Purpose
This seminar is a place for graduate students to learn classical analytic number theory together. The goal for Fall 2025 is to learn fundamental estimation techniques including the Dirichlet hyperbola method and summation by parts, and to start learning about the role of L-functions through Dirichlet's theorem on primes in arithmetic progressions. Time permitting, we may also discuss a proof of the Prime Number Theorem.
Meeting Details:
There will be weekly meetings, currently scheduled for Mondays, 12:45-1:45 PM
Meetings will take place in The Graduate Center Math Thesis Room (4214.03)
Meetings will also be broadcasted via Zoom. Please contact Nathaniel Kingsbury-Neuschotz for the zoom link.
September 2
Speaker: Nathaniel Kingsbury-Neuschotz
Topic: Recap from Last Semester and Organizational Meeting
Agenda: Nathaniel will review some of what we learned last semester. We will set a rough schedule of presenters over the next few weeks.
Meeting Details:
There will be weekly meetings, currently scheduled for Tuesdays, 12-1 PM
Meetings will take place in The Graduate Center Math Thesis Room (4214.03)
Meetings will also be broadcasted via Zoom. Please contact Nathaniel Kingsbury-Neuschotz for the zoom link.
September 2
Speaker: Nathaniel Kingsbury-Neuschotz
Topic: Organizational First Meeting
Agenda: Nathaniel will discuss the purpose of the seminar. We will set a rough schedule of presenters over the next few weeks. Nathaniel will discuss the average order of the divisor counting function as a toy problem for the techniques we'll learn.
September 9
Speaker: Jeremy Weissman
Topic: The Möbius and Euler Totient Functions
Agenda: Jeremy will introduce the notion of an Arithmetical Function, and will discuss the Euler Totient and Möbius functions as key examples thereof. He will derive a formula for the Euler Totient function in terms of the Möbius function, and discuss a proof by way of the Principle of Inclusion-Exclusion. This corresponds to section 2.1-2.5 of Apostol's book.
September 16
Speaker: Jeremy Weissman and Abe Radin
Topic: The Dirichlet Product, the von Mangoldt Function, and Multiplicative Functions
Agenda: Jeremy will discuss the Dirichlet Product and the structure of the monoid of arithmetical functions under this product. He will derive the Möbius inversion formula as a corollary, before characterizing the units of this monoid. Abe will then introduce the von Mangoldt function, define multiplicative and completely multiplicative functions, and prove some basic properties thereof. This corresponds to sections 2.6-2.9 of Apostol's book.
September 23 - No Meeting, No Classes Scheduled
September 30
Speaker: Abe Radin
Topic: Dirichlet Products of Multiplicative Functions, and more Arithmetical Functions
Agenda: Abe will discuss the behavior of multiplicative functions under the Dirichlet product, proving that the form a subgroup of the units of the monoid of arithmetical functions. He will then derive a simple formula for the Dirichlet inverse of a completely multiplicative function, before introducing several more important examples of multiplicative functions and proving some Dirichlet product identities that they satisfy. This corresponds to sections 2.10-2.13 of Apostol's book.
October 7 - No Meeting due to Sukkot
October 14 - CUNY Monday, No Meeting
October 21
Speaker: Nathaniel Kingsbury-Neuschotz
Topic: Generalized Convolutions, Generating Functions, and "Derivatives" of Arithmetical Functions
Agenda: Nathaniel will discuss an important generalization of the Dirichlet product and its basic properties, before introducing generating functions and sketching their role in analytic number theory. He will then discuss the notion of the derivative of an arithmetical function, motivated by way of Dirichlet series, and will use this concept to quickly derive Selberg's identity. This corresponds to sections 2.14-2.18 of Apostol's book.
October 28
Speaker: N/A
Topic: Chapter 2 exercises
Agenda: We will solve some of the exercises in Chapter 2 of Apostol's book.
November 4
Speaker: Vievie
Topic: Asymptotics for Partial Sums of Series
Agenda: Vieview will introduce two key tools for giving asymptotics for partial sums of arithmetical functions: Euler's summation formula, and Dirichlet's hyperbola method. She will use these results to give asymptotics for the partial sums of the divisor-counting function. This corresponds to sections 3.1-3.5 of Apostol's book.