F. Ge

activity

20092023

most citedSome multidimensional integrals in number theory and connections with the Painlevé V equation

9 citations · 19 across the 14 of their papers we have counts for

collaborators

14 papers

math-ph2023★ 1 cited

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

Fan Ge

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…

math.NT2022★ 1 cited

On the logarithmic derivative of characteristic polynomials for random unitary matrices

Fan Ge

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…

math.NT2022

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

Fan Ge

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 st…

math.CO2020

Solution to the index conjecture in zero-sum theory

Fan Ge

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 sol…

math.NT2020

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

Fan Ge, Ade Irma Suriajaya

Assuming the Riemann hypothesis, we prove that $N_{k} (T) = \frac{T}{2 π} lo g \frac{T}{4 π e} + O_{k} (\frac{lo g T}{lo g lo g T}),$ where $N_{k} (T)$ is the number of zeros of…

math.CO2020★ 5 cited

A combinatorial identity and the finite dual of infinite dihedral group algebra

Fan Ge, Gongxiang Liu

In this note, we find a combinatorial identity which is closely related to the multi-dimensional integral $γ_{m}$ in the study of divisor functions. As an application, we determine…

math.NT2018★ 9 cited

Some multidimensional integrals in number theory and connections with the Painlevé V equation

Estelle Basor, Fan Ge, Michael O. Rubinstein

We study piecewise polynomial functions $γ_{k} (c)$ that appear in the asymptotics of averages of the divisor sum in short intervals. Specifically, we express these polynomials as the…

math.NT2016

Note on the index conjecture in zero-sum theory and its connection to a Dedekind-type sum

Fan Ge

Let $S = (a_{1}) \dots (a_{k})$ be a minimal zero-sum sequence over a finite cyclic group $G$ . The index conjecture states that if $k = 4$ and $g cd (∣ G ∣, 6) = 1$ , then $S$ has index 1. In this…

math.NT2016★ 3 cited

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

Fan Ge

Let $S = (a_{1}) \dots (a_{k})$ be a minimal zero-sum sequence over a finite cyclic group $G$ . The index conjecture states that if $k = 4$ and $g cd (∣ G ∣, 6) = 1$ , then $S$ has index $1$ . In th…

math.NT2016

Conditional estimates on small distances between ordinates of zeros of $ζ (s)$ and $ζ^{'} (s)$

Fan Ge

Let $β^{'} + i γ^{'}$ be a zero of $ζ^{'} (s)$ . In \cite{GYi} Garaev and Yıldırım proved that there is a zero $β + iγ$ of $ζ (s)$ with $γ^{'} - γ ≪ ∣ β^{'} - 1/2∣$ . Assuming RH, we improve this bou…

math.NT2016

On a permutation problem for finite abelian groups

Fan Ge, Zhi-Wei Sun

Let $G$ be a finite additive abelian group with exponent $n > 1$ , and let $a_{1}, \dots, a_{n - 1} \in G$ . We show that there is a permutation $σ \in S_{n - 1}$ such that all the elements $sa…

math.NT2015

The Number of Zeros of $ζ^{'} (s)$

Fan Ge

Assuming the Riemann Hypothesis, we prove that $N_{1} (T) = \frac{T}{2 π} lo g \frac{T}{4 π e} + O (\frac{lo g T}{lo g lo g T}),$ where $N_{1} (T)$ is the number of zeros of $ζ'(…

math.NT2015

The distribution of zeros of $ζ^{'} (s)$ and gaps between zeros of $ζ (s)$

Fan Ge

Assume the Riemann Hypothesis, and let $γ^{+} > γ > 0$ be ordinates of two consecutive zeros of $ζ (s)$ . It is shown that if $γ^{+} - γ < v / lo g γ$ with $v < c$ for some absolute positive const…

math.NT2009

On some universal sums of generalized polygonal numbers

Fan Ge, Zhi-Wei Sun

For $m = 3, 4, \dots$ those $p_{m} (x) = (m - 2) x (x - 1) /2 + x$ with $x \in Z$ are called generalized $m$ -gonal numbers. Sun [13] studied for what values of positive integers $a, b, c$ the s…

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

On the logarithmic derivative of characteristic polynomials for random unitary matrices

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

Solution to the index conjecture in zero-sum theory

Note on the number of zeros of ζ(k)(s)

A combinatorial identity and the finite dual of infinite dihedral group algebra

Some multidimensional integrals in number theory and connections with the Painlevé V equation

Note on the index conjecture in zero-sum theory and its connection to a Dedekind-type sum

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

Conditional estimates on small distances between ordinates of zeros of ζ(s) and ζ′(s)

On a permutation problem for finite abelian groups

The Number of Zeros of ζ′(s)

The distribution of zeros of ζ′(s) and gaps between zeros of ζ(s)

On some universal sums of generalized polygonal numbers

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

Conditional estimates on small distances between ordinates of zeros of $ζ (s)$ and $ζ^{'} (s)$

The Number of Zeros of $ζ^{'} (s)$

The distribution of zeros of $ζ^{'} (s)$ and gaps between zeros of $ζ (s)$