Limit Theorems For L-Functions In Analytic Number Theory, 2024 The Graduate Center, City University of New York

#### Limit Theorems For L-Functions In Analytic Number Theory, Asher Roberts

*Dissertations, Theses, and Capstone Projects*

We use the method of Radziwill and Soundararajan to prove Selberg’s central limit theorem for the real part of the logarithm of the Riemann zeta function on the critical line in the multivariate case. This gives an alternate proof of a result of Bourgade. An upshot of the method is to determine a rate of convergence in the sense of the Dudley distance. This is the same rate Selberg claims using the Kolmogorov distance. We also achieve the same rate of convergence in the case of Dirichlet L-functions. Assuming the Riemann hypothesis, we improve the rate of convergence by using …

Relating Elasticity And Other Multiplicative Properties Among Orders In Number Fields And Related Rings, 2024 Clemson University

#### Relating Elasticity And Other Multiplicative Properties Among Orders In Number Fields And Related Rings, Grant Moles

*All Dissertations*

This dissertation will explore factorization within orders in a number ring. By far the most well-understood of these orders are rings of algebraic integers. We will begin by examining how certain types of subrings may relate to the larger rings in which they are contained. We will then apply this knowledge, along with additional techniques, to determine how the elasticity in an order relates to the elasticity of the full ring of algebraic integers. Using many of the same strategies, we will develop a corresponding result in the rings of formal power series. Finally, we will explore a number of …

Some Experiments In Additive Number Theory, 2024 Clemson University

#### Some Experiments In Additive Number Theory, Yunan Wang

*All Dissertations*

This dissertation explores fundamental conjectures in number theory, focusing on the distribution patterns of representation functions in prime pairs. The work concentrates on twin primes, cousin primes, and primes separated by six units, offering a fresh heuristic interpretation of the Hardy-Littlewood correction factor. The analysis progresses to investigate the partition function for prime pairs in the form $(p, p+k)$, specifically for $k = 2, 4, 6$. The study culminates in the derivation of a general formula for prime pairs $(p, p+d)$, where $d$ is an even integer. Drawing on the insights gleaned from examining the correction factor, this dissertation proposes …

Bivariate Polynomials Of Low Degree And Small Mahler Measure, 2024 Institut Elie Cartan de Lorraine, University of Lorraine, Metz, France

#### Bivariate Polynomials Of Low Degree And Small Mahler Measure, Souad El Otmani

*BAU Journal - Science and Technology*

In this work, we highlight that many of the known limit points of the Mahler measure of univariate polynomials can be obtained as the Mahler measure of *low-degree* bivariate polynomials. To this end, we provide for each relevant measure the corresponding original bivariate polynomial found in the literature, along with the corresponding low-degree polynomial with an analogous measure.

Combinatorial Problems On The Integers: Colorings, Games, And Permutations, 2024 University of Denver

#### Combinatorial Problems On The Integers: Colorings, Games, And Permutations, Collier Gaiser

*Electronic Theses and Dissertations*

This dissertation consists of several combinatorial problems on the integers. These problems fit inside the areas of extremal combinatorics and enumerative combinatorics.

We first study monochromatic solutions to equations when integers are colored with finitely many colors in Chapter 2. By looking at subsets of {1, 2, . . . , *n*} whose least common multiple is small, we improved a result of Brown and Rödl on the smallest integer *n* such that every 2-coloring of {1, 2, . . . , *n*} has a monochromatic solution to equations with unit fractions. Using a recent result of Boza, …

Bifurcations And Resultants For Rational Maps And Dynatomic Modular Curves In Positive Characteristic, 2024 The Graduate Center, City University of New York

#### Bifurcations And Resultants For Rational Maps And Dynatomic Modular Curves In Positive Characteristic, Colette Lapointe

*Dissertations, Theses, and Capstone Projects*

No abstract provided.

Explicit Composition Identities For Higher Composition Laws In The Quadratic Case, 2024 The Graduate Center, City University of New York

#### Explicit Composition Identities For Higher Composition Laws In The Quadratic Case, Ajith A. Nair

*Dissertations, Theses, and Capstone Projects*

The theory of Gauss composition of integer binary quadratic forms provides a very useful way to compute the structure of ideal class groups in quadratic number fields. In addition to that, Gauss composition is also important in the problem of representations of integers by binary quadratic forms. In 2001, Bhargava discovered a new approach to Gauss composition which uses 2x2x2 integer cubes, and he proved a composition law for such cubes. Furthermore, from the higher composition law on cubes, he derived four new higher composition laws on the following spaces - 1) binary cubic forms, 2) pairs of binary quadratic …

