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Linji Long

Publications and source records attributed to Linji Long.

8 recordsLinked to original sources
arXiv2026

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

Let $N$ be a sufficiently large real number. In this paper, it is proved that, for $1 0$, the Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c+p_4^c-N\big|<(\log N)^{-E} \end{equation*} is solvable in prime variables $p_1,p_2,p_3,p_4$ such that each of the numbers $p_i+2\,(i=1,2,3,4)$ has, counted according to multiplicity, at most $[\frac{31250}{5179-3909c}]$ prime factors, which constitutes a large improvement upon the previous results of Dimitrov [13] and Li-Xue-Zhang [22], respectively.

math.NT↗
arXiv2026

On a variant of the Piatetski-Shapiro prime number theorem

Denote by $Π_γ(x)$ the number of primes $p \leqslant x$ such that $[p^{1/γ}]$ is also a prime. In this paper, it is proved that the expected asymptotic formula \begin{equation*} Π_γ(x)=\frac{γx}{(\log x)^2} (1+o(1)) \end{equation*} holds for $γ> 15/13$, which constitutes an improvement upon the previous result of Balog [2] who showed that the above asymptotic formula holds for $γ> 6/5$.

math.NT↗
arXiv2026

On Piatetski-Shapiro primes from almost primes

Denote by $\mathcal{P}_r$ an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this manuscript, it is established that, for any fixed $0.98353<γ<1$, there exist infinitely many primes of the form $p=[n^{1/γ}]$, where $n$ is an almost-prime $\mathcal{P}_7$. This result constitutes an improvement upon the previous result of Baker, Banks, Guo and Yeager [1], who showed that there exist infinitely many primes $p$ such that $p=[n^{1/γ}]$ with $n\in\mathcal{P}_8$ for $γ$ near to one.

math.NT↗
arXiv2026

On a system of two Diophantine inequalities with six prime variables

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 $N_1$ and $N_2$ subject to $α\leqslant N_2/N_1^{d/c}\leqslantβ$, the following Diophantine inequalities system \begin{align*} \begin{cases} |p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c-N_1|<\varepsilon_1 (N_1) \\ |p_1^d+p_2^d+p_3^d+p_4^d+p_5^d+p_6^d-N_2|<\varepsilon_2 (N_2) \end{cases} \end{align*} is solvable in prime variables $p_1, p_2, p_3, p_4, p_5, p_6$, where \begin{align*} \begin{cases} \varepsilon_1 (N_1)=N_1^{-(1/c)(79/71-c)} (\log N_1)^{201}, \\ \varepsilon_2 (N_2)=N_2^{-(1/d)(79/71-d)} (\log N_2)^{201} . \end{cases} \end{align*} This result constitutes an improvement upon the previous result of Han-Liu-Zhang [5].

math.NT↗
arXiv2026

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

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 $n\equiv5\pmod{24}$ can be represented as the sum of five squares of primes, i.e., \begin{equation*} n=p_1^2+p_2^2+p_3^2+p_4^2+p_5^2, \end{equation*} such that $p_i=\lfloor m_i^{1/γ_i}\rfloor$ for some $m_i\in\mathbb{N}^+$ for each $1\leqslant i\leqslant 5$. This result constitutes an improvement upon the previous result of Zhang and Zhai [29].

math.NT↗
arXiv2025

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

Suppose that $a$ and $b$ are positive integers subject to $(a,b)=1$. For $n\in\mathbb{Z}^+$, denote by $τ_{a,b}(n;\ell_1,M_1,l_2,M_2)$ the asymmetric two--dimensional divisor function with congruence conditions, i.e., \begin{equation*} τ_{a,b}(n;\ell_1,M_1,l_2,M_2)=\sum_{\substack{n=n_1^an_2^b\\ n_1\equiv\ell_1\!\!\!\!\!\pmod{M_1}\\ n_2\equiv\ell_2\!\!\!\!\!\pmod{M_2}}}1. \end{equation*} In this paper, we shall establish an asymptotic formula of the mean square of the error term of the sum $\sum_{n\leqslant M_1^aM_2^bx}τ_{a,b}(n;\ell_1,M_1,l_2,M_2)$. This result constitutes an enhancement upon the previous result of Zhai and Cao [16].

math.NT↗
arXiv2025

Cubic Waring-Goldbach problem with Piatetski-Shapiro primes

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 form $[n^{1/γ}]$. This result constitutes an improvement upon the previous result of Akbal and Güloğlu [1].

math.NT↗