DORIAN GOLDFELD

    Professor
       
Mathematics Building, Room 422
       
Columbia University
       
New York, NY 10027 
     
       Telephone:
(212) 854-4304
       Fax: (212) 854-8962
       e-mail: goldfeld@columbia.edu

 

Research Interests:  Number Theory, curriculum vitae

Some Preprints: (pdf files)
Voronoi Formulas on GL(n) (with X. Li)
Second Moments of GL(2) L-Functions (with A. Diaconu)
Counting Congruence Subgroups (with A. Lubotzky and L. Pyber)
Multiple Dirichlet Series and Moments of Zeta and L-Functions
                                                 (with A. Diaconu and J. Hoffstein)
Modular Forms, Elliptic Curves, and the ABC Conjecture
The Gauss Class Number problem for Imaginary Quadratic Fields
The Elementary proof of the Prime Number Theorem,  An Historical Perspective        
                    Links:
                    L-functions and Automorphic forms (Goldfeld Fest)     (Conference photos)
                   
Automorphic Forms and L-Functions for the Group GL(n,R)
                    Joint COLUMBIA-CUNY-NYU Number Theory Seminar
                    Braid Group Cryptography
                    Bretton Woods Workshop on Multiple Dirichlet Series(2005)    (Conference photos)
                    Edinburgh Workshop on Multiple Dirichlet Series(2008)
                    Decision Regarding World Record Musky Challenge
                    Very accurate clock
            

Teaching, Spring 2009:

MATH W4007 Analytic Number Theory 3 pts.
(Tuesday/Thursday 2:40-3:55pm, Room 520 Math Building)


TA: Min Lee (Help Room Hours: Monday 11:00-12:00,  Wednesday 11:00-12:00 and 3:00-4:00)

Course Description: A one semeser course covering the theory of modular forms, zeta functions, L -functions, and the Riemann hypothesis. Particular topics covered include the Riemann zeta function, the prime number theorem, Dirichlet characters, Dirichlet L-functions, Siegel zeros, prime number theorem for arithmetic progressions, SL (2, Z) and subgroups, quotients of the upper half-plane and cusps, modular forms, Fourier expansions of modular forms, Hecke operators, L-functions of modular forms.

Homework:
#1: (due Tuesday, February 3)
Let G, G' be two cyclic groups which contain n, n' elements, respectively.  Construct all the characters of the direct product G x G' and show that these characters themselves form a group which is isomorphic to G x G'. In the same manner explicitly construct all possible Dirichlet characters (mod 35).

#2: (due Tuesday, February 10) (I) Let χ0 be the trivial character mod q, and let q1 be some factor of q. For  any character χ1 (mod q1) there is a character χ (mod q) defined by χ = χ0 χ1. Express L(s, χ) in terms of L(s, χ1). Conclude that L(1, χ) = 0 if and only if L(1, χ1) = 0. (II) A character (mod q) that cannot be obtained in this way from any character  mod a proper factor q1 | q (a factor other than q itself ) is called primitive. Show  that any Dirichlet character χ comes from a unique primitive character χ.  (III) Prove that if s has real part σ > 1 then ζ (2σ)/ζ (σ) < |L(s, χ)|  ≤ ζ (σ) for all Dirichlet characters χ.

#3: (due Tuesday, February 17)   HW3.pdf

#4: (due Tuesday, February 24) Show that   Γ(1/2) = √π  where Γ(s)  is the Gamma function. Show that  Γ(s)  has a simple pole at  s = -5. What is the residue? Use the previous two results (together with the functional equation of the Riemann zeta function ζ (s)) to evaluate ζ (−10).

#5: (due Tuesday, March 3) Assume the prime number theorem. Using summation by parts, show that the sum  1/2 + 1/3 + 1/5 + 1/7 + 1/11 + 1/13 + 1/17 + . . .  of the recipocals of the primes diverges. 

#6: (due Tuesday, March 10)   HW6.pdf

#7: (due Tuesday, March 31)   HW7.pdf

#8: (due Tuesday, April 7)   HW8.pdf

#9: (due Tuesday, April 21)   HW9.pdf

TAKE HOME FINAL EXAM (due Tuesday, May 5 at 3:00 P.M.)  FinalExam.pdf






...........
MATH G6660 Automorphic Representations 4.5 pts.
(Tuesday/Thursday 11:00-12:15, Room 405 Math Building)


Course Description: This course presents the Jacquet-Langlands theory of automorphic representations for GL(n). The stress will be to develop the theory with many concrete explicit examples.   Local theory - principal series representations, special representations, (g, K)-modules, the supercuspidal representations, tensor products of local representations, L-functions, Godement-Jacquet proof of the analytic continuation and functional equation of L-functions.