Min Zhang

A quaternary Diophantine inequality with prime numbers of a special form (II)

Yuhua Zhao, Jinjiang Li, Linji Long +1

Let $N$ be a sufficiently large real number. In this paper, it is proved that, for $1 < c < \frac{5179}{3909}$ and for any arbitrary large number $E > 0$ , the Diophantine inequality \beg…

On a variant of the Piatetski-Shapiro prime number theorem

Linji Long, Jinjiang Li, Min Zhang

Denote by $Π_{γ} (x)$ the number of primes $p ⩽ x$ such that $[p^{1/ γ}]$ is also a prime. In this paper, it is proved that the expected asymptotic formula \begin{equation*} Π_…

math.PR2026

Weak universality of global dynamics for the fractional hyperbolic $Φ_{3}^{4}$ -model

Ruoyuan Liu

We continue the study of weak universality for the fractional hyperbolic $Φ_{3}^{4}$ -model initiated by Tzvetkov, Wang, and the author (2023). In this work, we establish weak universal…

On Piatetski-Shapiro primes from almost primes

Yuhua Zhao, Jinjiang Li, Linji Long +1

Denote by $P_{r}$ an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this manuscript, it is established that, for any fixed $0.98353 < γ < 1$ …

On the quadratic Waring-Goldbach problem with primes in Piatetski-Shapiro sets

Meng Gao, Jinjiang Li, Linji Long +1

In this paper, it is proved that, for any $γ_{1}, γ_{2}, γ_{3}, γ_{4}, γ_{5} \in (\frac{28}{29}, 1)$ , every sufficiently large integer $n$ subject to $n \equiv 5 (mod 24)$ can be represented as the s…

Cubic Waring-Goldbach problem with Piatetski-Shapiro primes

Linji Long, Jinjiang Li, Min Zhang +1

In this paper, it is proved that, for $γ \in (\frac{317}{320}, 1)$ , every sufficiently large odd integer can be written as the sum of nine cubes of primes, each of which is of the for…

Vinogradov's three primes theorem in the intersection of multiple Piatetski-Shapiro sets

Xiaotian Li, Jinjiang Li, Min Zhang

Vinogradov's three primes theorem indicates that, for every sufficiently large odd integer $N$ , the equation $N = p_{1} + p_{2} + p_{3}$ is solvable in prime variables $p_{1}, p_{2}, p_{3}$ . In this p…

On Higher-Power Moments of $Δ_{a} (x)$ for $- 1/2 < a < 0$

Yi Cai, Jinjiang Li, Yankun Sui +2

Let $- 1/2 < a < 0$ be a fixed real number and \begin{equation*} Δ_{a}(x)=\sideset{}{'}\sum_{n\leq x} σ_a(n)-ζ(1-a)x-\frac{ζ(1+a)}{1+a}x^{1+a}+\frac{1}{2}ζ(-a). \end{equation*} In this…

On the mean square of the error term for the asymmetric two-dimensional divisor problem with congruence conditions

Zhen Guo, Jinjiang Li, Linji Long +1

Suppose that $a$ and $b$ are positive integers subject to $(a, b) = 1$ . For $n \in Z^{+}$ , denote by $τ_{a, b} (n; ℓ_{1}, M_{1}, l_{2}, M_{2})$ the asymmetric two--dimensional divisor funct…

On a system of two Diophantine inequalities with six prime variables

Linji Long, Jinjiang Li, Min Zhang +1

Suppose that $c, d, α, β$ are real numbers satisfying the inequalities $1 < d < c < 79/71$ and $1 < α < β < 6^{1 - d / c}$ . In this paper, it is proved that, for sufficiently large real numbers $N_{1}$ …

math.NT2024

Almost primes of the form $[p^{1/ γ}]$

Fei Xue, Jinjiang Li, Min Zhang

Let $P_{r}$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that, for $0.989 < γ < 1$ , there exist infinit…

math.NT2024

On Waring-Goldbach problem for one square and seventeen fifth powers of primes

Min Zhang, Jinjiang Li, Fei Xue

In this paper, it is established that every sufficiently large positive integer $n$ subject to $n \equiv 0 (mod 2)$ can be represented as a sum of one square of prime and seventeen fif…

math.NT2023

On the distribution of $α p^{2}$ modulo one in the intersection of two Piatetski--Shapiro sets

Junyi Chu, Jinjiang Li, Min Zhang

Let $⌊ t ⌋$ denote the integer part of $t \in R$ and $∥ x ∥$ the distance from $x$ to the nearest integer. Suppose that $1/2 < γ_{2} < γ_{1} < 1$ are two fixed constants. I…

math.NT2022

A generalization of Piatetski-Shapiro sequences (II)

Jinjiang Li, Jinyun Qi, Min Zhang

Suppose that $α, β \in R$ . Let $α ⩾ 1$ and $c$ be a real number in the range $1 < c < 12/11$ . In this paper, it is proved that there exist infinitely many primes in the ge…

math.NT2020

On Waring-Goldbach Problem for Squares, Cubes and Higher Powers

Min Zhang, Jinjiang Li

Let $P_{r}$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, we generalize the result of Vaughan for ternary admissi…

math.NT2020

An Additive Problem over Piatetski-Shapiro Primes and Almost-primes

Jinjiang Li, Min Zhang

Let $P_{r}$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, we establish a theorem of Bombieri-Vinogradov type for…

math.NT2018

A Diophantine inequality with four prime variables

Min Zhang, Jinjiang Li

Let $N$ be a sufficiently large real number. In this paper, it is proved that, for $1 < c < \frac{1193}{889}$ , the following Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p…

math.NT2018

On two Diophantine inequalities over primes (II)

Yuetong Zhao, Jinjiang Li, Min Zhang

Let $1 < c < \frac{26088036}{12301745}, c \neq = 2$ and $N$ be a sufficiently large real number. In this paper, it is proved that, for almost all $R \in (N, 2 N]$ , the Diophantine inequality…

math.NT2018★ 1 cited

On a Diophantine equation with five prime variables

Jinjiang Li, Min Zhang

Let $[x]$ denote the integral part of the real number $x$ , and $N$ be a sufficiently large integer. In this paper, it is proved that, for $1 < c < \frac{4109054}{1999527}, c \neq = 2$ , th…

On the Fourth Power Moment of the Error Term for the Divisor Problem with Congruence Conditions

Jinjiang Li, Min Zhang

Let $d (n; ℓ_{1}, M_{1}, ℓ_{2}, M_{2})$ denote the number of factorizations $n = n_{1} n_{2}$ , where each of the factors $n_{i} \in N$ belongs to a prescribed congruence class $\ell_i\bmod…

Waring-Goldbach Problem: One Square, Four Cubes and Higher Powers

Jinjiang Li, Min Zhang

Let $P_{r}$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that, for $12 ⩽ b ⩽ 35$ and…

A Remark on the Piatetski-Shapiro-Hua Theorem

Jinjiang Li, Min Zhang

In this paper, we prove that for any fixed $205/243 < γ ⩽ 1$ , every sufficiently large $N$ satisfying $N \equiv 5 (mod 24)$ can be represented as five squares of primes with o…