Sums of nine almost equal prime cubes

We prove that each sufficiently large odd integer N can be written as sum of the form N = p1^3 ＋p2^3 ＋... ＋p9^3 with [pj - （N/9）^1/31 ≤ N^（1/3）-θ, where pj, j = 1,2,...,9, are primes and θ = （1/51） -ε.     

Sums of five almost equal prime squares II

It is proved that every large integerN≡5 (mod 24) can be written as with each primep _j satisfying , which gives a short interval version of a classical theorem of Hua. Project supported by the Trans-Century Training Programme Foundation for the Talents by the State Education Commission and the National Natural Science Foundation of China (Grant No. 19701019).     

几乎相等的三次华林-哥德巴赫问题的例外集

本文证明了对5≤s≤8,几乎所有的满足某些同余条件的正整数N都可以表示为N=p31+···+p3s,|pi-(N/s)1/3|≤N1/3-θs,其中θ5=7261-2ε,θ6=5159-ε,θ7=11333-ε,θ8=19561-ε.     

Sums of four cubes

An upper bound for the numberE(N) of numbers not exceedingN and not being the sum of four cubes is given; namelyE(N)N ^131/147+e.     

Sums of cubes of square-free numbers

Almost all positive integers may be represented as the sum of four cubes of positive square-free integers.     

Sums of four cubes of primes

It is conjectured that all sufficiently large integers satisfying some necessary congruence conditions are the sum of four cubes of primes. Using the circle method and sieves, we prove that the conjecture is true for at least 1.5% of the positive integers satisfying the necessary conditions.     

Sums of prime powers

It is shown that iff is a suitable polynomial then every sufficiently large positive integer is a sum of distinct summands taken from the sequence {f(p)}, wherep runs through rational primes.     

Sums of cubes of square-free numbers II

Letr _s (n) denote the number of representations ofn as the sum ofs cubes of square-free numbers. We prove an asymptotic formula forr _s (n) and a strong lower bound forr ₇ (n).     

Sums of cubes of primes in short intervals

In this paper, we are able to sharpen Hua??s result by proving that almost all integers satisfying some necessary congruence conditions can be represented as $$N=p_1^3+ \cdots+p_s^3 \quad \mbox{with } \biggl \vert p_j-\sqrt[3]{\frac {N}{s}}\biggr \vert \leqslant U, j=1,\ldots, s, $$ where p _j are primes and $U=N^{\frac{1}{3}-\delta_{s}+\varepsilon }$ with $\delta_{s}=\frac{s-4}{6s+72}$ , where s=5,6,7,8.     

On almost prime k-tuples

Let k?2 and ai,bi(1?i?k) be integers such that ai>0 and _∏1?i<j?k(aibj−ajbi)≠0. Let Ω(m) denote the total number of prime factors of m. Suppose has no fixed prime divisors. Results of the form where rk is asymptotic to klogk have been obtained by using sieve methods, in particular weighted sieves. In this paper, we use another kind of weighted sieve due to Selberg to obtain improved admissible values for rk.     

Sums over numbers with restricted prime factors

For k a non-negative integer, let P_k(n) denote the kth largest prime factor of n where P₀(n) = +∞ and if the number of prime factors of n is less than k, then P_k(n) = 1. We shall study the asymptotic behavior of the sum Ψ_k(x, y; g) = Σ_{1 ≤ n ≤ x, Pk(n) ≤ y}g(n), where g(n) is an arithmetic function satisfying certain general conditions regarding its behavior on primes. The special case where g(n) = μ(n), the Möbius function, is discussed as an application.     

关于连续素数上取值的Legendre符号和的估计

利用Karatsuba的方法,研究了关于连续素数上取值的Legendre符号和∑p≤N(p q)的估计问题,得到了当q0.75+ω相似文献   

On almost regular automorphisms of prime order

Novosibirsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 5, pp. 206–208, September–October, 1992.     

Sums over positive integers with few prime factors

The author has presented estimates for sums of multiplicative functions, satisfying certain conditions, extended over positive integers n such that n is less than or equal to x^t, the greatest prime factor of n is at most equal to x, and n is relatively prime to a natural number k. These estimates are uniform in t, x, and k.