Number Theory Commons^™

Lecture 8, Kannan Soundararajan

NSF-CBMS Conference: L-functions and Multiplicative Number Theory

Continuation of Extreme values of L-functions.

Lecture 7, Kannan Soundararajan

NSF-CBMS Conference: L-functions and Multiplicative Number Theory

Extreme values of L-functions.

Lecture 6, Kannan Soundararajan

NSF-CBMS Conference: L-functions and Multiplicative Number Theory

Continuation of Progress towards moment conjectures -- upper and lower bounds.

Lecture 5, Kannan Soundararajan

NSF-CBMS Conference: L-functions and Multiplicative Number Theory

Progress towards moment conjectures -- upper and lower bounds.

High Moments Of L-Functions, Vorrapan Chandee

NSF-CBMS Conference: L-functions and Multiplicative Number Theory

Abstract: Moments of L-functions on the critical line (Re(s) = 1/2) have been extensively studied due to numerous applications, for example, bounds for L-functions, information on zeros of L-functions, and connections to the generalized Riemann hypothesis. However, the current understanding of higher moments is very limited. In this talk, I will give an overview how we can achieve asymptotic and bounds for higher moments by enlarging the size of various families of L-functions and show some techniques that are involved.

Moments Of Cubic L-Functions Over Function Fields, Alexandra Florea

NSF-CBMS Conference: L-functions and Multiplicative Number Theory

Abstract: I will talk about some recent work with Chantal David and Matilde Lalin about the mean value of L-functions associated to cubic characters over F_q[t] when q=1 (mod 3). I will explain how to obtain an asymptotic formula which relies on obtaining cancellation in averages of cubic Gauss sums over functions fields. I will also talk about the corresponding non-Kummer case when q=2 (mod 3) and I will explain why this setting is somewhat easier to handle than the Kummer case, which allows us to prove some better results.

Lecture 4, Kannan Soundararajan

NSF-CBMS Conference: L-functions and Multiplicative Number Theory

Larger values of L-functions on critical line -- moments, conjectures.

Lecture 3, Kannan Soundararajan

NSF-CBMS Conference: L-functions and Multiplicative Number Theory

Continuation of Selberg's central limit theorem and analogues in families of L-functions (typical size of values on critical line).

An Effective Chebotarev Density Theorem For Families Of Fields, With An Application To Class Groups, Caroline Turnage-Butterbaugh

NSF-CBMS Conference: L-functions and Multiplicative Number Theory

This talk will present an effective Chebotarev theorem that holds for all but a possible zero-density subfamily of certain families of number fields of fixed degree. For certain families, this work is unconditional, and in other cases it is conditional on the strong Artin conjecture and certain conjectures on counting number fields. As an application, we obtain nontrivial average upper bounds on ℓ-torsion in the class groups of the families of fields.

Landau-Siegel Zeros And Their Illusory Consequences, Kyle Pratt

NSF-CBMS Conference: L-functions and Multiplicative Number Theory

Updated time

Abstract: Researchers have tried for many years to eliminate the possibility of LandauSiegel zeros—certain exceptional counterexamples to the Generalized Riemann Hypothesis. Often one thinks of these zeros as being a severe nuisance, but there are many situations in which their existence allows one to prove spectacular, though illusory, results. I will review some of this history and some of these results. In the latter portion of the talk I will discuss recent work, joint with H. M. Bui and Alexandru Zaharescu, in which we show that the existence of Landau-Siegel zeros has implications for the behavior of Dirichlet …

Lecture 2, Kannan Soundararajan

NSF-CBMS Conference: L-functions and Multiplicative Number Theory

Selberg's central limit theorem and analogues in families of L-functions (typical size of values on critical line).

Lecture 1, Kannan Soundararajan

NSF-CBMS Conference: L-functions and Multiplicative Number Theory

Introduction to the rest of lectures + value distribution of L-functions away from critical line.

Arecibo Message, Joshua P. Tan

Open Educational Resources

This two week assignment asks students to interpret and analyze the 1974 Arecibo Message sent by Drake and Sagan. Week 1 introduces the concepts behind the construction of the message and engages with a critical analysis of the architecture and the contents of the message. Week 2 asks students to develop software in a Jupyter Notebook (available for free from the Anaconda Python Distribution) to interpret messages that were similar to those produced by Drake and Sagan.

Greatest Common Divisor: Algorithm And Proof, Mary K. Flagg

Number Theory

No abstract provided.

