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Chaohua Jia

Publications and source records attributed to Chaohua Jia.

12 recordsLinked to original sources
arXiv2020

On the conditional bounds for Siegel zeros

Under a weakened version of Hardy-Littlewood Conjecture on the number of representations in Goldbach problem, J. H. Fei proved bounds for the Siegel zeros. Recently G. Bhowmik and K. Halupczok generalized Fei's result under a weaker conjecture. In the first version of this paper on arXiv, we pointed out a defect in the paper of G. Bhowmik and K.Halupczok, and assumed a new conjecture and used new discussion to overcome this defect to recover their result. Afterwards, in the second version of their paper, G. Bhowmik and K. Halupczok assumed previous conjecture and used our new discussion to get a weaker result. But they did not mention our paper so that we have to make some explanation now.

math.NT↗
arXiv2020

On the generalization of Golomb's conjecture

Let $p$ be a sufficiently large prime number, $r$ be any given positive integer. Suppose that $a_1,\,\dots,\,a_r$ are pairwise distinct and not zero modulo $p$. Let $N(a_1,\,\dots,\,a_r;\,p)$ denote the number of $α_1,\,\dots,\,α_r,\,β$, which are primitive roots modulo $p$, such that $α_1+β\equiv a_1,\,\dots,\,α_r+β\equiv a_r\,({\rm mod}\,p).$ In the first version of this paper, we proved an asymptotic formula for $N(a_1,\,\dots,\,a_r;\,p)$ so that we could answer an open problem of Wenpeng Zhang and Tingting Wang. But we found that our result had been included in a paper of L. Carlitz in 1956, which is explained in the additional remark below.

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arXiv2019

On power residues modulo a prime

Let $p$ be a sufficiently large prime number, $n$ be a positive odd integer with $n|\,p-1$ and $n>p^\varepsilon $, where $\varepsilon$ is a sufficiently small constant. Let $k(p,\,n)$ denote the least positive integer $k$ such that for $x=-k,\,\dots,\,-1,\,1,\,2,\,\dots,\,k$, the numbers $x^n\pmod p$ yield all the non-zero $n$-th power residues modulo $p$. In this paper, we shall prove $$ k(p,\,n)=O(p^{1-δ}), $$ which improves a result of S. Chowla and H. London in the case of large $n$.

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arXiv2014

Shifted Character Sums with Multiplicative Coefficients

Let $f(n)$ be a multiplicative function satisfying $|f(n)|\leq 1$, $q$ $(\leq N^2)$ be a prime number and $a$ be an integer with $(a,\,q)=1$, $χ$ be a non-principal Dirichlet character modulo $q$. In this paper, we shall prove that $$ \sum_{n\leq N}f(n)χ(n+a)\ll {N\over q^{1\over 4}}\log\log(6N)+q^{1\over 4}N^{1\over 2}\log(6N)+{N\over \sqrt{\log\log(6N)}}. $$ We shall also prove that \begin{align*} &\sum_{n\leq N}f(n)χ(n+a_1)\cdotsχ(n+a_t)\ll {N\over q^{1\over 4}}\log\log(6N)\\ &\quad+q^{1\over 4}N^{1\over 2}\log(6N)+{N\over \sqrt{\log\log(6N)}}, \end{align*} where $t\geq 2$, $a_1,\,\cdots,\,a_t$ are pairwise distinct integers modulo $q$.

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arXiv2014

A note on the number of coefficients of automorphic $L-$functions for $GL_m$ with same signs

Let $π$ be an irreducible unitary cuspidal representation of $GL_m({\Bbb A}_{\Bbb Q})$ and $L(s,\,π)$ be the global $L-$function attached to $π$. If ${\rm Re}(s)>1$, $L(s,\,π)$ has a Dirichlet series expression. When $π$ is self-contragradient, all the coefficients of Dirichlet series are real. In this note, we shall give non-trivial lower bounds for the number of positive and negative coefficients respectively, which is an improvement on the recent work of Jianya Liu and Jie Wu.

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arXiv2014

The Mean Square of Divisor Function

Let $d(n)$ be the divisor function. In 1916, S. Ramanujan stated but without proof that $$\sum_{n\leq x}d^2(n)=xP(\log x)+E(x), $$ where $P(y)$ is a cubic polynomial in $y$ and $$ E(x)=O(x^{{3\over 5}+ε}), $$ where $ε$ is a sufficiently small positive constant. He also stated that, assuming the Riemann Hypothesis(RH), $$ E(x)=O(x^{{1\over 2}+ε}). $$ In 1922, B. M. Wilson proved the above result unconditionally. The direct application of the RH would produce $$ E(x)=O(x^{1\over 2}(\log x)^5\log\log x). $$ In 2003, K. Ramachandra and A. Sankaranarayanan proved the above result without any assumption. In this paper, we shall prove $$ E(x)=O(x^{1\over 2}(\log x)^5). $$

