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Hieu T. Ngo

Publications and source records attributed to Hieu T. Ngo.

9 recordsLinked to original sources
arXiv2025

Simultaneous nonvanishing of Dirichlet $L$-functions in Galois orbits

Under the Generalized Riemann Hypothesis, we prove that given any two distinct imprimitive Dirichlet characters $η_1, η_2$ modulo $q=p^k$, a positive proportion of characters $χ$ modulo $q$ in a fixed Galois orbit of primitive characters satisfies the nonvanishing property that $L(1/2,χη_1) L(1/2,χη_2) \neq 0$, as $k \to \infty$ (with $p$ fixed). Previously, only a positive proportion of nonvanishing result was available in Galois orbits (as opposed to simultaneously nonvanishing), due to work of Khan, Milićević and Ngo. The main ingredients are obtaining a sharp upper bound on the mollified fourth moment over the Galois orbit using an Euler product mollifier, and obtaining a lower bound for the mollified second moment, which relies on using results from Diophantine approximation (such as the $p$-adic Roth theorem). We also unconditionally compute the second moments for $L$--functions associated to primitive Dirichlet characters in full orbits and thinner orbits.

math.NT↗
arXiv2022

Expanders on matrices over a finite chain ring, I

In this work and its sequel, we study the expanding phenomenon of matrices over a finite chain ring of large residue field. A sum-product estimate is proved. It is showed that $x+yz$ is a moderate expander on $n\times n$ matrices with exponent $\frac{n+1}{6}$. These results generalise the main theorems in a recent work of Xie and Ge. The proofs use spectral graph theory and elementary divisor theory.

math.CO↗
arXiv2021

On roots of quadratic congruences

The equidistribution of roots of quadratic congruences with prime moduli depends crucially upon effective bounds for a special Weyl linear form. Duke, Friedlander and Iwaniec discovered a strong estimate for this Weyl linear form when the quadratic polynomial has negative discriminant. Tóth established an analogous but weaker bound when the quadratic polynomial has positive discriminant. We obtain a stronger estimate for the Weyl linear form for quadratics of positive discriminants.

math.NT↗
arXiv2020

Nonvanishing of Dirichlet L-functions, II

We show that for at least $\frac{5}{13}$ of the primitive Dirichlet characters $χ$ of large prime modulus, the central value $L(\frac{1}{2},χ)$ does not vanish, improving on the previous best known result of $\frac{3}{8}$.

math.NT↗
arXiv2015

Non-vanishing of Dirichlet L-functions in Galois orbits

A well known result of Iwaniec and Sarnak states that for at least one third of the primitive Dirichlet characters to a large modulus q, the associated L-functions do not vanish at the central point. When q is a large power of a fixed prime, we prove the same proportion already among the primitive characters of any given order. The set of primitive characters modulo q of a given order can be described as an orbit under the action of the Galois group of the corresponding cyclotomic field. We also prove a positive proportion of nonvanishing within substantially shorter orbits generated by intermediate Galois groups as soon as they are larger than roughly the square-root of the prime-power conductor.

math.NT↗
arXiv2013

Renormalization and quantum modular forms, part I: Maass wave forms

Sander Zwegers showed that Ramanujan's mock theta functions are $q$-hypergeometric series, whose $q$-expansion coefficients are half of the Fourier coefficients of a non-holomorphic modular form. George Andrews, Henri Cohen, Freeman Dyson, and Dean Hickerson found a pair of $q$-hypergeometric series each of which contains half of the Fourier coefficients of Maass waveform of eigenvalue $1/4$. This series of papers shows that a $q$-series construction, called ``renormalization'', yields the other half of the Fourier coefficients from a series which contains half of them. This construction unifies examples associated with mock theta functions and examples associated with Maass waveforms. Thus confirming a conviction of Freeman Dyson. This construction is natural in the context of Don Zagier's quantum modular forms. Detailed discussion of the role quantum modular forms play in this construction is given. New examples associated to Maass waveforms are given in Part I. Part II contains new examples associated with mock theta functions, and classical modular forms. Part II contains an extensive survey of the ``renormalization'' construction. A large number of examples and open questions which share similarities to the main examples, but remain mysterious, are given.

math.NT↗
arXiv2013

Renormalization and quantum modular forms, part II: Mock theta functions

Sander Zwegers showed that Ramanujan's mock theta functions are $q$-hypergeometric series, whose $q$-expansion coefficients are half of the Fourier coefficients of a non-holomorphic modular form. George Andrews, Henri Cohen, Freeman Dyson, and Dean Hickerson found a pair of $q$-hypergeometric series each of which contains half of the Fourier coefficients of Maass waveform of eigenvalue $1/4$. This series of papers shows that a $q$-series construction, called ``renormalization'', yields the other half of the Fourier coefficients from a series which contains half of them. This construction unifies examples associated with mock theta functions and examples associated with Maass waveforms. Thus confirming a conviction of Freeman Dyson. This construction is natural in the context of Don Zagier's quantum modular forms. Detailed discussion of the role quantum modular forms play in this construction is given. New examples associated to Maass waveforms are given in Part I. Part II contains new examples associated with mock theta functions, and classical modular forms. Part II contains an extensive survey of the ``renormalization'' construction. A large number of examples and open questions which share similarities to the main examples, but remain mysterious, are given.

math.NT↗