Extension Of Soft Set To Hypersoft Set, And Then To Plithogenic Hypersoft Set, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper, we generalize the soft set tothe hypersoft set by transforming the function F into a multi-attribute function. Then we introduce the hybrids of Crisp, Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic Hypersoft Set.

Congruence Relations Mod 2 For (2 X 4^T + 1)-Colored Partitions, Nicholas Torello

Senior Theses

Let p_r(n) denote the difference between the number of r-colored partitions of n into an even number of distinct parts and into an odd number of distinct parts. Inspired by proofs involving modular forms of the Hirschhorn-Sellers Conjecture, we prove a similar congruence for p_r(n). Using the Jacobi Triple Product identity, we discover a much stricter congruence for p_3(n).

New Experimental Investigations For The 3𝑥+1 Problem: The Binary Projection Of The Collatz Map, Benjamin Bairrington, Aaron Okano

Rose-Hulman Undergraduate Mathematics Journal

The 3x + 1 Problem, or the Collatz Conjecture, was originally developed in the early 1930's. It has remained unsolved for over eighty years. Throughout its history, traditional methods of mathematical problem solving have only succeeded in proving heuristic properties of the mapping. Because the problem has proven to be so difficult to solve, many think it might be undecidable. In this paper we brie y follow the history of the 3x + 1 problem from its creation in the 1930's to the modern day. Its history is tied into the development of the Cosper Algorithm, which maps binary sequences …

Pascal's Triangle Modulo N And Its Applications To Efficient Computation Of Binomial Coefficients, Zachary Warneke

Honors Program: Senior Projects (Public)

In this thesis, Pascal's Triangle modulo n will be explored for n prime and n a prime power. Using the results from the case when n is prime, a novel proof of Lucas' Theorem is given. Additionally, using both the results from the exploration of Pascal's Triangle here, as well as previous results, an efficient algorithm for computation of binomial coefficients modulo n (a choose b mod n) is described, and its time complexity is analyzed and compared to naive methods. In particular, the efficient algorithm runs in O(n log(a)) time (as opposed to …

Experience Of A Noyce-Student Learning Assistant In An Inquiry-Based Learning Class, Melissa Riley

UNO Student Research and Creative Activity Fair

This presentation refers to an undergraduate course called introduction to abstract mathematics at the University of Nebraska at Omaha. During the academic year 2017-2018, undergraduate, mathematics student Melissa Riley was a Noyce-student learning assistant for the Inquiry Based Learning (IBL) section of the course. She assisted the faculty-in-charge with all aspects of the course. These included: materials preparation, class organization, teamwork, class leading, presentations, and tutoring. This presentation shall address some examples of how the IBL approach can be used in this type of class including: the structure of the course, the activities and tasks performed by the students, learning …

Bounding The Number Of Compatible Simplices In Higher Dimensional Tournaments, Karthik Chandrasekhar

Theses and Dissertations--Mathematics

A tournament graph G is a vertex set V of size n, together with a directed edge set E ⊂ V × V such that (i, j) ∈ E if and only if (j, i) ∉ E for all distinct i, j ∈ V and (i, i) ∉ E for all i ∈ V. We explore the following generalization: For a fixed k we orient every k-subset of V by assigning it an orientation. That is, every facet of the (k − 1)-skeleton of the ( …

Hidden Symmetries In Classical Mechanics And Related Number Theory Dynamical System, Mohsin Md Abdul Karim

Masters Theses

Classical Mechanics consists of three parts: Newtonian, Lagrangian and Hamiltonian Mechanics, where each part is a special extension of the previous part. Each part has explicit symmetries (the explicit Laws of Motion), which, in turn, generate implicit or hidden symmetries (like the Law of Conservation of Energy, etc). In this Master's Thesis, different types of hidden symmetries are considered; they are reflected in the Noether Theorem and the Poincare Recurrence Theorem applied to Lagrangian and Hamiltonian Systems respectively.

The Poincare Recurrence Theorem is also applicable to some number theory problems, which can be considered as dynamical systems. In …

On A Generalization Of A Theorem Of Ibukiyama, Brad Isaacson

Publications and Research

We generalize a theorem of Ibukiyama and express periodic generalized Bernoulli functions by generalized Bernoulli numbers. As a corollary, we obtain formulas expressing these character sums by generalized Bernoulli numbers using only elementary methods from algebra and number theory.

