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Fan Ge

Publications and source records attributed to Fan Ge.

15 recordsLinked to original sources
arXiv2023

Real moments of the logarithmic derivative of characteristic polynomials in random matrix ensembles

We prove asymptotics for real moments of the logarithmic derivative of characteristic polynomials evaluated at $1-\frac{a}{N}$ in unitary, even orthogonal, and symplectic ensembles, where $a>0$ and $a=o(1)$ as the size $N$ of the matrix goes to infinity. Previously, such asymptotics were known only for integer moments (in the unitary ensemble by the work of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith, and in orthogonal and symplectic ensembles by the work of Alvarez and Snaith), except that in the odd orthogonal ensemble real moments asymptotics were obtained by Alvarez, Bousseyroux and Snaith. Our proof is new and does not make use of the aforementioned integer moments results, and is different from the method in the work of Alvarez et al for the odd orthogonal ensemble.

math-ph↗
arXiv2022

On the logarithmic derivative of characteristic polynomials for random unitary matrices

Let $U\in U(N)$ be a random unitary matrix of size $N$, distributed with respect to the Haar measure on $U(N)$. Let $P(z)=P_U(z)$ be the characteristic polynomial of $U$. We prove that for $z$ close to the unit circle, $ \frac{P'}{P}(z) $ can be approximated using zeros of $P$ very close to $z$, with a typically controllable error term. This is an analogue of a result of Selberg for the Riemann zeta-function. We also prove a mesoscopic central limit theorem for $ \frac{P'}{P}(z) $ away from the unit circle, and this is an analogue of a result of Lester for zeta.

math.NT↗
arXiv2022

Mean values of the logarithmic derivative of the Riemann zeta-function near the critical line

Assume the Riemann Hypothesis and a hypothesis on small gaps between zeta zeros, we prove a conjecture of Bailey, Bettin, Blower, Conrey, Prokhorov, Rubinstein and Snaith, which states that for any positive integer $K$ and real number $a>0$, \begin{align*} \lim_{a \to 0^+}\lim_{T \to \infty} \frac{(2a)^{2K-1}}{T (\log T)^{2K}} \int_{T}^{2T} \left|\frac{\zeta'}{\zeta}\left(\frac{1}{2}+\frac{a}{\log T}+it\right)\right|^{2K} dt = \binom{2K-2}{K-1}. \end{align*} When $K=1$, this was essentially a result of Goldston, Gonek and Montgomery.

math.NT↗
arXiv2021

Keywords Guided Method Name Generation

High quality method names are descriptive and readable, which are helpful for code development and maintenance. The majority of recent research suggest method names based on the text summarization approach. They take the token sequence and abstract syntax tree of the source code as input, and generate method names through a powerful neural network based model. However, the tokens composing the method name are closely related to the entity name within its method implementation. Actually, high proportions of the tokens in method name can be found in its corresponding method implementation, which makes it possible for incorporating these common shared token information to improve the performance of method naming task. Inspired by this key observation, we propose a two-stage keywords guided method name generation approach to suggest method names. Specifically, we decompose the method naming task into two subtasks, including keywords extraction task and method name generation task. For the keywords extraction task, we apply a graph neural network based model to extract the keywords from source code. For the method name generation task, we utilize the extracted keywords to guide the method name generation model. We apply a dual selective gate in encoder to control the information flow, and a dual attention mechanism in decoder to combine the semantics of input code sequence and keywords. Experiment results on an open source dataset demonstrate that keywords guidance can facilitate method naming task, which enables our model to outperform the competitive state-of-the-art models by margins of 1.5\%-3.5\% in ROUGE metrics. Especially when programs share one common token with method names, our approach improves the absolute ROUGE-1 score by 7.8\%.

cs.SE↗
arXiv2020

Solution to the index conjecture in zero-sum theory

A problem in zero-sum theory is to determine all pairs $(k,n)$ for which every minimal zero-sum sequence of length $k$ modulo $n$ has index $1$. While all other cases have been solved more than a decade ago, the case when $k$ equals $4$ and $n$ is coprime to $6$ remains open. Precisely, The Index Conjecture in this subject states that if $n$ is coprime to $6$ then every minimal zero-sum sequence of length $4$ modulo $n$ has index $1$. In this paper, we prove an equivalent version of this conjecture for all $n>N$ for some absolute constant $N$.

