About

Dorian Goldfeld is a professor in the Mathematics Department at Columbia University, located in New York, NY. His research interests focus on Number Theory, a branch of pure mathematics devoted to the study of the integers and integer-valued functions. Professor Goldfeld has contributed to the field through his work on trace formulae for SL(2,R), as indicated by his authored book on the subject. He is actively involved in the academic community, participating in and organizing seminars and conferences such as the Joint COLUMBIA-CUNY-NYU Number Theory Seminar and the JNT Biennial Conferences held in 2019, 2022, and 2024. Additionally, he has engaged with workshops like the Bretton Woods Workshop on Multiple Dirichlet Series in 2005. Beyond research, Professor Goldfeld is also dedicated to teaching, offering courses such as MATH UN3025, Making, Breaking Codes, demonstrating his commitment to educating students in mathematical theory and applications.

Research topics

• Computer Science
• Mathematics
• Pure mathematics
• Computer Security
• Theoretical computer science
• Geology
• Physics
• Philosophy
• Paleontology
• Geometry
• Quantum mechanics
• Computer engineering
• Mathematical analysis
• Computer hardware
• Algorithm
• Computer network

Selected publications

• Multiplicativity of Fourier coefficients of Maass forms for $\operatorname{SL}(n,\mathbb{Z})$

  Rendiconti Lincei Matematica e Applicazioni · 2026-03-04

  articleOpen access1st authorCorresponding

  The Fourier coefficients of a Maass form \phi for \operatorname{SL}(n,\mathbb{Z}) are complex numbers A_{\phi}(M) , where M=(m_{1},m_{2},\dots,m_{n-1}) and m_{1},m_{2},\dots,m_{n-1} are non-zero integers. It is well known that coefficients of the form A_{\phi}(m_{1},1,\dots,1) are eigenvalues of the Hecke algebra and are multiplicative. We prove that the more general Fourier coefficients A_{\phi}(m_{1},\dots,m_{n-1}) are also eigenvalues of the Hecke algebra and satisfy the multiplicativity relations A_{\phi}(m_{1}m_{1}',m_{2}m_{2}',\dots,m_{n-1}m_{n-1}')=A_{\phi}(m_{1},m_{2},\dots,m_{n-1})\cdot A_{\phi}(m_{1}',m_{2}',\dots,m_{n-1}') provided the products \prod_{i=1}^{n-1}m_{i} and \prod_{i=1}^{n-1}m_{i}' are relatively prime to each other.

  PublisherDOI

• An asymptotic orthogonality relation forGL(n, ℝ)

  Algebra & Number Theory · 2025-09-14

  articleOpen access1st authorCorresponding
  PublisherOA PDFDOI

• Multiplicativity of Fourier Coefficients of Maass Forms for SL($n,\mathbb Z$)

  ArXiv.org · 2025-02-05

  preprintOpen access1st authorCorresponding

  The Fourier coefficients of a Maass form $ϕ$ for SL$(n,\mathbb Z)$ are complex numbers $A_ϕ(M)$, where $M=(m_1,m_2,\ldots,m_{n-1})$ and $m_1,m_2,\ldots ,m_{n-1}$ are nonzero integers. It is well known that coefficients of the form $A_ϕ(m_1,1,\ldots,1)$ are eigenvalues of the Hecke algebra and are multiplicative. We prove that the more general Fourier coefficients $A_ϕ(m_1,\ldots,m_{n-1})$ are also eigenvalues of the Hecke algebra and satisfy the multiplicativity relations $$A_ϕ\big(m_1m_1',\;m_2m_2', \;\ldots\; m_{n-1}m_{n-1}'\big) = A_ϕ\big(m_1,m_2,\ldots,m_{n-1})\cdot A_ϕ(m_1',m_2',\ldots,m_{n-1}'\big)$$ provided the products $\prod\limits_{i=1}^{n-1} m_i$ and $\prod\limits_{i=1}^{n-1} m_i'$ are relatively prime to each other.

