Mathematics Building, Room 422
Columbia University
New York, NY 10027 
(212) 854-4304
       Fax: (212) 854-8962


Research Interests:  Number Theory curriculum vitae
Some Preprints: (pdf files
An additive prime divisor function of Alladi and Erdős
A standard zero free region for Rankin-Selberg L-functions (with X. Li)

A converse theorem for double Dirichlet series and Shintani zeta functions
                                                   (with N. Diamantis)
On the determination of the Plancherel measure for Lebedev-Whittaker transforms on GL(n)
(with A. Kontorovich)
Fourier expansions of GL(2) newforms at various cusps
                                        (with J. Hundley and M. Lee)
Natural boundaries and the correct notion of integral moments of L-functions
                                                            (with A. Diaconu and P. Garrett)
Moments for L-functions for GL(r) x GL(r-1) (with A. Diaconu and P. Garrett)  
A converse theorem for double Dirichlet series (with N. Diamantis)
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 

Automorphic Representations and L-Functions for the General Linear Group, Volume 1
Automorphic Representations and L-Functions for the General Linear Group, Volume 2
Automorphic Forms and L-Functions for the Group GL(n,R) 
Blog with corrections of Errors in the above books

First  JNT Biennial Conference   (July 22 - 26, 2019)
                    (Goldfeld Conference photos)

                    Joint COLUMBIA-CUNY-NYU Number Theory Seminar
                    Bretton Woods Workshop on Multiple Dirichlet Series(2005)    (Conference photos)
                    AAG key exchange
                    Edinburgh Workshop on Multiple Dirichlet Series(2008)
                    Decision Regarding World Record Musky Challenge
                    My Daughter's art work (including my portrait above)
                    Some Interesting Chemistry
                    Very accurate clock


  TR 1:10 - 2:25  Location: Room 507 Mathematics Building

RECOMMENDED (not required) Textbooks:

Automorphic forms and L-functions for the group GL(n, R),  by D. Goldfeld
Cambridge University Press (2015).

This course will focus on recent developments in the spectral theory of automorphic forms
on GL(n,R) with n > 2.  I will begin by reviewing the basic theory of automorphic forms
and L-functions on the upper half plane H^n.  One of the main goals is to present new
methods to derive the Fourier expansion of Langlands Eisenstein series twisted by cusp
forms of lower rank and then obtain spectral expansions of L^2 automorphic forms into
cusp forms, Langlands Eisenstein series, and residues of Langlands Eisenstein series.  We
will then give applications that have been developed recently to the trace formula, spectral
reciprocity, special values of L-functions, and other topics.