Zheng-Yu Tao

activity

20222026

collaborators

8 papers

math.NT2026

The weak Chinburg conjecture on Mahler measures

Xuejun Guo, Zhengyu Tao

For every negative fundamental discriminant $- f$ and every $k \geq 1$ , we construct a rational function $R_{f, 2 k} \in Q (x_{1}, \dots, x_{2 k})$ and a constant $r_{f,2k}\in\mathbb{…

math.NT2026

Density of Vanishing of Certain Eigenspaces of Cyclotomic Class Groups

Xuejun Guo, Zhengyu Tao

For an odd prime $p$ , let $A_{j}$ be the $ω^{j}$ -eigenspace of the $p$ -primary class group of $Q (ζ_{p})$ . Fix an even integer $d \geq 4$ , put $N = (p - 1) / d$ , and let $U_d=(\mathbb Z/d\…

math.CO2026

Counterexamples to Stanley's conjecture on dimer coverings

Xuejun Guo, Zhengyu Tao

Let $Q_{k} (x)$ be Stanley's explicit denominator for the dimer-covering generating function $F_{k} (x) = \sum_{n \geq 0} A_{k, n} x^{n}$ of $k \times n$ rectangles. Stanley conjectured in 1985 tha…

math.NT2024

Determinants of Mahler measures and special values of $L$ -functions

Detchat Samart, Zhengyu Tao

We consider Mahler measures of two well-studied families of bivariate polynomials, namely $P_{t} = x + x^{- 1} + y + y^{- 1} + t$ and $Q_{t} = x^{3} + y^{3} + 1 - 3 t x y$ , where $t$ is a complex…

math.NT2023

CM points, class numbers, and the Mahler measures of $x^{3} + y^{3} + 1 - k x y$

Zhengyu Tao, Xuejun Guo

We study the Mahler measures of the polynomial family $Q_{k} (x, y) = x^{3} + y^{3} + 1 - k x y$ using the method previously developed by the authors. An algorithm is implemented to search for CM…

math.NT2022

Mahler measures and $L$ -values of elliptic curves over real quadratic fields

Zhengyu Tao, Xuejun Guo, Tao Wei

A famous formula of Rodriguez Villegas shows that the Mahler measures $m (k)$ of $P_{k} (x, y) = x + 1/ x + y + 1/ y + k$ can be written as a Kronecker-Eisenstein series. We prove that the degree o…

math.NT2022

Some new Ramanujan-Sato series for $1/ π$

Tao Wei, Zhengyu Tao, Xuejun Guo

We derive 10 new Ramanujan-Sato series of $1/ π$ by using the method of Huber, Schultz and Ye. The levels of these series are 14, 15, 16, 20, 21, 22, 26, 35, 39.

math.CO2022

The eigenvectors-eigenvalues identity and Sun's conjectures on determinants and permanents

Xuejun Guo, Xin Li, Zhengyu Tao +1

In this paper, we prove a conjecture raised by Zhi-Wei Sun in 2018 by the eigenvectors-eigenvalues identity found by Denton, Parke, Tao and X. Zhang in 2019.

The weak Chinburg conjecture on Mahler measures

Density of Vanishing of Certain Eigenspaces of Cyclotomic Class Groups

Counterexamples to Stanley's conjecture on dimer coverings

Determinants of Mahler measures and special values of L-functions

CM points, class numbers, and the Mahler measures of x3+y3+1−kxy

Mahler measures and L-values of elliptic curves over real quadratic fields

Some new Ramanujan-Sato series for 1/π

The eigenvectors-eigenvalues identity and Sun's conjectures on determinants and permanents

Determinants of Mahler measures and special values of $L$ -functions

CM points, class numbers, and the Mahler measures of $x^{3} + y^{3} + 1 - k x y$

Mahler measures and $L$ -values of elliptic curves over real quadratic fields

Some new Ramanujan-Sato series for $1/ π$