arXiv subjects

Boqing Xue

Publications and source records attributed to Boqing Xue.

24 records · Page 2Linked to original sources

arXiv2019

On a Hilbert Space Reformulation of Riemann Hypothesis

We explore Hilbert space reformulations of Riemann Hypothesis developed by Nyman, Beurling, Báez-Duarte, et. al. with a weighted Bergman space $\mathcal{H}=A_1^2(\mathbb{D})$, i.e., Riemann hypothesis holds if and only if the Hilbert subspace $\mathcal{H}_0$ spanned by a certain family of functions coincides with $\mathcal{H}$. A condition that a function does not belong to $\mathcal{H}_0^\bot$ is given. Moreover, it is proved that the von-Neumann algebra generated by a certain monoid $T_\mathbb{N}=\{T_k:\, k\in \mathbb{N}\}$ of operators is exactly $B(\mathcal{H})$. As a result, Riemann hypothesis is true if and only if $\mathcal{H}_0$ is $T_k^\ast$-invariant for all $k\in \mathbb{N}$.

math.NT↗

arXiv2019

Natural Monoids and Non-commutative Arithmetics

We introduce several classes of monoids satisfying up to five axioms and establish basic theories on their arithmetics. The one satisfying all the axioms is named natural monoid. Two typical examples are 1) the monoid $\mathbb{N}$ of natural numbers in the group of positive rationals and 2) a certain monoid $\mathbb{S}$ in one of Thompson's groups. The latter one is non-abelian, which serves as an important example for non-commutative arithmetics. Defining primes in a non-abelian monoid $S$ is highly non-trivial, which relies on a concept we called `castling'. Three types of castlings are essential to grasp the arithmetics on $S$. Multiplicative and completely multiplicative functions are defined. In particular, Möbius function is multiplicative, and Liouville function on a natural monoid is completely multiplicative. The divisor function has a sub-multiplicative property, which induces a non-trivial quantity $τ_0(u)=\lim\nolimits_{n\rightarrow \infty}(τ(u^n))^{1/n}$ in a non-abelian monoid $S$. Moreover, the quantity $Ç(S)=\sup\nolimits_{1\neq u\in S}τ_0(u)/τ(u)$ describes the complexity for castlings in $S$. We show that $Ç(\mathbb{N})=1/2$ and $Ç(\mathbb{S})=1$. The reduced $C^\ast$-algebra of $S$, on which a particular trace can be defined, is also studied. Furthermore, we prove that a natural monoid having finitely many primes is amenable.

math.NT↗

arXiv2015

General divisor functions in arithmetic progressions to large moduli

We prove a result on the distribution of the general divisor functions in arithmetic progressions to smooth moduli which exceed the square root of the length.

math.NT↗

arXiv2013

On sums of sparse prime subsets

For arbitrary $c_0>0$, if $A$ is a subset of the primes less than $x$ with cardinality $δx (\log x)^{-1}$ with $δ\geq (\log x)^{-c_0}$, then there exists a positive constant $c$ such that the cardinality of $A+A$ is larger than $c\, δx (\log\log x)^{-1}$.

math.NT↗

arXiv2013

Rational points on cubic hypersurfaces that split off two forms

We show that if $X\subseteq \mathbb{P}^{n-1}$, defined over $\mathbb{Q}$ by a cubic form that splits off two forms, with $n\geq 11$, then $X(\mathbb{Q})$ is non-empty. The same holds for an $(m_1,m_2)$-form with $m_1\geq 4$ and $m_2\geq 5$.

math.NT↗

arXiv2012

On sums of subsets of Chen primes

In this paper we show that if $A$ is a subset of Chen primes with positive relative density $α$, then $A+A$ must have positive upper density at least $cαe^{-c^\prime\log(1/α)^{2/3}(\log\log(1/α))^{1/3}}$ in the natural numbers.

math.NT↗