两个素数和一个素数平方和的丢番图逼近(英文)

We show that if λ1 , λ2 , λ3 are non-zero real numbers, not all of the same sign, η is real and λ1 /λ2 is irrational, then there are infinitely many ordered triples of primes (p1 , p2 , p3 ) for which |λ1 p1 + λ2 p2 + λ3 p2 3 + η| < (max pj )- 1/40 (log max pj ) 4 .     

关于两个素数和一个素数k次幂的丢番图不等式

令■设λ_1,λ_2,λ_3是不全同号的非零实数,且满足λ_1/λ_2为无理数,则对于任意实数η和ε 0,不等式■有无穷多组素数解p_1,p_2,p_3.该结果改进了Gambini,Languasco和Zaccagnini的结果.     

One Cubic Diophantine Inequality

Suppose that G(x) is a form, or homogeneous polynomial, of odddegree d in s variables, with real coefficients. Schmidt [15]has shown that there exists a positive integer s₀(d), whichdepends only on the degree d, so that if s s₀(d), then thereis an x Z^s\{0} satisfying the inequality |G(x)|<1. (1) In other words, if there are enough variables, in terms of thedegree only, then there is a nontrivial solution to (1). Lets₀(d) be the minimum integer with the above property. In thecourse of proving this important result, Schmidt did not explicitlygive upper bounds for s₀(d). His methods do indicate how todo so, although not very efficiently. However, in fact muchearlier, Pitman [13] provided explicit bounds in the case whenG is a cubic. We consider a general cubic form F(x) with realcoefficients, in s variables, and look at the inequality |F(x)|<1. (2) Specifically, Pitman showed that if s(1314)²⁵⁶–1, (3) then inequality (2) is non-trivially soluble in integers. Wepresent the following improvement of this bound.     

一个加性混合幂丢番图不等式Ⅲ

证明了:如果λ1,…,λ11,μ是非零实数,并且不同一符号,至少有一个λi/λj是无理数,那么对任意实数η和ε>0,不等式λ1x14 … λ11x141 μy2 η<ε有无穷多正整数解x1,…,x11,y.     

一个混合幂为2和4的丢番图不等式

In this paper,it is shown that:if λ1,..,λ8 are nonzero real numbers,not all of the same sign,such that λ1/λ2 is irrational,then for any real numbex,η and ε＞0 the inequality,|λ1x21 λ2x22 λ3x43 … λ8x48 η|＜ε has infinitely many solutions in positive integers x1,…,x8.     

一个加性混合幂丢番图不等式(Ⅰ)

本文证明了如果λ1,…,λ6是非零实数,并且不同一符号,至少有一个λi/λj(1≤i,j≤3)是无理数,那么对任意实数η和ε＞0,不等式|λ1x21+λ2x22+λ3x23+λ4x44+λ5x45+λ6x46+η|＜ε有无穷多正整数解x1,…,x6.     

四个素数的平方及2的若干次幂和丢番图逼近

Under certain condition, the inequality |λ_1p_1~2 λ_2p_2~2 λ_3p_3~2 λ_4p_4~2 μ_12~(x1) … μ_s2~(xs) γ|＜ηhas infinitely many solutions in primes p_1,p_2,p_3,p_4 and positive integers x_1,…,x_s.     

Groups with One or Two Super-Brauer Character Theories

A super-Brauer character theory of a group G and a prime p is a pair consisting of a partition of the irreducible p-Brauer characters and a partition of the p-regular elements of G that satisfy certain properties.We classify the groups and primes that have exactly one super-Brauer character theory.We discuss the groups with exactly two super-Brauer character theories.     

Groups with One or Two Super-Brauer Character Theories

A super-Brauer character theory of a group G and a prime p is a pair consisting of a partition of the irreducible p-Brauer characters and a partition of the p-regular elements of G that satisfy certain properties. We classify the groups and primes that have exactly one super-Brauer character theory. We discuss the groups with exactly two super-Brauer character theories.     

Diophantine approximation with multiplicative functions

Let σ(n) denote the sum of divisors function. Our main result shows that, given any real α > 1 there are infinitely many integers n such that

Diophantine inequalities with mixed powers

It is shown that if λ₁,…, λ₆ are nonzero real numbers, not all of the same sign, such that $\text{λ}{}_1\text{λ}{}_2$ is irrational, then the values taken by λ₁x₁² + λ₂x₂² + λ₃x₃³ + λ₄x₄³ + λ₅x₅⁵ + λ₆x₆⁵ for integral x₁,…, x₆ are everywhere dense on the real line. Similar results are proved with other combinations in place of the two fifth powers.     

