Research

Honors and Awards

Bateman Prize in Number Theory (outstanding research in number theory, 2017)

Bateman Fellowship in Number Theory (outstanding research in number theory, 2017)

Large bias for integers with prime factors in arithmetic progressions

2016, Urbana, IL

We study the distribution of the product of $k ~(\geq 2)$ distinct primes $p_1\cdots p_k\leq x$ with each prime factor in an arithmetic progression $p_j\equiv a_j \bmod q$, $(a_j, q)=1$ $(q \geq 3, 1\leq j\leq k)$. We show that, there are large biases towards some certain arithmetic progressions $\boldsymbol{a}=(a_1, \cdots, a_k)$, and such biases have connections with Mertens theorem and the least prime in arithmetic progressions.

Chebyshev's bias for products of k primes

2015-2016, Urbana, IL

For any k ≥ 1, we study the distribution of the difference between the number of integers n ≤ x with ѡ(n)=k or Ω(n)=k in two different arithmetic progressions, where ѡ(n) is the number of distinct prime factors of n and Ω(n) is the number prime factors of n counted with multiplicity. Under some reasonable assumptions, we show that, if k is odd, the integers with Ω(n)=k have preference for quadratic non-residue classes; and if k is even, such integers have preference for quadratic residue classes. This result confirms a conjecture of Hudson. However, the integers with ѡ(n)=k always have preference for quadratic residue classes. Moreover, as k increases, the biases become smaller and smaller for both of the two cases. This research is under the support of the Research Assistantship from my advisor.

Simultaneous distribution of the fractional parts of Riemann zeta zeros

2014-2015, Urbana, IL

Studied the limiting distribution of the fractional parts of the zeros of the Riemann zeta-function. Joint work with Prof. Ford and Prof. Zaharescu. Under the support of the Research Assistantship from my advisor.
The following graphs are the histograms of ({αγ}, {βγ}) for some special α's and β's.

The distribution of k-free numbers

2013-2014, Urbana, IL

Studied the distribution of the error term of the counting function for k-free numbers, proved an equivalent statement for the analog of weak Mertens conjecture, and made a conjecture on the maximal order of the error term. Research is under the support of the Research Assistantship from my advisor.

Farey Series and Mayer-Erdös Phenomenon

2013, Urbana, IL

Studied and generalized the Mayer-Erdös phenomenon to linear form in more than two variables. Joint work with Prof. Zaharescu.

Irrational factor of order k and its connection to k-free number

2013, Urbana, IL

Generalized the irrational factor to the general case of order k and gave asymptotic formulas for the average of them. Joint work with Dong Dong.

Research on Standard L-functions attached to Ikeda Lift of Siegel Cusp Forms and Zeta-functions of Graphs.

2010-2012, Jinan, Shandong, P. R. China