math.CO↗
arXiv2020

Note on the number of zeros of $\zeta^{(k)}(s)$

Assuming the Riemann hypothesis, we prove that $$ N_k(T) = \frac{T}{2\pi}\log \frac{T}{4\pi e} + O_k\left(\frac{\log{T}}{\log\log{T}}\right), $$ where $N_k(T)$ is the number of zeros of $\zeta^{(k)}(s)$ in the region $0<\Im s\le T$. We further apply our method and obtain a zero counting formula for the derivative of Selberg zeta functions, improving earlier work of Luo.

math.NT↗
arXiv2018

Some multidimensional integrals in number theory and connections with the Painlev\'e V equation

We study piecewise polynomial functions $\gamma_k(c)$ that appear in the asymptotics of averages of the divisor sum in short intervals. Specifically, we express these polynomials as the inverse Fourier transform of a Hankel determinant that satisfies a Painlev\'e V equation. We prove that $\gamma_k(c)$ is very smooth at its transition points, and also determine the asymptotics of $\gamma_k(c)$ in a large neighbourhood of $k=c/2$. Finally, we consider the coefficients that appear in the asymptotics of elliptic Aliquot cycles.

math.NT↗
arXiv2016

On the index conjecture in zero-sum theory: singular case

Let $S=(a_1)\cdots(a_k)$ be a minimal zero-sum sequence over a finite cyclic group $G$. The index conjecture states that if $k=4$ and $\gcd(|G|,6)=1$, then $S$ has index $1$. In this paper we prove that if $S$ is singular then the index of $S$ is $1$.

math.NT↗
arXiv2016

On a permutation problem for finite abelian groups

Let $G$ be a finite additive abelian group with exponent $n>1$, and let $a_1,\ldots,a_{n-1}\in G$. We show that there is a permutation $\sigma\in S_{n-1}$ such that all the elements $sa_{\sigma(s)}\ (s=1,\ldots,n-1)$ are nonzero if and only if $$\left|\left\{1\le s<n:\ \frac{n}{d}a_s\ne 0\right\}\right|\ge d-1\ \ \textrm{ for every positive divisor }\ d\ \textrm{ of }\ n.$$ When $G$ is the cyclic group $\mathbb Z/n\mathbb Z$, this confirms a conjecture of Z.-W. Sun.

math.NT↗
arXiv2015

The Number of Zeros of $\zeta'(s)$

Assuming the Riemann Hypothesis, we prove that $$ N_1(T) = \frac{T}{2\pi}\log \frac{T}{4\pi e} + O\bigg(\frac{\log T}{\log\log T}\bigg), $$ where $N_1(T)$ is the number of zeros of $\zeta'(s)$ in the region $0<\Im s\le T$.

math.NT↗
arXiv2015

The distribution of zeros of $\zeta'(s)$ and gaps between zeros of $\zeta(s)$

Assume the Riemann Hypothesis, and let $\gamma^+>\gamma>0$ be ordinates of two consecutive zeros of $\zeta(s)$. It is shown that if $\gamma^+-\gamma < v/ \log \gamma $ with $v<c$ for some absolute positive constant $c$, then the box $$ \{s=\sigma+it: 1/2<\sigma<1/2+v^2/4\log\gamma, \gamma\le t\le \gamma^+\} $$ contains exactly one zero of $\zeta'(s)$. In particular, this allows us to prove half of a conjecture of Radziwi{\l}{\l} in a stronger form. Some related results on zeros of $\zeta(s)$ and $\zeta'(s)$ are also obtained.

math.NT↗
arXiv2009

On some universal sums of generalized polygonal numbers

For $m=3,4,\ldots$ those $p_m(x)=(m-2)x(x-1)/2+x$ with $x\in\mathbb Z$ are called generalized $m$-gonal numbers. Sun [13] studied for what values of positive integers $a,b,c$ the sum $ap_5+bp_5+cp_5$ is universal over $\mathbb Z$ (i.e., any $n\in\mathbb N=\{0,1,2,\ldots\}$ has the form $ap_5(x)+bp_5(y)+cp_5(z)$ with $x,y,z\in\mathbb Z$). We prove that $p_5+bp_5+3p_5\,(b=1,2,3,4,9)$ and $p_5+2p_5+6p_5$ are universal over $\mathbb Z$, as conjectured by Sun. Sun also conjectured that any $n\in\mathbb N$ can be written as $p_3(x)+p_5(y)+p_{11}(z)$ and $3p_3(x)+p_5(y)+p_7(z)$ with $x,y,z\in\mathbb N$; in contrast, we show that $p_3+p_5+p_{11}$ and $3p_3+p_5+p_7$ are universal over $\mathbb Z$. Our proofs are essentially elementary and hence suitable for general readers.

math.NT↗