Number Theory
Instructor: Yu Liu and Dingxin Zhang
Course Introduction: This course studies how prime numbers are spread out among the integers. It starts from basic properties of numbers and uses only elementary methods, avoiding complex analysis, to understand patterns in the primes.
Contents:
1. Arithmetical and Generating Functions: Introduction to functions such as the Möbius function, Euler's totient function, and von Mangoldt function, and how generating functions help translate number problems into analytic ones.
2. Analytic Tools: Use of calculus, the Riemann zeta function in its elementary form, and estimation techniques to study arithmetical functions.
3. Elementary Theorems on Prime Distribution: Chebyshev’s bounds, introduction to the sieve method, the Shapiro Tauberian theorem, and estimates for the sum of reciprocals of primes.