On A Generalization Of A Theorem Of Ibukiyama To Evaluate Three Imprimitive Character Sums, 2024 CUNY New York City College of Technology

#### On A Generalization Of A Theorem Of Ibukiyama To Evaluate Three Imprimitive Character Sums, Brad Isaacson

*Publications and Research*

In a previous paper, we expressed three families of character sums by certain generalized Bernoulli functions which in turn were expressed by generalized Bernoulli numbers via a complicated and indirect process. In this paper, we generalize a theorem of Ibukiyama to directly express these generalized Bernoulli functions by generalized Bernoulli numbers. As a result, we can express the three families of character sums by generalized Bernoulli numbers in a more elegant fashion than was done before.

Boolean Group Structure In Class Groups Of Positive Definite Quadratic Forms Of Primitive Discriminant, 2024 University of Mary Washington

#### Boolean Group Structure In Class Groups Of Positive Definite Quadratic Forms Of Primitive Discriminant, Christopher Albert Hudert Jr.

*Student Research Submissions*

It is possible to completely describe the representation of any integer by binary quadratic forms of a given discriminant when the discriminant’s class group is a Boolean group (also known as an elementary abelian 2-group). For other discriminants, we can partially describe the representation using the structure of the class group. The goal of the present project is to find whether any class group with 32 elements and a primitive positive definite discriminant is a Boolean group. We find that no such class group is Boolean.

Local Converse Theorem For 2-Dimensional Representations Of Weil Groups, 2024 University of Maine

#### Local Converse Theorem For 2-Dimensional Representations Of Weil Groups, William Lp Johnson

*Electronic Theses and Dissertations*

A local converse theorem is a theorem which states that if two representations \chi_1, \chi_2 have equal \gamma-factors for all twists by representations \sigma coming from a certain class, then \chi_1 and \chi_2 are equivalent in some way. We provide a direct proof of a local converse theorem in two distinct settings. Previous proofs published in the literature for these settings were indirect proofs making use of various correspondences between representations of other groups. We first prove a Gauss sum local converse theorem for representations of (F_{p^2})^{\times} twisted by representations of F_p^{\times}. We then apply this theorem to tamely ramified …

Murmurations And Root Numbers, 2024 University of Connecticut

#### Murmurations And Root Numbers, Alexey Pozdnyakov

*University Scholar Projects*

We report on a machine learning investigation of large datasets of elliptic curves and *L*-functions. This leads to the discovery of murmurations, an unexpected correlation between the root numbers and Dirichlet coefficients of *L*-functions. We provide a formal definition of murmurations, describe the connection with 1-level density, and provide three examples for which the murmuration phenomenon has been rigorously proven. Using our understanding of murmurations, we then build new machine learning models in search of a polynomial time algorithm for predicting root numbers. Based on our models and several heuristic arguments, we conclude that it is unlikely for …

Hilbert Reciprocity Over Number Fields, 2024 University of Connecticut

#### Hilbert Reciprocity Over Number Fields, Dillon Snyder

*Honors Scholar Theses*

A Hilbert symbol has the value 1 or −1 depending on the existence of solutions to a certain quadratic equation in a local field, R, or C. Hilbert reciprocity states that for a number field F and two nonzero a and b in F, the product of Hilbert symbols associated to a and b at all the places of F is 1. That is, these Hilbert symbols are −1 for a finite, even number of places of F . Hilbert reciprocity when F = Q is equivalent to the classical quadratic reciprocity law, so Hilbert reciprocity in number fields can …

Rsa Algorithm, 2024 Arkansas Tech University

#### Rsa Algorithm, Evalisbeth Garcia Diazbarriga

*ATU Research Symposium*

I will be presenting about the RSA method in cryptology which is the coding and decoding of messages. My research will focus on proving that the method works and how it is used to communicate secretly.

Finite Monodromy And Artin Representations, 2024 Louisiana State University

#### Finite Monodromy And Artin Representations, Emma Lien

*LSU Doctoral Dissertations*

Artin representations, which are complex representations of finite Galois groups, appear in many contexts in number theory. The Langlands program predicts that Galois representations like these should arise from automorphic representations and many examples of this correspondence have been found such as in the proof of Fermat's Last Theorem. This dissertation aims to make an analysis of explicitly computable examples of Artin representations from both sides of this correspondence. On the automorphic side, certain weight 1 modular forms have been shown to be related to Artin representations and an explicit analysis of their Fourier coefficients allows us to identify the …

Birkhoff Summation Of Irrational Rotations: A Surprising Result For The Golden Mean, 2024 Portland State University

#### Birkhoff Summation Of Irrational Rotations: A Surprising Result For The Golden Mean, Heather Moore

*University Honors Theses*

This thesis presents a surprising result that the difference in a certain sums of constant rotations by the golden mean approaches *exactly* 1/5. Specifically, we focus on the Birkhoff sums of these rotations, with the number of terms equal to squared Fibonacci numbers. The proof relies on the properties of continued fraction approximants, Vajda's identity and the explicit formula for the Fibonacci numbers.

