Math 687R: Topics in Analytic Number Theory.

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Catalog Information


Topics in Analytic Number Theory.

Credit Hours



Math 352, 487; or equivalents.


Current topics of research interest.

Desired Learning Outcomes

Students should gain a familiarity with a particular area of analytic number theory selected by the instructor.


A knowledge of complex analysis and a first course in number theory, at the level provided by Math 352, Math 487, should suffice.

Minimal learning outcomes

These cannot be specified uniquely for a topics course. The following example gives a clear indication of the level of difficulty appropriate to the course. Students should know the technical terms, and be able to implement the methods taught in the course to work associated problems, including proving results of suitable accessibility.

  1. Riemann's memoir on the zeta function.
  2. The functional equation of the L functions.
  3. Properties of the gamma function.
  4. Integral functions of order 1.
  5. The infinite products for ξ(s) and ξ(s).
  6. Zero free regions for ζ(s) and L(s, χ).
  7. The counting functions N(T) and N(T, χ).
  8. The explicit formula for ψ(x) and the prime number theorem.
  9. The explicit formula for ψ(x, χ) and the prime number theorem for arithmetic progression.
  10. Siegel's theorem and application to prime numbers in arithmetic progressions.
  11. Vaughan's identity.
  12. The large sieve.
  13. The Bombieri-Vinogradov theorem.
  14. The Barban-Davenport-Halberstram theorem.
  15. Modular Forms


Possible textbooks for this course include (but are not limited to):

Additional topics

Courses for which this course is prerequisite