On exponential sums over primes in short intervals

Let Λ(n) be the von Mangoldt function, x real and y small compared with x. This paper gives a non-trivial estimate on the exponential sum over primes in short intervals for all α ∈ [0,1] whenever . This result is as good as what was previously derived from the Generalized Riemann Hypothesis.     

Exponential sums over primes in short intervals

In this paper we establish one new estimate on exponential sums over primes in short intervals. As an application of this result, we sharpen Hua's result by proving that each sufficiently large integer N congruent to 5 modulo 24 can be written as N = p12 p22 p32 p42 p52, with |pj-(N/5)~(1/2)|≤U = N1/2-1/20 ε, where pj are primes. This result is as good as what one can obtain from the generalized Riemann hypothesis.     

Bombieri-type theorems for exponential sums over primes in short intervals and its application

We treat the estimate of mean value of exponential sums over primes in short intervals and obtain the related Bombieri-type theorems, which generalize the result of Liu and Zhan (Acta Arith LXXXII(3):197–227, 1997). Moreover, we present some application to the distribution of primes in the ternary quadratic form in short intervals.     

On certain exponential sums over primes

Let f(x) be a real valued polynomial in x of degree k?4 with leading coefficient α. In this paper, we prove a non-trivial upper bound for the quantity

     

On Cochrane sums over short intervals

The main purpose of this paper is to study the mean square value problem of Cochrane sums over short intervals by using the properties of Gauss sums and Kloosterman sums, and finally give a sharp asymptotic formula.     

Short cubic exponential sums over primes

For y ≥ x ^4/5 L ^8B+151 (where L = log(xq) and B is an absolute constant), a nontrivial estimate is obtained for short cubic exponential sums over primes of the form S ₃(α; x, y) = ∑ _x?y<n≤x Λ(n)e(αn ³), where α = a/q + θ/q ², (a, q) = 1, L ^32(B+20) < q ≤ y ⁵ x ^?2 L ^?32(B+20), |θ| ≤ 1, Λ is the von Mangoldt function, and e(t) = e ^2πit.     

On exponential sums over primes and application in Waring-Goldbach problem

In this paper, we prove the following estimate on exponential sums over primes: Let κ≥1,βκ=1/2 log κ/log2, x≥2 and α=a/q λsubject to (a, q) = 1, 1≤a≤q, and λ∈R. Then As an application, we prove that with at most O(N2/8 ε) exceptions, all positive integers up to N satisfying some necessary congruence conditions are the sum of three squares of primes. This result is as strong as what has previously been established under the generalized Riemann hypothesis.     

Explicit upper bounds for exponential sums over primes

We give explicit upper bounds for linear trigonometric sums over primes.

     

小区间内一个素数和三个素数的平方和问题

研究了把一个满足必要条件的自然数在小区间内分解成一个素数和三个素数平方和的问题,利用刘建亚和展涛处理扩大了的主区间的新方法,成功的缩短了小区间的长度.     

Almost primes in short intervals

In this paper,we prove that the short interval(x-x101/232,x] contains at least an almost prime P2 for sufficiently large x,where P2 denotes an integer having at most two prime factors counted with multiplicity.     

On the normal density of primes in short intervals

Selberg has shown on the basis of the Riemann hypothesis that for every ε > 0 most intervals |x,x+x^?| of length x^? contain approximately $\text{x}{}^{\mathrm{?}}\text{log}\text{x}$ primes. Here by “most” we mean “for a set of values of x of asymptotic density one.” Prachar has extended Selberg's result to primes in arithmetic progressions. Both authors noted that if we assume the quasi Riemann hypothesis, that ζ(s) has no zeros in the domain {$\text{σ>}\text{1}\text{2}\text{+δ}$} for some $\text{δ<}\text{1}\text{2}$, then the same conclusions hold, provided that ε > 2 δ. Here we give a simple proof of these theorems in a general context, where an arbitrary signed measure takes the place of d[ψ(x)?x]. Then we show by a counterexample that this general theorem is the best of its kind: the condition ε > 2δ cannot be replaced by ε = 2δ. In our example, the associated Dirichlet integral is an entire function which remains bounded on the domain {$\text{σ≥}\text{1}\text{2}\text{+δ}$}. Thus its growth and regularity properties are better than those of $\text{ζ′(s)}\text{ζ(s)}$. Nevertheless the corresponding signed measure behaves badly.     

Character sums over shifted primes

We obtain a new bound for sums of a multiplicative character modulo an integer q at shifted primes p + a over primes p ≤ N. Our bound is nontrivial starting with N ≥ q ^8/9+? for any ? > 0. This extends the range of the bound of Z. Kh. Rakhmonov that is nontrivial for N ≥ q ^1+? .     

Sums of primes and squares of primes in short intervals

Let H₂{\mathcal H_2} denote the set of even integers n \not o 1 mod 3{n \not\equiv 1 \pmod 3} . We prove that when H ≥ X ^0.33, almost all integers n ? H₂ ?(X, X + H]{n \in \mathcal H_2 \cap (X, X + H]} can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes.     

On the mean value of a sum analogous to character sums over short intervals

The main purpose of this paper is to study the mean value properties of a sum analogous to character sums over short intervals by using the mean value theorems for the Dirichlet L-functions, and to give some interesting asymptotic formulae. This work is supported by the N.S.F. (60472068) of P.R. China.     

A note on primes in short intervals

This paper is concerned with the number of primes in short intervals. We prove that , for θ > 1/2, with the assumption of an heuristic hypothesis weaker than the Lindel?f hypothesis. Received: 8 October 2007, Revised: 14 April 2008     

小区间上的三素数定理

设N是充分大的奇数,本文在广义Riemman 假设下证明了方程N=p1 p2 p3,pi-(N/3)≤U, i=1,2,3, U≥N(1/2)logN3 ε有解,此处pi是素数,并得到了方程解数的渐进式.     

算术数列中的素变数指数和问题

旨在应用初等方法研究指数和问题,给出了算术数列中素变数非线性指数和的一个上界估计.