Schedule

The seminar will discuss topics in "analytic" number theory. It will consist mostly of presentations by the students.

Possible Topics:

• Arithmetic functions and function field arithmetic

  1. Arithmetic functions, Dirichlet convolution, Moebius inversion
  2. The Prime Number Theorem for F_q[x].

• Binary quadratic forms

  1. binary quadratic forms, the modular group, reduction theory, the class number.
  2. Unique factorization in imaginary quadratic fields.
  3. Dirichlet's class number formula.

• Primes in arithmetic progression

  1. Dirichlet characters, orthogonality relations; Dirichlet L-functions, analytic continuation to Re(s)>0.
  2. Infinite products and the Euler product for ζ(s) and L(s, χ ).
  3. Primes in arithmetic progressions and the connection to L(1, χ ).

• Sieve methods

  1. The sieve of Erathosthenes
  2. Selberg's upper bound sieve
  3. Applications: An upper bound for the number of twin primes up to x.

• The zeros of Riemann's zeta function

  1. The Gamma function and its properties, Stirling's formula.
  2. The Fourier transform and Poisson summation.
  3. The Riemann zeta function: analytic continuation and functional equation.
  4. Entire functions of finite order and Weierstrass products
  5. The zeros of ζ(s)- basics.
  6. The asymptotic formula for N(T), the number of zeros up to height T.
  7. Relating zeros and primes: Riemann's explicit formula
  8. The prime number theorem - an overview

Prerequisites:

1. Introduction to Number Theory.
2. complex variables

References

• H. Davenport, Multiplicative Number Theory
• T. Apostol, Introduction to Analytic Number Theory
• M. Ram Murty, Problems in Analytic Number Theory
• D. Cox, Primes of the form x² + n y²
• H. Cohn, Advanced Number Theory

Office : Schreiber 316, tel: 640-7806