arXiv subjects

Guang-Liang Zhou

Publications and source records attributed to Guang-Liang Zhou.

4 recordsLinked to original sources

arXiv2026

Large values of the Hurwitz zeta function with rational parameter

In this paper, we establish lower bounds for large values of the Hurwitz zeta function with rational parameter when the real part $σ\in [1/2,1]$. These results improve the earlier results of Ramachandra and Sankaranarayanan in 1989. On the critical line, our result recovers the corresponding lower bound of de la Bretèche and Tenenbaum (2019) for the Riemann zeta function, while for $1/2< σ\leq 1$, our lower bounds attain the same order as the corresponding lower bounds for large values of the Riemann zeta function. Our proofs are based on the resonance method.

math.NT↗

arXiv2026

Representations of positive integers by three almost-prime squares

Let $P_r$ denote an integer with at most $r$ prime factors, counted with multiplicity. It is known that every sufficiently large integer $N$ satisfying $N \equiv 3 \pmod{24}$ and $5 \nmid N$, can be written in the form $N= x_1^2+x_2^2+x_3^2$ where $x_1,x_2,x_3$ are integers. In this paper, we prove that the above representation in the following two different forms (i) $x_1x_2x_3$ is a $P_{67}$-number; (ii) each $x_i$ is a $P_{27}$-number. This result improves on the previous result of Waibel\cite{Wa}, in which $P_{72}$ was obtained in place of $P_{67}$. The proofs combine the higher-dimensional sieve, a Richert-type weighted sieve method introduced by Cai \cite{Cai} with a Bombieri-Vinogradov type result given by Waibel\cite{Wa}. Applying the same method in a one dimensional sieve setting, we also show that every sufficiently large $N$ not of the form $4^k(8l+7)$ can be written in the form \[ N = x^{2} + y^{2} + (2^{a} z)^{2}, \] where $x,y,a,z$ are non-negative integers and $z$ is a $P_{18}$-number. This improves upon a result of Banerjee \cite{Ban} who obtained $P_{118}$ in place of $P_{18}$.

math.NT↗

arXiv2021

Numbers represented by restricted sums of four squares

In this paper, we prove some results of restricted sums of four squares using arithmetic of quaternions in the ring of Lipschitz integers. For example, we show that every nonnegative integer $n$ can be written as $x^{2}+y^{2}+z^{2}+t^{2}$ where $x,y,z,t$ are integers and $x+y+2z+2t$ is a square or a cube.

math.NT↗

arXiv2018

On sums and products in a field

In this paper we study sums and products in a field. Let $F$ be a field with ${\rm ch}(F)\not=2$, where ${\rm ch}(F)$ is the characteristic of $F$. For any integer $k\ge4$, we show that each $x\in F$ can be written as $a_1+\ldots+a_k$ with $a_1,\ldots,a_k\in F$ and $a_1\ldots a_k=1$ if ${\rm ch}(F)\not=3$, and that for any $α\in F\setminus\{0\}$ we can write each $x\in F$ as $a_1\ldots a_k$ with $a_1,\ldots,a_k\in F$ and $a_1+\ldots+a_k=α$. We also prove that for any $x\in F$ and $k\in\{2,3,\ldots\}$ there are $a_1,\ldots,a_{2k}\in F$ such that $a_1+\ldots+a_{2k}=x=a_1\ldots a_{2k}$.

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