Number Theory Commons^™

Hermitian Maass Lift For General Level, An Hoa Vu

Dissertations, Theses, and Capstone Projects

For an imaginary quadratic field $K$ of discriminant $-D$, let $\chi = \chi_K$ be the associated quadratic character. We will show that the space of special hermitian Jacobi forms of level $N$ is isomorphic to the space of plus forms of level $DN$ and nebentypus $\chi$ (the hermitian analogue of Kohnen's plus space) for any integer $N$ prime to $D$. This generalizes the results of Krieg from $N = 1$ to arbitrary level. Combining this isomorphism with the recent work of Berger and Klosin and a modification of Ikeda's construction we prove the existence of a lift from the space …

Zeta Functions Of Classical Groups And Class Two Nilpotent Groups, Fikreab Solomon Admasu

Dissertations, Theses, and Capstone Projects

This thesis is concerned with zeta functions and generating series associated with two families of groups that are intimately connected with each other: classical groups and class two nilpotent groups. Indeed, the zeta functions of classical groups count some special subgroups in class two nilpotent groups.

In the first chapter, we provide new expressions for the zeta functions of symplectic groups and even orthogonal groups in terms of the cotype zeta function of the integer lattice. In his paper on universal $p$-adic zeta functions, J. Igusa computed explicit formulae for the zeta functions of classical algebraic groups. These zeta functions …

A Few Firsts In The Epsilon Years Of My Career, Heidi Goodson

Journal of Humanistic Mathematics

In this essay, I describe the unexpected ways I achieved some milestones in the early years of my career.

The Last Digits Of Infinity (On Tetrations Under Modular Rings), William Stowe

Celebration of Learning

A tetration is defined as repeated exponentiation. As an example, 2 tetrated 4 times is 2^(2^(2^2)) = 2^16. Tetrated numbers grow rapidly; however, we will see that when tetrating where computations are performed mod n for some positive integer n, there is convergent behavior. We will show that, in general, this convergent behavior will always show up.

Inquiry In Inquiry: A Classification Of The Learning Theories Underlying Inquiry-Based Undergraduate Number Theory Texts, Rebecca L. Butler

Honors Projects

While undergraduate inquiry-based texts in number theory share similar approaches with respect to learning as the embodiment of professional practice, this does not entail that these texts all operate from the same fundamental understanding of what it means to learn mathematics. In this paper, the instructional design of several texts of the aforementioned types are analyzed to assess the theory of learning under which they operate. From this understanding of the different theories of learning employed in an inquiry-based mathematical setting, one can come to understand the popular model of what it is to learn number theory in a meaningful …

On The Lucas Difference Sequence Spaces Defined By Modulus Function, Murat Karakaş, Tayfur Akbaş, Ayşe M. Karakaş

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, firstly, we define the Lucas difference sequence spaces by the help of Lucas sequence and a sequence of modulus function. Besides, we give some inclusion relations and examine geometrical properties such as Banach-Saks type p, weak fixed point property.

Lecture 10, Kannan Soundararajan

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

Continuation of Fyodorov--Keating conjectures, connections with random multiplicative functions.

Lecture 9, Kannan Soundararajan

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

Fyodorov--Keating conjectures, connections with random multiplicative functions.

The Weyl Bound For Dirichlet L-Functions, Matthew P. Young

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

Abstract: In the 1960's, Burgess proved a subconvexity bound for Dirichlet L-functions. However, the quality of this bound was not as strong, in terms of the conductor, as the classical Weyl bound for the Riemann zeta function. In a major breakthrough, Conrey and Iwaniec established the Weyl bound for quadratic Dirichlet L-functions. I will discuss recent work with Ian Petrow that generalizes the Conrey-Iwaniec bound for more general characters, in particular arbitrary characters of prime modulus.

Extension Of A Positivity Trick And Estimates Involving L-Functions At The Edge Of The Critical Strip, Xiannan Li

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

Updated schedule

Abstract: I will review an old trick, and relate this to some modern results involving estimates for L-functions at the edge of the critical strip. These will include a good bound for automorphic L-functions and Rankin-Selberg L-functions as well as estimates for primes which split completely in a number field.

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 …