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Full-Text Articles in Number Theory
Murmurations And Root Numbers, Alexey Pozdnyakov
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, Dillon Snyder
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 …
The Generalized Riemann Hypothesis And Applications To Primality Testing, Peter Hall
The Generalized Riemann Hypothesis And Applications To Primality Testing, Peter Hall
University Scholar Projects
The Riemann Hypothesis, posed in 1859 by Bernhard Riemann, is about zeros
of the Riemann zeta-function in the complex plane. The zeta-function can be repre-
sented as a sum over positive integers n of terms 1/ns when s is a complex number
with real part greater than 1. It may also be represented in this region as a prod-
uct over the primes called an Euler product. These definitions of the zeta-function
allow us to find other representations that are valid in more of the complex plane,
including a product representation over its zeros. The Riemann Hypothesis says that
all …
Polynomial Factoring Algorithms And Their Computational Complexity, Nicholas Cavanna
Polynomial Factoring Algorithms And Their Computational Complexity, Nicholas Cavanna
Honors Scholar Theses
Finite fields, and the polynomial rings over them, have many neat algebraic properties and identities that are very convenient to work with. In this paper we will start by exploring said properties with the goal in mind of being able to use said properties to efficiently irreducibly factorize polynomials over these fields, an important action in the fields of discrete mathematics and computer science. Necessarily, we must also introduce the concept of an algorithm’s speed as well as particularly speeds of basic modular and integral arithmetic opera- tions. Outlining these concepts will have laid the groundwork for us to introduce …