Optimizing Buying Strategies In Dominion, 2024 Georgia Southern University

#### Optimizing Buying Strategies In Dominion, Nikolas A. Koutroulakis

*Rose-Hulman Undergraduate Mathematics Journal*

Dominion is a deck-building card game that simulates competing lords growing their kingdoms. Here we wish to optimize a strategy called Big Money by modeling the game as a Markov chain and utilizing the associated transition matrices to simulate the game. We provide additional analysis of a variation on this strategy known as Big Money Terminal Draw. Our results show that player's should prioritize buying provinces over improving their deck. Furthermore, we derive heuristics to guide a player's decision making for a Big Money Terminal Draw Deck. In particular, we show that buying a second Smithy is always more optimal …

Inexact Fixed-Point Proximity Algorithm For The ℓ₀ Sparse Regularization Problem, 2024 Old Dominion University

#### Inexact Fixed-Point Proximity Algorithm For The ℓ₀ Sparse Regularization Problem, Ronglong Fang, Yuesheng Xu, Mingsong Yan

*Mathematics & Statistics Faculty Publications*

We study inexact fixed-point proximity algorithms for solving a class of sparse regularization problems involving the ℓ₀ norm. Specifically, the ℓ₀ model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the ℓ₀ norm regularization term. Such an ℓ₀ model is non-convex. Existing exact algorithms for solving the problems require the availability of closed-form formulas for the proximity operator of convex functions involved in the objective function. When such formulas are not available, numerical computation of the proximity operator becomes inevitable. This leads to inexact iteration algorithms. We investigate in this …

Bridging Theory And Application: A Journey From Minkowski's Theorem To Ggh Cryptosystems In Lattice Theory, 2024 Claremont McKenna College

#### Bridging Theory And Application: A Journey From Minkowski's Theorem To Ggh Cryptosystems In Lattice Theory, Danzhe Chen

*CMC Senior Theses*

This thesis provides a comprehensive exploration of lattice theory, emphasizing its dual significance in both theoretical mathematics and practical applications, particularly within computational complexity and cryptography. The study begins with an in-depth examination of the fundamental properties of lattices and progresses to intricate lattice-based problems such as the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP). These problems are analyzed for their computational depth and linked to the Subset Sum Problem (SSP) to highlight their critical roles in understanding computational hardness. The narrative then transitions to the practical applications of these theories in cryptography, evaluating the shift from …

Solving Robert Wilson’S 𝑡 ≠ 2 Conjecture On Graham Sequences, 2024 Claremont Colleges

#### Solving Robert Wilson’S 𝑡 ≠ 2 Conjecture On Graham Sequences, Krishna Rajesh

*HMC Senior Theses*

Ron Graham's sequence is a surprising bijection from the natural numbers to the non-prime integers, which is constructed by looking at sequences whose product is square. In this thesis we will resolve a 22-year-old conjecture about this bijection, by construction of explicit sequences in a modified number theoretic context. Additionally, we will discuss the history of this problem, and give computational techniques for computing this bijection, levering ideas from linear algebra over the finite field of two elements.

Pairs Of Quadratic Forms Over P-Adic Fields, 2024 University of Kentucky

#### Pairs Of Quadratic Forms Over P-Adic Fields, John Hall

*Theses and Dissertations--Mathematics*

Given two quadratic forms $Q_1, Q_2$ over a $p$-adic field $K$ in $n$ variables, we consider the pencil $\mathcal{P}_K(Q_1, Q_2)$, which contains all nontrivial $K$-linear combinations of $Q_1$ and $Q_2$. We define $D$ to be the maximal dimension of a subspace in $K^n$ on which $Q_1$ and $Q_2$ both vanish. We define $H$ to be the maximal number of hyperbolic planes that a form in $\mathcal{P}_K(Q_1, Q_2)$ splits off over $K$. We will determine which values for $(D, H)$ are possible for a nonsingular pair of quadratic forms over a $p$-adic field $K$.