Victor Z. Guo

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20152026

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math.NT2026

An explicit elementary proof of no Landau--Siegel zeros

Victor Zhenyu Guo

We give an elementary proof of the absence of Landau--Siegel zeros with the explicit constant $c = 1/100$ . For every primitive quadratic Dirichlet character $χ$ of conductor $q \geq 3$ ,…

math.NT2022

Consecutive Piatetski-Shapiro primes based on the Hardy-Littlewood conjecture

Victor Z. Guo, Yuan Yi

The Piatetski-Shapiro sequences are of the form $N^{(c)} := (⌊ n^{c} ⌋)_{n = 1}^{\infty}$ with $c > 1, c \neq \in N$ . In this paper, we study the distribu…

math.NT2021

Beatty primes from fractional powers of almost-primes

Victor Zhenyu Guo, Jinjiang Li, Min Zhang

Let $α > 1$ be irrational and of finite type, $β \in R$ . In this paper, it is proved that for $R ⩾ 13$ and any fixed $c \in (1, c_{R})$ , there exist infinitely many primes in…

math.NT2021

Piatetski-Shapiro primes in the intersection of multiple Beatty sequences

Victor Zhenyu Guo, Jinjiang Li, Min Zhang

Suppose that $α_{1}, α_{2}, β_{1}, β_{2} \in R$ . Let $α_{1}, α_{2} > 1$ be irrational and of finite type such that $1, α_{1}^{- 1}, α_{2}^{- 1}$ are linearly independent over $Q$ . Le…

math.NT2016

Consecutive primes and Beatty sequences

William D. Banks, Victor Z. Guo

Fix irrational numbers $α, α^> 1$ of finite type and real numbers $β, β^\geq 0$ , and let $B$ and $B^$ be the Beatty sequences $$ B:=(\lfloorαm+β\rfloor)_{m\ge 1}\quad\text{an…

math.NT2015

Quadratic nonresidues below the Burgess bound

William D. Banks, Victor Z. Guo

For any odd prime number $p$ , let $(\cdot ∣ p)$ be the Legendre symbol, and let $n_{1} (p) < n_{2} (p) < \dots$ be the sequence of positive nonresidues modulo $p$ , i.e., $(n_{k} ∣ p) = - 1$ for each…

math.NT2015

Some arithmetic properties of numbers of the form $⌊ p^{c} ⌋$

William D. Banks, Victor Z. Guo, Igor E. Shparlinski

Let $P^{c} = (⌊ p^{c} ⌋)_{p \in P} (c > 1, c \neq \in N),$ where $P$ is the set of prime numbers, and $⌊ \cdot ⌋$ is the…

math.NT2015

Piatetski-Shapiro Primes in a Beatty Sequence

Victor Z. Guo

Let $α, β$ be real numbers such that $α > 1$ is irrational and of finite type, and let $c$ be a real number in the range $1 < c < \frac{14}{13}$ . In this paper, it is shown that there are…

An explicit elementary proof of no Landau--Siegel zeros

Consecutive Piatetski-Shapiro primes based on the Hardy-Littlewood conjecture

Beatty primes from fractional powers of almost-primes

Piatetski-Shapiro primes in the intersection of multiple Beatty sequences

Consecutive primes and Beatty sequences

Quadratic nonresidues below the Burgess bound

Some arithmetic properties of numbers of the form ⌊pc⌋

Piatetski-Shapiro Primes in a Beatty Sequence

Some arithmetic properties of numbers of the form $⌊ p^{c} ⌋$