Primes in Special Intervals and Additive Problems with Such Numbers

We study primes in a special set E which is naturally described by the fractional part of p^a, where a<1 is a noninteger. An asymptotic formula with power lowering in the remainder of the trigonometric sum over primes from the set E is obtained. We study several applications of this result to problems of the distribution of primes from E in arithmetic progressions and to additive problems with primes from E.     

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   

More Precise Pair Correlation of Zeros and Primes in Short Intervals

Goldston and Montgomery proved that the strong pair correlationconjecture and two second moments of primes in short intervalsare equivalent to each other under the Riemann hypothesis. Thepaper obtains the second main terms for each of the above, andit is shown that they are almost equivalent to each other.     

Estimation of Exponential Sums over Primes in Short Intervals I

 Let and be the von Mangoldt function and M?bius function, respectively, x real and y“small” compared with x. This paper gives, for the first time, a non-trivial estimate of the sum

Estimation of Exponential Sums over Primes in Short Intervals I

 Let and be the von Mangoldt function and M?bius function, respectively, x real and y“small” compared with x. This paper gives, for the first time, a non-trivial estimate of the sum

On the Exceptional Set of Goldbach’s Problem in Short Intervals

Denote by E[X,X+H] the set of even integers in [X,X+H] that are not a sum of two primes (i.e. that are not Goldbach numbers). Here we prove that there exists a (small) positive constant H 3 X ^[7/24]+7dH\ge X^{\,{7\over24}+7\delta} we have |E(X,H) | << H^1-d/600\vert E(X,H) \vert \ll H^{1-\delta/600} .     

Powers of Rational Numbers Modulo 1 Lying in Short Intervals

Let p > q > 1 be two coprime integers. We construct some positive numbers ξ such that the numbers ξ(p/q) ⁿ , n = 0, 1, 2, . . . , modulo 1 all lie in a short interval. Our results imply, for instance, that there exist three positive real numbers ξ, ζ, τ such that the inequalities ||ξ(5/3) ⁿ || < 2/5, ||ζ (5/3) ⁿ || > 1/10 and ||τ (3/2)²ⁿ || < 14/45 hold for each integer ${n \geqslant 0}$ .     

On Estimations of Trigonometric Sums over Primes in Short Intervals(III)

In this paper the following result is proved: There is an absolute positive integer c such that for every large odd integer N the Diophantine equation with prime variables $N=p_1+p_2+p_3,N/3-U 相似文献   

The Ternary Goldbach Problem with Primes in Arithmetic Progressions

In this paper, we give an explicit numerical upper bound for the moduli of arithmetic progressions, in which the ternary Goldbach problem is solvable. Our result implies a quantitative upper bound for the Linnik constant. Received November 4, 1998, Accepted March 6, 2001     

稀疏素数集中的Waring-Goldbach问题

本文研究了Ｗａｒｉｎｇ Ｇｏｌｄｂａｃｈ问题（ｋ＝２）与Ｐｉａｔｅｔｓｋｉ Ｓｈａｐｉｒｏ素数定理的混合问题，从而进一步深化了华罗庚教授的经典结果     

Lattice Points Close to a Smooth Curve and Squarefull Numbers in Short Intervals

An approach of Swinnerton-Dyer is extended to obtain new upperbounds for the number of lattice points close to a smooth curve.One consequence of these bounds is a new asymptotic result forthe distribution of squarefull numbers in short intervals.     

A Note on the Associated Primes of Local Cohomology Modules

In this note we give a simple proof of the following result: Let R be a commutative Noetherian ring,  an ideal of R and M a finite R-module, if H_ ⁱ (M) has finite support for all i < n, then Ass(H_ ⁿ (M)) is finite.     

A Note on Log-Convexity of q-Catalan Numbers

The q-Catalan numbers studied by Carlitz and Riordan are polynomials in q with nonnegative coefficients. They evaluate, at q = 1, to the Catalan numbers: 1, 1, 2, 5, 14,…, a log-convex sequence. We use a combinatorial interpretation of these polynomials to prove a q-log-convexity result. The sequence of q-Catalan numbers is not q-log-convex in the narrow sense used by other authors, so our work suggests a more flexible definition of q-log convex be adopted. Received January 2, 2007     

Bombieri's Theorem in Short Intervals

Under the assumption of sixth power large sieve mean-value of Dirichlet L-function,we improve Bombieri's theorem in short intervals by virtue of the large sieve method and Heath-Brown's identity.     

A Note on the Kissing Numbers of Superballs

The lower bounds for the translative kissing numbers of superballs are studied in this note. We improve the bound given by Larman and Zong. Furthermore, we give a constructive bound based on algebraic-geometry codes that also improves the bound by Larman and Zong in almost all cases.     

A Note on Bernoulli Numbers and Shintani Generalized Bernoulli Polynomials

Generalized Bernoulli polynomials were introduced by Shintani in 1976 in order to express the special values at non-positive integers of Dedekind zeta functions for totally real numbers. The coefficients of such polynomials are finite combinations of products of Bernoulli numbers which are difficult to get hold of. On the other hand, Zagier was able to get the explicit formula for the special values in cases of real quadratic number fields.

In this paper, we shall improve Shintani's formula by proving that the special values can be determined by a finite set of polynomials. This provides a convenient way to evaluate the special values of various types of Dedekind functions. Indeed, a much broader class of zeta functions considered by the author [4] admits a similar formula for its special values. As a consequence, we are able to find infinitely many identities among Bernoulli numbers through identities among zeta functions. All these identities are difficult to prove otherwise.

     

A Short Note on Containment of Cores

We show that the operation of taking the core of an ideal does not preserve inclusions.     

Hua's Theorem on Prime Squares in Short Intervals

It is proved that every large integer N≡ 5(mod24) can be written as N = p ² ₁ + ... + p ² ₅ with each prime p _j satisfying This gives a short interval version of Hua's theorem on the quadratic Waring-Goldbach problem. Received October 14, 1999, Accepted April 17, 2000