Open Access. Powered by Scholars. Published by Universities.®
- Discipline
- Keyword
-
- Additive number theory (1)
- Algebraic Number Theory (1)
- Associated subring (1)
- Commutative Algebra (1)
- Conductor (1)
-
- Eichler-Selberg trace formula (1)
- Eisenstein series (1)
- Factorization (1)
- Fundamental domain (1)
- HFD (1)
- Hardy-Littlewood correction factor (1)
- Ideal-preserving (1)
- Identifying eigenforms (1)
- Jacobi theta function (1)
- Maeda's conjecture (1)
- Modular forms (1)
- Power series (1)
- Products of eigenforms (1)
- Q-series (1)
- Valence formula (1)
Articles 1 - 5 of 5
Full-Text Articles in Number Theory
Monomial Quadratic Identities Of Hecke Eigenforms, Trevor Vilardi
Monomial Quadratic Identities Of Hecke Eigenforms, Trevor Vilardi
All Dissertations
Duke and Ghate independently studied the question of when it is possible for the product of two eigenforms to be an eigenform. In this dissertation, we take up a generalization of that question, namely is it possible for the product of two eigenforms to be equal to a different product of two eigenforms? Under this formulation, the question becomes closer to one about unique factorization, i.e., how closely do eigenforms work like irreducible elements? Our conjecture is that there are only finitely many cases where the product of two eigenforms is equal to a different product of two eigenforms, and …
Some Experiments In Additive Number Theory, Yunan Wang
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 …
Relating Elasticity And Other Multiplicative Properties Among Orders In Number Fields And Related Rings, Grant Moles
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 …
Zeros Of Modular Forms, Daozhou Zhu
Zeros Of Modular Forms, Daozhou Zhu
All Dissertations
Let $E_k(z)$ be the normalized Eisenstein series of weight $k$ for the full modular group $\text{SL}(2, \mathbb{Z})$. It is conjectured that all the zeros of the weight $k+\ell$ cusp form $E_k(z)E_\ell(z)-E_{k+\ell}(z)$ in the standard fundamental domain lie on the boundary. Reitzes, Vulakh and Young \cite{Reitzes17} proved that this statement is true for sufficiently large $k$ and $\ell$. Xue and Zhu \cite{Xue} proved the cases when $\ell=4,6,8$ with $k\geq\ell$, all the zeros of $E_k(z)E_\ell(z)-E_{k+\ell}(z)$ lie on the arc $|z|=1$. For all $k\geq\ell\geq10$, we will use the same method as \cite{Reitzes17} to locate these zeros on the arc $|z|=1$, and improve the …
Algebraic And Integral Closure Of A Polynomial Ring In Its Power Series Ring, Joseph Swanson
Algebraic And Integral Closure Of A Polynomial Ring In Its Power Series Ring, Joseph Swanson
All Dissertations
Let R be a domain. We look at the algebraic and integral closure of a polynomial ring, R[x], in its power series ring, R[[x]]. A power series α(x) ∈ R[[x]] is said to be an algebraic power series if there exists F (x, y) ∈ R[x][y] such that F (x, α(x)) = 0, where F (x, y) ̸ = 0. If F (x, y) is monic, then α(x) is said to be an integral power series. We characterize the units of algebraic and integral power series. We show that the only algebraic power series with infinite radii of convergence are …