Fei Xue

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 paper, we investigate the higher--power moments of $Δ_a(x)$ and give the corresponding asymptotic formula for the integral $\int_{1}^{T}Δ_a^k(x)\mathrm{d}x$, which constitutes an improvement upon the previous result of Zhai [9] for $k=3,4,5$ and an enlargement of the upper bound of $k$ to $7$.

In this paper, it is established that every sufficiently large positive integer $n$ subject to $n\equiv0\pmod2$ can be represented as a sum of one square of prime and seventeen fifth powers of primes, which gives an enhancement upon the previous result of Brüdern and Kawada [1].

In this paper, we establish a new mean value theorem of Bombieri-Vinogradov type over Piatetski-Shapiro sequence. Namely, it is proved that for any given constant $A>0$ and any sufficiently small $\varepsilon>0$, there holds \begin{equation*} \sum_{\substack{d\leqslant x^ξ\\ (d,l)=1}}\Bigg|\sum_{\substack{A_1(x)\leqslant a<A_2(x)\\ (a,d)=1}}g(a) \Bigg(\sum_{\substack{ap\leqslant x\\ ap\equiv l\!\pmod d \\ ap=[k^{1/γ}]}}1 -\frac{1}{\varphi(d)}\sum_{\substack{ap\leqslant x\\ ap=[k^{1/γ}] }} 1\Bigg)\Bigg|\ll\frac{x^γ}{(\log x)^A}, \end{equation*} provided that $1\leqslant A_1(x)<A_2(x)\leqslant x^{1-\varepsilon}$ and $g(a)\ll τ_r^s(a)$, where $l\not=0$ is a fixed integer and \begin{equation*} ξ:=ξ(γ)=\frac{2^{38}+17}{38}γ-\frac{2^{38}-1}{38}-\varepsilon \end{equation*} with \begin{equation*} 1-\frac{18}{2^{38}+17}<γ<1. \end{equation*} Moreover, for $γ$ satisfying \begin{equation*} 1-\frac{0.03208}{2^{38}+17}<γ<1, \end{equation*} we prove that there exist infinitely many primes $p$ such that $p+2=\mathcal{P}_2$ with $\mathcal{P}_2$ being Piatetski-Shapiro almost-primes of type $γ$, and there exist infinitely many Piatetski-Shapiro primes $p$ of type $γ$ such that $p+2=\mathcal{P}_2$. These results generalize the result of Pan and Ding [37] and constitutes an improvement upon a series of previous results of [29,31,39,47].

Let $ \mathcal{P}_r $ denote an almost–prime with at most $ r $ prime factors, counted according to multiplicity. In this paper, it is proved that for every sufficiently large even integer $ N $, the following equation $ \begin{equation*} N = p_1^2+p_2^2+x^3+p_3^3+p_4^3+p_5^3 \end{equation*} $ is solvable with $ x $ being an almost–prime $ \mathcal{P}_7 $ and the other variables primes. This result constitutes a deepening upon that of previous results.