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arXiv2014

Kloosterman Sums with Multiplicative Coefficients

Let $f(n)$ be a multiplicative function satisfying $|f(n)|\leq 1$, $q$ $(\leq N^2)$ be a positive integer and $a$ be an integer with $(a,\,q)=1$. In this paper, we shall prove that $$\sum_{\substack{n\leq N\\ (n,\,q)=1}}f(n)e({a\bar{n}\over q})\ll\sqrt{τ(q)\over q}N\log\log(6N)+q^{{1\over 4}+{ε\over 2}}N^{1\over 2}(\log(6N))^{1\over 2}+{N\over \sqrt{\log\log(6N)}},$$ where $\bar{n}$ is the multiplicative inverse of $n$ such that $\bar{n}n\equiv 1\,({\rm mod}\,q),\,e(x)=\exp(2πix),\,τ(q)$ is the divisor function.

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arXiv2012

On sumsets in ${\Bbb F}_2^n$

Let ${\Bbb F}_2$ be the finite field of two elements, ${\Bbb F}_2^n$ be the vector space of dimension $n$ over ${\Bbb F}_2$. For sets $A,\,B\subseteq{\Bbb F}_2^n$, their sumset is defined as the set of all pairwise sums $a+b$ with $a\in A,\,b\in B$. Ben Green and Terence Tao proved that, let $K\geq 1$, if$A,\,B\subseteq{\Bbb F}_2^n$ and $|A+B|\leq K|A|^{1\over 2}|B|^{1\over 2}$, then there exists a subspace $H\subseteq{\Bbb F}_2^n$ with $$ |H|\gg\exp(-O(\sqrt{K}\log K))|A| $$ and $x,\,y\in{\Bbb F}_2^n$ such that $$ |A\cap(x+H)|^{1\over 2}|B\cap(y+H)|^{1\over 2}\geq{1\over 2K}|H|. $$ In this note, we shall use the method of Green and Tao with some modification to prove that if $$ |H|\gg\exp(-O(\sqrt{K}))|A|, $$ then the above conclusion still holds true.

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arXiv2011

Mean Value from Representation of Rational Number as Sum of Two Egyptian Fractions

For given positive integers $n$ and $a$, let $R(n;\,a)$ denote the number of positive integer solutions $(x,\,y)$ of the Diophantine equation $$ {a\over n}={1\over x}+{1\over y}. $$ Write $$ S(N;\,a)=\sum_{\substack{n\leq N (n,\,a)=1}}R(n;\,a). $$ Recently Jingjing Huang and R. C. Vaughan proved that for $4\leq N$ and $a\leq 2N$, there is an asymptotic formula $$ S(N;\,a)={3\over π^2a}\prod_{p|a}{p-1\over p+1}\cdot N(\log^2N+c_1(a) \log N+c_0(a))+Δ(N;\,a). $$ In this paper, we shall get a more explicit expression with better error term for $c_0(a)$.

math.NT↗
arXiv2011

On the estimate for a mean value relative to 4/p=1/n_1+1/n_2+1/n_3

For the positive integer $n$, let $f(n)$ denote the number of positive integer solutions $(n_1, n_2, n_3)$ of the Diophantine equation $$ {4\over n}={1\over n_1}+{1\over n_2}+{1\over n_3}. $$ For the prime number $p$, $f(p)$ can be split into $f_1(p)+f_2(p),$ where $f_i(p)(i=1, 2)$ counts those solutions with exactly $i$ of denominators $n_1, n_2, n_3$ divisible by $p.$ Recently Terence Tao proved that $$ \sum_{p< x}f_1(p)\ll x\exp({c\log x\over \log\log x}) $$ with other results. In this paper we shall improve it to $$ \sum_{p< x}f_1(p)\ll x\log^5x\log\log^2x. $$

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arXiv2011

A Note on Terence Tao's Paper "On the Number of Solutions to 4/p=1/n_1+1/n_2+1/n_3"

For the positive integer $n$, let $f(n)$ denote the number of positive integer solutions $(n_1,\,n_2,\,n_3)$ of the Diophantine equation $$ {4\over n}={1\over n_1}+{1\over n_2}+{1\over n_3}. $$ For the prime number $p$, $f(p)$ can be split into $f_1(p)+f_2(p),$ where $f_i(p)(i=1,\,2)$ counts those solutions with exactly $i$ of denominators$n_1,\,n_2,\,n_3$ divisible by $p.$ Recently Terence Tao proved that $$ \sum_{p< x}f_2(p)\ll x\log^2x\log\log x $$ with other results. But actually only the upper bound $x\log^2x\log\log^2x$ can be obtained in his discussion. In this note we shall use an elementary method to save a factor $\log\log x$ and recover the above estimate.

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