arXiv subjects

Victor Z. Guo

Publications and source records attributed to Victor Z. Guo.

6 recordsLinked to original sources

arXiv2025

Improvements on exponential sums related to Piatetski-Shapiro primes

We prove a new bound to the exponential sum of the form $$ \sum_{h \sim H}δ_h \mathop{\sum_{m\sim M}\sum_{n\sim N}}_{mn\sim x}a_{m}b_{n}\e\big(αmn + h(mn + u)^γ\big), $$ by a new approach to the Type I sum. The sum can be applied to many problems related to Piatetski-Shapiro primes, which are primes of the form $\lfloor n^c \rfloor$. In this paper, we improve the admissible range of the Balog-Friedlander condition, which leads to an improvement to the ternary Goldbach problem with Piatetski-Shapiro primes. We also investigate the distribution of Piatetski-Shapiro primes in arithmetic progressions, Piatetski-Shapiro primes in the intersection of multiple Beatty sequences and so on.

math.NT↗

arXiv2025

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

The Piatetski-Shapiro sequences are of the form ${\mathcal{N}}^{(c)} := (\lfloor n^c \rfloor)_{n=1}^\infty$ with $c > 1, c \not\in \mathbb{N}$. In this paper, we study the distribution of pairs $(p, p^{\#})$ of consecutive primes such that $p \in {\mathcal{N}}^{(c_1)}$ and $p^{\#} \in {\mathcal{N}}^{(c_2)}$ for $c_1, c_2 > 1$ and give a conjecture with the prime counting functions of the pairs $(p, p^{\#})$. We give a heuristic argument to support this prediction which relies on a strong form of the Hardy-Littlewood conjecture. Moreover, we prove a proposition related to the average of singular series with a weight of a complex exponential function.

math.NT↗

arXiv2016

Consecutive primes and Beatty sequences

Fix irrational numbers $α,\hatα>1$ of finite type and real numbers $β,\hatβ\ge 0$, and let $B$ and $\hat B$ be the Beatty sequences $$ B:=(\lfloorαm+β\rfloor)_{m\ge 1}\quad\text{and}\quad\hat B:=(\lfloor\hatαm+\hatβ\rfloor)_{m\ge 1}. $$ In this note, we study the distribution of pairs $(p,p^\sharp)$ of consecutive primes for which $p\in B$ and $p^\sharp\in\hat B$. Under a strong (but widely accepted) form of the Hardy-Littlewood conjectures, we show that $$ \big|\{p\le x:p\in B\text{ and }p^\sharp\in\hat B\}\big|=(α\hatα)^{-1}π(x)+O\big(x(\log x)^{-3/2+ε}\big), $$ where $π(x)$ is the prime counting function.

math.NT↗

arXiv2015

Quadratic nonresidues below the Burgess bound

For any odd prime number $p$, let $(\cdot|p)$ be the Legendre symbol, and let $n_1(p) 0$. In this paper, we prove that the stronger bound $$ n_k(p)\ll p^{(4\sqrt{e})^{-1}}\exp\big(\sqrt{e^{-1}\log p\log\log p}\,\big) $$ holds for all odd primes $p$, where the implied constant is absolute, provided that $$ k\le p^{(8\sqrt{e})^{-1}} \exp\big(\tfrac12\sqrt{e^{-1}\log p\log\log p}-\tfrac12\log\log p\big). $$ For fixed $ε\in(0,\frac{π-2}{9π-2}]$ we also show that there is a number $c=c(ε)>0$ such that for all odd primes $p$ and either choice of $θ\in\{\pm 1\}$, there are $\gg_εy/(\log y)^ε$ natural numbers $n\le y$ with $(n|p)=θ$ provided that $$ y\ge p^{(4\sqrt{e})^{-1}}\exp\big(c(\log p)^{1-ε}\big). $$

math.NT↗

arXiv2015

Some arithmetic properties of numbers of the form $\lfloor p^c\rfloor$

Let $${\mathbb P}^c=(\lfloor p^c\rfloor)_{p\in{\mathbb P}} \qquad (c>1,\ c\not\in {\mathbb N}), $$ where ${\mathbb P}$ is the set of prime numbers, and $\lfloor\cdot\rfloor$ is the floor function. We show that for every such $c$ there are infinitely many members of ${\mathbb P}^c$ having at most $R(c)$ prime factors, giving explicit estimates for $R(c)$ when $c$ is near one and also when $c$ is large.

math.NT↗

arXiv2015

Piatetski-Shapiro Primes in a Beatty Sequence

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 infinitely many Piatetski-Shapiro primes $p = \left\lfloor n^c \right\rfloor$ in the non-homogenous Beatty sequence $\big(\left\lfloorαm+β\right\rfloor\big)_{m=1}^\infty$.

math.NT↗