Special Issue: New Types Of Neutrosophic Set/Logic/Probability, Neutrosophic Over-/Under-/ Off-Set, Neutrosophic Refined Set, And Their Extension To Plithogenic Set/Logic/ Probability, With Applications, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

A Short Remark On Gödel Incompleteness Theorem And Its Self-Referential Paradox From Neutrosophic Logic Perspective, Florentin Smarandache, Victor Christianto

Branch Mathematics and Statistics Faculty and Staff Publications

It is known from history of mathematics, that Gödel submitted his two incompleteness theorems, which can be considered as one of hallmarks of modern mathematics in 20th century. Here we argue that Gödel incompleteness theorem and its self-referential paradox have not only put Hilbert’s axiomatic program into question, but he also opened up the problem deep inside the then popular Aristotelian Logic. Although there were some attempts to go beyond Aristotelian binary logic, including by Lukasiewicz’s three-valued logic, here we argue that the problem of self-referential paradox can be seen as reconcilable and solvable from Neutrosophic Logic perspective. Motivation of …

Special Issue: Algebraic Structures Of Neutrosophic Triplets, Neutrosophic Duplets, Or Neutrosophic Multisets, Vol. Ii, Florentin Smarandache, Xiaohong Zhang, Mumtaz Ali

Branch Mathematics and Statistics Faculty and Staff Publications

No abstract provided.

Primes In Arithmetical Progression, Edward C. Wessel

Honors Theses

This thesis will tackle Dirichlet’s Theorem on Primes in Arithmetical Progressions. The majority of information that follows below will stem from Tom M. Apostol’s Introduction to Analytical Number Theory. This is the main source of all definitions, theorems, and method. However, I would like to assure the reader that prior knowledge of neither the text nor analytical number theory in general is needed to understand the result. A rough background in Abstract Algebra and a moderate grasp on Complex and Real Analysis are more than sufficient. In fact, my project’s intent is to introduce Dirichlet’s ideas to the mathematics student …

Fibonacci And Lucas Differential Equations, Esra Erkus-Duman, Hakan Ciftci

Applications and Applied Mathematics: An International Journal (AAM)

The second-order linear hypergeometric differential equation and the hypergeometric function play a central role in many areas of mathematics and physics. The purpose of this paper is to obtain differential equations and the hypergeometric forms of the Fibonacci and the Lucas polynomials. We also write again these polynomials by means of Olver’s hypergeometric functions. In addition, we present some relations between these polynomials and the other well-known functions.

Simplifying Coefficients In A Family Of Ordinary Differential Equations Related To The Generating Function Of The Laguerre Polynomials, Feng Qi

Applications and Applied Mathematics: An International Journal (AAM)

In the paper, by virtue of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the Lah inversion formula, the author simplifies coefficients in a family of ordinary differential equations related to the generating function of the Laguerre polynomials.

Triangular Cubic Hesitant Fuzzy Einstein Hybrid Weighted Averaging Operator And Its Application To Decision Making, Florentin Smarandache, Aliya Fahmi, Fazli Amin, Madad Khan, Nasruddin Hassan

Branch Mathematics and Statistics Faculty and Staff Publications

In this paper, triangular cubic hesitant fuzzy Einstein weighted averaging (TCHFEWA) operator, triangular cubic hesitant fuzzy Einstein ordered weighted averaging (TCHFEOWA) operator and triangular cubic hesitant fuzzy Einstein hybrid weighted averaging (TCHFEHWA) operator are proposed. An approach to multiple attribute group decision making with linguistic information is developed based on the TCHFEWA and the TCHFEHWA operators. Furthermore, we establish various properties of these operators and derive the relationship between the proposed operators and the existing aggregation operators. Finally, a numerical example is provided to demonstrate the application of the established approach

Sums Involving The Number Of Distinct Prime Factors Function, Tanay Wakhare

Rose-Hulman Undergraduate Mathematics Journal

We find closed form expressions for finite and infinite sums that are weighted by $\omega(n)$, where $\omega(n)$ is the number of distinct prime factors of $n$. We then derive general convergence criteria for these series. The approach of this paper is to use the theory of symmetric functions to derive identities for the elementary symmetric functions, then apply these identities to arbitrary primes and values of multiplicative functions evaluated at primes. This allows us to reinterpret sums over symmetric polynomials as divisor sums and sums over the natural numbers.