Diophantine approximation with multiplicative functions

Let σ(n) denote the sum of divisors function. Our main result shows that, given any real α > 1 there are infinitely many integers n such that $$\left|\frac{\sigma(n)}{n}-\alpha \right| < n^{-0.52}.$$ We prove this result by modifying an argument given by Wolke (Monatsh Math 83:163–166, 1977) which in its original form could not produce an exponent greater than 0.5. We also explain how the exponent can be improved to 0.61 on the Riemann Hypothesis.     

Diophantine inequalities with mixed powers

Let {λ_i}_{i = 1}^s (s ≥ 2) be a finite sequence of non-zero real numbers, not all of the same sign and in which not all the ratios $\text{λ}{}_{\mathrm{i}}\text{λ}{}_{\mathrm{j}}$ are rational. A given sequence of positive integers {n_i}_{i = 1}^s is said to have property (P) (($\text{P}{}^1$) respectively) if for any {λ_i}_{i = 1}^s and any real number η, there exists a positive constant σ, depending on {λ_i}_{i = 1}^s and {n_i}_{i = 1}^s only, so that the inequality |η + Σ_{i = 1}^sλ_ix_iⁿⁱ| < (max x_i)^?σ has infinitely many solutions in positive integers (primes respectively) x₁, x₂,…, x_s. In this paper, we prove the following result: Given a sequence of positive integers {n_i}_{i = 1}^∞, a necessary and sufficient condition that, for any positive integer j, there exists an integer s, depending on {n_i}_{i = j}^∞ only, such that {n_i}_{i = j}^{j + s ? 1} has property (P) (or ($\text{P}{}^1$)), is that Σ_{i = 1}^∞n_i^?1 = ∞. These are parallel to some striking results of G. A. Fre?man, E. J. Scourfield and K. Thanigasalam.     

Diophantine inequalities with mixed powers

It is shown that λ₁, λ₂,…, λ₆, μ are not all of the same sign and at least one ratio $\text{λ}{}_{\mathrm{i}}\text{λ}{}_{\mathrm{j}}$ is irrational then the values taken by $\text{λ}{}_1\text{x}{}_1{}^3\text{+ ? + λ}{}_6\text{x}{}_6{}^3\text{+ μy}{}^3$ for integer values of x₁ ,…, x₆, y are everywhere dense on the real line. A similar result holds for expressions of the form $\text{λ}{}_1\text{x}{}_1{}^3\text{+ ? + λ}{}_4\text{x}{}_4{}^3\text{+ μ}{}_1\text{y}{}_1{}^2\text{+ μ}{}_2\text{y}{}_2{}^3$.     

Diophantine approximation with sign constraints

Let $\alpha $ and $\beta $ be real numbers such that $1$ , $\alpha $ and $\beta $ are linearly independent over $\mathbb {Q}$ . A classical result of Dirichlet asserts that there are infinitely many triples of integers $(x_0,x_1,x_2)$ such that $|x_0+\alpha x_1+\beta x_2| < \max \{|x_1|,|x_2|\}^{-2}$ . In 1976, Schmidt asked what can be said under the restriction that $x_1$ and $x_2$ be positive. Upon denoting by $\gamma \cong 1.618$ the golden ratio, he proved that there are triples $(x_0,x_1,x_2) \in \mathbb {Z}^3$ with $x_1,x_2>0$ for which the product $|x_0 + \alpha x_1 + \beta x_2| \max \{|x_1|,|x_2|\}^\gamma $ is arbitrarily small. Although Schmidt later conjectured that $\gamma $ can be replaced by any number smaller than $2$ , Moshchevitin proved very recently that it cannot be replaced by a number larger than $1.947$ . In this paper, we present a construction of points $(1,\alpha ,\beta )$ showing that the result of Schmidt is in fact optimal. These points also possess strong additional Diophantine properties that are described in the paper.     

Diophantine approximation with mild divisibility constraints

Given an irrational number α and a sequence B of coprime positive integers with the sum of inverses convergent, we investigate the problem of finding small values of nα, with n B-free.     

On a Diophantine Inequality with Reciprocals

We generalize Chebotarev’s density theorem to Weil groups. Since the Artin–Weil conjecture on the integrality of the Artin–Hecke L-functions, constructed by A. Weil, has not been completely proved so far, we estimate the character sums both under and without the assumption of the validity of that conjecture.