Zeta Functions Of Classical Groups And Class Two Nilpotent Groups, 2019 The Graduate Center, City University of New York

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

*All 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 ...

A Few Firsts In The Epsilon Years Of My Career, 2019 Brooklyn College (CUNY)

#### 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), 2019 Augustana College, Rock Island Illinois

#### 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, 2019 Seattle Pacific University

#### 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 ...

Lecture 10, 2019 University of Mississippi

#### 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, 2019 University of Mississippi

#### 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, 2019 Texas A&M University

#### 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, 2019 Kansas State University

#### 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, 2019 University of Mississippi

#### Lecture 8, Kannan Soundararajan

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

Continuation of Extreme values of *L*-functions.

Lecture 7, 2019 University of Mississippi

#### Lecture 7, Kannan Soundararajan

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

Extreme values of *L*-functions.

Lecture 6, 2019 University of Mississippi

#### 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, 2019 University of Mississippi

#### 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, 2019 Kansas State University

#### 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, 2019 Columbia University

#### 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, 2019 University of Mississippi

#### 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, 2019 University of Mississippi

#### 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, 2019 Carleton College

#### 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, 2019 University of Illinois

#### 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 ...

Lecture 2, 2019 University of Mississippi

#### 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, 2019 University of Mississippi

#### 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.