Introduction to Analytic Number Theory
Course description:
Analytic number theory studies the distribution of prime numbers via mathematical (real & complex) analysis tools. In this course, we study arithmetic functions, but from an analytic perspective, dwelling more on their asymptotic expansion. Moreover, we study the distribution of prime numbers which is still an active research topic today. One of the key figures for this endeavor is the celebrated Riemann zeta function which can be regarded as an important tool in connection with prime numbers. Some of its properties have been studied intensively over the past century. One of the most famous open questions remains the Riemann Hypothesis. Also, another central figure in this course is the prime number theorem which encodes important information on the distribution of prime numbers. Last but not least, we discuss primes in arithmetic progressions.
By the end of this course, students should develop fundamental knowledge and skills involving basic concepts of the topics covered in this course. Overall, this course will serve as an essential ingredient for further more advanced (graduate level courses) in analysis and number theory.
Analytic number theory studies the distribution of prime numbers via mathematical (real & complex) analysis tools. In this course, we study arithmetic functions, but from an analytic perspective, dwelling more on their asymptotic expansion. Moreover, we study the distribution of prime numbers which is still an active research topic today. One of the key figures for this endeavor is the celebrated Riemann zeta function which can be regarded as an important tool in connection with prime numbers. Some of its properties have been studied intensively over the past century. One of the most famous open questions remains the Riemann Hypothesis. Also, another central figure in this course is the prime number theorem which encodes important information on the distribution of prime numbers. Last but not least, we discuss primes in arithmetic progressions.
By the end of this course, students should develop fundamental knowledge and skills involving basic concepts of the topics covered in this course. Overall, this course will serve as an essential ingredient for further more advanced (graduate level courses) in analysis and number theory.
Lecturer
Date
22nd September, 2025 ~ 12th January, 2026
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday | 13:30 - 16:05 | A3-3-301 | ZOOM 11 | 435 529 7909 | BIMSA |
Prerequisite
Topics covered: arithmetic and multiplicative functions, Abel summation and Möbius inversion, the Mellin transformation and Perron’s formula, Dirichlet series and Euler products, Dirichlet characters, Riemann zeta function, Euler’s gamma and beta functions, distribution of prime numbers, prime number theorem, Dirichlet’s hyperbola method, Dirichlet’s L-functions, primes in arithmetic progression.
Syllabus
Course Outline:
• Chapter 0. What is analytic number theory? The big picture.
• Chapter 1. The fundamental theorem of Arithmetic. A recap of elementary number theory.
• Chapter 2. Arithmetical functions and Dirichlet multiplication.
• Chapter 3. Asymptotics and averages of arithmetical functions.
• Chapter 4. Elementary results on the distribution of primes.
• Chapter 5. The prime number theorem. The Riemann zeta function.
• Chapter 6. Dirichlet series.
• Chapter 7. Primes in arithmetic progression. Dirichlet’s theorem.
Course Calendar:
• Lecture 1: The big picture of analytic number theory (2025/09/22)
• Lecture 2: A recap of elementary number theory. (2025/09/29)
• Lecture 3: Arithmetic functions and Dirichlet multiplication I (2025/10/13)
• Lecture 4: Arithmetic functions and Dirichlet multiplication II (2025/10/20)
• Lecture 5: Asymptotics and averages of arithmetic functions I (2025/10/27)
• Lecture 6: Asymptotics and averages of arithmetic functions II (2025/11/3)
• Lecture 7: Asymptotics and averages of arithmetic functions III (2025/11/10)
• Lecture 8: Elementary results on the distribution of primes I (2025/11/17)
• Lecture 9: Elementary results on the distribution of primes II (2025/11/24)
• Lecture 10: The prime number theorem I (2025/12/1)
• Lecture 11: The prime number theorem II (2025/12/8)
• Lecture 12: The prime number theorem III (2025/12/15)
• Lecture 13: Dirichlet series I (2025/12/22)
• Lecture 14: Dirichlet series II (2025/12/29)
• Lecture 15: Primes in arithmetic progression I (2026/1/5)
• Lecture 16: Primes in arithmetic progression II (2026/1/12)
• Chapter 0. What is analytic number theory? The big picture.
