arXiv subjects
Publications and source records attributed to Genheng Zhao.
Let $E(X)$ denote the number of even integers below $X$ which are not a sum of two primes. We prove the bound $E(X)=O(X^{\frac{7}{10}})$, where the implicit constant is ineffective. The method applied here also leads to $P(q)=O(q^5)$, where $P(q)$ denotes the least prime, if it exists, in any arithmetic progression modulo $q$.
Let $\chi$ be a real non-principal character modulo a prime $q$ and $L(s,\chi)$ be the corresponding $L$-function. We prove that for any real number $s\geq 1$ there holds $$ -\frac{L'(s,\chi )}{L(s,\chi)}\leq c \log q,$$ where $c$ can be taken arbitrarily close to $1/4$ if we assume $q$ is sufficiently large depending upon it. As a consequence, for all large $q$, there are at least $q^{3/50}$ primes $p$ smaller than $q$ such that $\chi(p)=-1$.
Let $A\subset [1,x]$ be a non-empty set of primes with $|A|= \alpha x(\log x)^{-1}$. We prove that there exist absolute constants $c_1,c_2>0$ such that, as $x$ gets sufficiently large, we have $|A+A|\geq c_1(\log x)(\log \log 3\alpha^{-1})^{-1}|A|$ if $\alpha \geq c_2(\log x)^{-1/2}\log \log x$ and otherwise $|A+A|\geq c_1(\log x) (\log 2\alpha^{-1})^{-1}|A|$.
Let $s\geq 8$ be an integer and $P$ be a set of primes with relative lower density greater than $\sqrt{1-\min\{s,16\}/32}$. We prove that every sufficiently large integer $n\equiv s({\rm mod}24)$ can be represented by a sum of $s$ squares of primes in $P$.