  PublisherOA PDFDOI

• An Analogue of the Dedekind Eta Function for Hecke Groups $H(\sqrt{D})$

  ArXiv.org · 2025-08-28

  preprintOpen access

  Let $D\equiv 1\bmod{4}$ be a fundamental discriminant of a real quadratic field. We construct an analogue of the classical Dedekind eta function for the Hecke group $H(\sqrt{D})$. This gives rise to a new family of holomorphic modular functions for $H(\sqrt{D})$ which vanish at the cusp at $\infty$. We establish results on the asymptotic growth and sign patterns of the Fourier coefficients associated to these modular forms.

  PublisherOA PDFDOI

• The shifted convolution L-function for Maass forms

  Research in Number Theory · 2024-10-16 · 1 citations

  article1st authorCorresponding
  PublisherDOI

• Preface to “Proceedings of the 2nd JNT Biennial Conference, 2022”

  Journal of Number Theory · 2024-12-03

  article1st authorCorresponding
  PublisherDOI

• ℵ-structures: One-way Actions via Holomorphs and Split Extensions with Cryptographic Applications

  Series on computers and operations research · 2023-06-01

  book-chapter
  PublisherDOI

• Number theoretical locally recoverable codes

  Journal of Algebra and Its Applications · 2023-09-16

  article

  In this paper, we give constructions for infinite sequences of finite nonlinear locally recoverable codes [Formula: see text] over a product of finite fields arising from basis expansions in algebraic number fields. The codes in our sequences have increasing length and size, constant rate, fixed locality, and minimum distance going to infinity.

  PublisherDOI

• The shifted convolution L-function for Maass forms

  arXiv (Cornell University) · 2023-11-11

  preprintOpen access1st authorCorresponding

  Let $Φ_1,Φ_2$ be Maass forms for $\text{SL}(2,\mathbb Z)$ with Fourier coefficients $C_1(n),C_2(n)$. For a positive integer $h$ the meromorphic continuation and growth in $s\in\mathbb C$ (away from poles) of the shifted convolution L-function $$L_h(s,{Φ_1,Φ_2})\, := \sum_{n \neq 0,-h} {C_1(n) C_2(n + h)} \cdot \big|n(n + h)\big|^{-\frac{1}{2}s}$$ is obtained. For ${\rm Re}(s) > 0$ it is shown that the only poles are possible simple poles at $\frac{1}{2} \pm ir_k$, where $\tfrac14+r_k^2$ are eigenvalues of the Laplacian. As an application we obtain, for $T\to\infty$, the asymptotic formula \begin{align*} & \underset{n \neq 0,-h}{\sum_{\sqrt{|n (n + h)|}

  PublisherOA PDFDOI

• The first coefficient of Langlands Eisenstein series for $\hbox{SL}(n,\mathbb Z)$

  arXiv (Cornell University) · 2023-03-09 · 1 citations

  preprintOpen access1st authorCorresponding

  Fourier coefficients of Eisenstein series figure prominently in the study of automorphic L-functions via the Langlands-Shahidi method, and in various other aspects of the theory of automorphic forms and representations. In this paper, we define Langlands Eisenstein series for ${\rm SL}(n,\mathbb Z)$ in an elementary manner, and then determine the first Fourier coefficient of these series in a very explicit form. Our proofs and derivations are short and simple, and use the Borel Eisenstein series as a template to determine the first Fourier coefficient of other Langlands Eisenstein series.

  PublisherOA PDFDOI

Recent grants

• Collaborative Research: FRG: Applications of Multiple Dirichlet Series to Analytic Number Theory

  NSF · $312k · 2004–2008

• Collaborative Research: Automorphic forms, representations and L-functions

  NSF · $195k · 2010–2014

Frequent coauthors

• Joseph Hundley

  University at Buffalo, State University of New York

  24 shared

• Iris Anshel

  22 shared

• Paul E. Gunnells

  18 shared

• Jeffrey Hoffstein

  Providence College

  14 shared

• Derek Atkins

  11 shared

• Michael Anshel

  9 shared

• Michael Woodbury

  Rutgers, The State University of New Jersey

  8 shared

• Giacomo Micheli

  7 shared

Education

• Ph.D., Number Theory

  Columbia University

  1969