Zhuo Zhang's webpage

A Uniform Chebotarev Density Theorem with Artin’s Holomorphy Conjecture, with J. Thorner. [pdf]

The Chebotarev Density Theorem is a generalization of the prime number theorem in arithmetic progressions and plays an important role in algebraic number theory. In this project we improved the uniformity and the rate of convergence of the asymptotic Chebotarev density theorem. Our proof relies on a new zero free region and a zero density estimate for Artin L-functions that satisfy the Artin holomorphy conjecture. As a corollary, we improved the best-known upper bound on the least norm of a prime ideal with given Artin symbol in Galois extensions of number fields.

On a Special Metric in Cyclotomic Fields, with with K. Saettone, A. Zaharescu. [pdf]

In our work, we introduced a special metric in the cyclotomic fields. We proved a number of good properties of this metric, such as invariance under the Galois group, as well as an analog of Krasner’s lemma in algebraic number theory. Finally, we proved that almost all points in the cyclotomic field are almost equi-distanced from each other under this metric.

Pattern formation statistics on Fermat quotients, with C. Cobeli, A. Zaharescu. [link will come soon]

Fermat quotients form an important sequence in number theory. Heath-Brown proved that they are uniformly distributed. In our work we used Heath-Brown’s bound, exponential sums, and Erdos-Turan type bounds to establish some further uniform distribution properties of Fermat quotients.

An elementary characterization of the Gauss-Kuzmin distribution in the theory of continued fractions, with A.J. Hildebrand, S. Singh. [link will come soon]

The Gauss-Kuzmin measure is a continuous probability measure on [0,1] that controls the frequency of any given string of positive integers in the continued fraction expansion of almost all real numbers. In our work, we proved that the Gauss-Kuzmin measure is uniquely characterized by a certain symmetry property. Furthermore, this is a provably optimal characterization. We also proved various facts regarding this characterization.

On the continued fraction expansion of almost all real numbers, with A.J. Hildebrand, A. Jin, S. Singh, accepted for publication in Involve. [pdf]

For certain subsets of the natural numbers, we establish simple explicit formulas for the frequency with which the continued fraction expansion of a random real number contains a digit from the set. A similar problem was solved for strings of consecutive identical digits. Finally, we compare the frequencies predicted by these results with actual frequencies found among the first 300 million continued fraction digits of π, and we provide strong statistical evidence that the continued fraction expansion of π behaves like that of a random real number.