序列[n~c]上多维除数函数的和

设[θ]表示θ的整数部分,k≥2,dk(n)为除数函数.证明了当实数c满足1相似文献   

Beatty序列中的除数问题

本文研究Beatty序列中的除数问题,证明了在Lebesgue测度意义下,对几乎所有的θ≥1,当k≥5时,一致地有Dk(θ;x)=∑n≤x/θdk([nθ])=θ-1Dk(1;x)+O(x4/5+ε).     

丢番图方程|(ε~n-"非汉字符号"~n)/(ε-"非汉字符号")|=1的解数

设 a、b、c、k是适合 a+b=ck,gcd(a,b) =1 ,c∈ { 1 ,2 ,4} ,k>1且 k在 c=1或 2时为奇数的正整数 ;又设 ε=(a + - b) / c ,ε=(a - - b) / c .本文证明了 :当 (a,b,c,k)≠ (1 ,7,4,2 )或 (3,5,4,2 )时 ,至多有 1个大于 1的正奇数 n适合 |(εn-εn) / (ε-ε) |=1 ,而且如此的 n必为满足 n<1 +(2 logπ) / logk+2 563.43(1 +(2 1 .96π) / logk)的奇素数 .     

一道试题的背景研究及其推广

定理如果一个虚数的三次方是实数,那么,这个虚数必有形式Aw或Aw2,其中,w是1的立方虚根,A∈R且A≠0.证法1设z=r(cosθ isinθ),r∈R且r≠0,sinθ≠0,ω=-12 32i=cos23π i sin2π3,则z3=r3(cos3θ i sin3θ)∈R,∴sin3θ=0.3θ=kπ,θ=kπ3,k∈Z.1)当k=6n(n∈Z,下同)时,θ=2nπ,     

有关殆素数的二元丢番图不等式

Pr表示最多r个素因子的正整数.作者证明了,对于任一足够大的实数N和1＜c＜c0,不等式|pc+Pcr-N|＜N9/10(1-c0/c)对素数p和Pr是可解的.特别地,当1＜c＜c0=1.03074432…,有r=6.这个结果改进了翟文广和曹晓东的结果.     

丢番图方程|(εn-ε-n)/(ε-ε-)|=1的解数

设a、b、c、k是适合a+b=ck,gcd(a,b)=1,c∈{1,2,4},k＞1且k在c=1或2时为奇数的正整数;又设ε=(()a+()-b)/()c,ε=(()n-()-b)/()c.本文证明了当(a,b,c,k)≠(1,7,4,2)或(3,5,4,2)时,至多有1个大于1的正奇数n适合|(εn-εn)/(ε-ε)|=1,而且如此的n必为满足n＜1+(2logn)/logk+2563.43(1+(21.96π)/logk)的奇素数.     

一类数项级数和的表示

求出函数f(x)=xk的Fourier系数并将其代人Parseval等式,继而利用第二数学归纳法可证明:数项级数∞∑n=1 1/n2k的和能够表示为π2k/dk的形式.其中对于任意确定的k值.dk以为一常数.证明过程同时给出了求解dk的方法.     

小区间上的素变数三角和估计

设A(n)为von Mangoldt函数且实数θ=95-83~(1/2)/121.当xθ+ε≤y≤x时,本文对于所有的α∈[0,1]给出了指数和S2(x,y;α)=∑x0,估计式∑x相似文献   

最值问题的六种解法

<正>题目(2014年浙江省高中数学竞赛试题)设实数x,y满足方程(x+2)²+y2+y²=1,则y/x的最大值为.解法1令x=-2+cosθ,y=sinθ,θ∈[0,2π),y/x=k.则y/x=sinθ/-2+cosθ=k,即kcosθ-sinθ=2k,     

关于f（n）模1的分布

1 引言 设a是一个实数,k是一个自然数。我们用[θ]表示实数θ到最近整数的距离。对k=1我们有狄立克莱定理。对任何N≤1均存在自然数n≤N使 |an|≤N~(-1)(1)对k=2,Heilbronn证明了:假如给定ε>0和N≥N(ε),那么存在自然数n≤N使得     

Study of the blow-up of solutions of systems of equations of hydrodynamic type

Mathematical Notes - We study the initial boundary-value problem for three-dimensional systems of equations of pseudoparabolic type. The system is similar to the Oskolkov system, but differs from...     

Distribution of types of symmetry of tensors of high degree

The asymptotic distribution of tensors of degree N in symmetry types is studied in this paper.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 155, pp. 181–186, 1986.     

Characterization of types of asymptotic discernibility of families of hypotheses

We give a characterization of the types of asymptotic discernibility of families of hypotheses in the case of hypothetical measures that are not, in general, mutually absolutely continuous. The case when the logarithm of the likelihood ratio admits an asymptotic expansion of the type of an expansion with local asymptotic normality is examined in detail. Examples are studied.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 64–71, 1987.     

Axiomatizability of reducts of algebras of relations

In this paper, we prove that any subreduct of the class of representable relation algebras whose similarity type includes intersection, relation composition and converse is a non-finitely axiomatizable quasivariety and that its equational theory is not finitely based. We show the same result for subreducts of the class of representable cylindric algebras of dimension at least three whose similarity types include intersection and cylindrifications. A similar result is proved for subreducts of the class of representable sequential algebras. Received October 7, 1998; accepted in final form September 10, 1999.     

Lengths of periods of continued fractions of square roots of integers

Empirical study of the period’s length T of the continued fractions of $\sqrt{Q}$ (for growing integers Q) shows several strange asymptotical results, for instance, $T\leq C\sqrt{Q}\ln{Q}$ . These results show important differences between the statistics of the elements of the continued fractions of random real numbers and of square roots of random integers.