arXiv subjects
Publications and source records attributed to Zhenchao Ge.
We prove a quantitative non-covering theorem for finite nilpotent groups. As a consequence, every finite covering of $\Z^d$ by affine full-rank sublattices with pairwise distinct indices contains a member of bounded index, uniformly in $d$. We also establish a summability criterion allowing repeated indices, which extends to finite groups containing a normal nilpotent subgroup of bounded index.
We describe a new family of Dirichlet L-functions that provably have an unusual value distribution. Experimentally, we find that the L-functions in this family have a significant and previously undetected bias in the distribution of gaps between their zeros. These phenomena appear to have an arithmetic explanation that corresponds to the nonvanishing of a certain Gauss-type sum. We give a complete classification of the characters for which this sum is nonzero and count them. This sum vanishes for 100% of primitive Dirichlet characters, so L-functions in our newly discovered family are rare (a set of density zero among all primitive characters). The nonvanishing of this sum implies that there is a main term when one integrates the Z-function associated to a Dirichlet L-function over a long interval. We prove that the L-functions in this family are the only GL(1)/Q or GL(2)/Q cuspidal automorphic L-functions whose Z-functions have a "large" mean value when integrated. We discuss analogous results for the Riemann zeta-function.
Let $\mathbb F_q$ be the finite field of $q$ elements having characteristic $p$, and denote by $\mathbb K_\infty=\mathbb F_q((1/t))$ the field of formal Laurent series in $1/t$. We consider the equidistribution in $\mathbb T=\mathbb K_\infty/\mathbb F_q[t]$ of the values of polynomials $f(u)\in \mathbb K_\infty [u]$ as $u$ varies over $\mathbb F_q[t]$. Let $\mathcal K$ be a finite set of positive integers, and suppose that $α_r\in \mathbb K_\infty$ for $r\in \mathcal K\cup \{0\}$. We show that the polynomial $\sum_{r\in \mathcal K\cup\{0\}}α_ru^r$ is equidistributed in $\mathbb T$ whenever $α_k$ is irrational for some $k\in \mathcal K$ satisfying $p\nmid k$, and also $p^vk\not\in \mathcal K$ for any positive integer $v$. This conclusion resolves in full a conjecture made jointly by the third, fourth and fifth authors.