• Chapter 1. The fundamental theorem of Arithmetic. A recap of elementary number theory.
• Chapter 2. Arithmetical functions and Dirichlet multiplication.
• Chapter 3. Asymptotics and averages of arithmetical functions.
• Chapter 4. Elementary results on the distribution of primes.
• Chapter 5. The prime number theorem. The Riemann zeta function.
• Chapter 6. Dirichlet series.
• Chapter 7. Primes in arithmetic progression. Dirichlet’s theorem.
Course Calendar:
• Lecture 1: The big picture of analytic number theory (2025/09/22)
• Lecture 2: A recap of elementary number theory. (2025/09/29)
• Lecture 3: Arithmetic functions and Dirichlet multiplication I (2025/10/13)
• Lecture 4: Arithmetic functions and Dirichlet multiplication II (2025/10/20)
• Lecture 5: Asymptotics and averages of arithmetic functions I (2025/10/27)
• Lecture 6: Asymptotics and averages of arithmetic functions II (2025/11/3)
• Lecture 7: Asymptotics and averages of arithmetic functions III (2025/11/10)
• Lecture 8: Elementary results on the distribution of primes I (2025/11/17)
• Lecture 9: Elementary results on the distribution of primes II (2025/11/24)
• Lecture 10: The prime number theorem I (2025/12/1)
• Lecture 11: The prime number theorem II (2025/12/8)
• Lecture 12: The prime number theorem III (2025/12/15)
• Lecture 13: Dirichlet series I (2025/12/22)
• Lecture 14: Dirichlet series II (2025/12/29)
• Lecture 15: Primes in arithmetic progression I (2026/1/5)
• Lecture 16: Primes in arithmetic progression II (2026/1/12)
Reference
[1] T. Apostol, Introduction to Analytic Number Theory, Springer Verlag, 1998.
[2] H.H. Chan, Analytic Number Theory for Undergraduates, World Scientific Publishing Co., 2009.
[3] J-M. De Konnick, F. Luca, Analytic Number Theory, Exploring the Anatomy of Integers, American Mathematical Society Press, 2012.
[4] H. Davenport, H.L. Montgomery, Multiplicative Number theory, Springer Verlag, 2000.
[5] A. Hildebrand, Introduction to Analytic Number Theory, lecture notes, MATH 531 course University of Illinois at Urbana Champaign, 2005.
[2] H.H. Chan, Analytic Number Theory for Undergraduates, World Scientific Publishing Co., 2009.
[3] J-M. De Konnick, F. Luca, Analytic Number Theory, Exploring the Anatomy of Integers, American Mathematical Society Press, 2012.
[4] H. Davenport, H.L. Montgomery, Multiplicative Number theory, Springer Verlag, 2000.
[5] A. Hildebrand, Introduction to Analytic Number Theory, lecture notes, MATH 531 course University of Illinois at Urbana Champaign, 2005.
Audience
Undergraduate
, Advanced Undergraduate
, Graduate
, Postdoc
, Researcher
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Cezar Lupu obtained his PhD degree from the University of Pittsburgh in 2018 with a thesis on special values of Riemann zeta and multiple zeta functions under the supervision of Piotr Hajlasz and William C. Troy. Between 2018-2021, he was a postdoctoral scholar at Texas Tech University under the mentorship of Razvan Gelca and Dermot McCarthy. In 2021, he moved to China as a postdoctoral fellow at the Beijing Institute of Mathematical Sciences and Applications (BIMSA) and Tsinghua University under the mentorship of Shing-Tung Yau until 2024. His main research interests are in the areas of number theory, analysis and special functions. Most of his recent research is centered around special values of L-functions and multiple zeta functions which play an important role at the interface of analysis, number theory, geometry and physics. He taught numerous courses at Pitt and TTU both undergraduate and graduate ranging from calculus and linear algebra to abstract algebra and real analysis. Also, he coached the best undergraduate students for the William Lowell Putnam Mathematical Competition. After moving to China, he taught courses at the Qiuzhen College, Tsinghua University. Together with other colleagues from Tsinghua University, he is organizing the Shadow Putnam Mathematical Competition at the Qiuzhen College. Moreover, starting 2023, he is the academic director of the International Mathematics Summer Camp (IMSC).