Let $l⩾2$ be an integer. Recently, Hu and Lü offered the asymptotic formula for the sum of the higher divisor function $$\sum _{1⩽n_1,n_2,...,n_l⩽x^{1/2}}{\tau }_k(n_1^2+n_2^2+\cdots +n_l^2),$$ where ${\tau }_k\left(n\right)$ represents the $k$th divisor function. We give the Goldbach-type analogy of their result. That is to say, we investigate the asymptotic behavior of the sum $$\sum _{1⩽p_1,p_2,...,p_l⩽x}{\tau }_k(p_1+p_2+\cdots +p_l),$$ where $p_1,p_2,\cdots ,p_l$ are prime variables.

Estimates are provided for sth moments of cubic smooth Weyl sums, when 4 ≤ s ≤ 8, by enhancing the author's iterative method that delivers estimates beyond classical convexity. As a consequence, an improved lower bound is presented for the number of integers not exceeding X that are represented as the sum of three cubes of natural numbers.

We give non-trivial upper bounds for the number of integral solutions, of given size, of a system of two quadratic form equations in five variables.

Suppose that ${\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4$ are nonzero real numbers, not all negative, $\delta >0$, $𝒱$ is a well-spaced set, and the ratio ${\lambda }_1/{\lambda }_2$ is algebraic and irrational. Denote by $E(𝒱,N,\delta )$ the number of $v\in 𝒱$ with $v\le N$ such that the inequality $$|{\lambda }_1{p}_1^2+{\lambda }_2{p}_2^3+{\lambda }_3{p}_3^4+{\lambda }_4{p}_4^5-v|<v^{-\delta }$$ has no solution in primes $p_1$, $p_2$, $p_3$, $p_4$. We show that $$E(𝒱,N,\delta )\ll N^{1+2\delta -1/72+\epsilon }$$ for any $\epsilon >0$.

1. Introduction. A positive number which is a sum of two odd primes is called a Goldbach number. Let E(x) denote the number of even numbers not exceeding x which cannot be written as a sum of two odd primes. Then the Goldbach conjecture is equivalent to proving that E(x) = 2 for every x ≥ 4. E(x) is usually called the exceptional set of Goldbach numbers. In [8] H. L. Montgomery and R. C. Vaughan proved that $E\left(x\right)=O\left(x^{1-\Delta }\right)$ for some positive constant Δ > 0$.In\left[3\right]ChenandPanprovedthatonecantake\Delta >0.01.In\left[6\right],weprovedthatE\left(x\right)=O\left(x^{0.921}\right)$. In this paper we prove the following result. Theorem....

A new method for counting primes in a Beatty sequence is proposed, and it is shown that an asymptotic formula can be obtained for the number of such primes in a short interval.