17}=\epsilon{19}=\epsilon_{23}=-1$. In this talk, we exhibit infinite families of congruences of the type given by Garvan. For example, we show for $p=11$ and $\ell=2$ that

$a_{11}(11^t\cdot n-5)\equiv 0\bmod 2$

whenever $\left(\frac{n}{22}\right)=-1$ and $n\not\not\equiv 0\bmod 2$.

**Tom Wright (Wofford College):** Siegel zeroes and primes in arithmetic progressions

In 2003, John Friedlander and Henrik Iwaniec proved that the existence of Siegel zeroes would imply equidistribution for primes up to x in (most) arithmetic progressions $\bmod q$ with $q\lt x^{233/462}$; this is, in some sense, better than GRH, since $233/462\gt 1/2$. In this talk, we build on this result, improving the exponent in the Friedlander-Iwaniec bound.

**Hui Xue (Clemson)**: Divisibility of an eigenform by an Eisenstein series

We will use properties of Bernoulli numbers to solve a special case in a joint work with Jeff Beyerl and Kevin James.

### Acknowledgements

SERMON 2023 is supported by the NSF, UNC, and the Department of Mathematics at UNCG.

### Local Organizers

Torre Caparatta, Sebastian Pauli, Filip Saidak, Kalani Thalagoda

### Other Contacts

- Clemson: Hui Xue
- USC: Matthew Boylan, Michael Filaseta, Frank Thorne