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

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   

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   

Mean Value of Kloosterman Sums over Short Intervals

This paper is concerned with a kind of mean value problem of Kloosterman sums, which will lead to a sum of Kloosterman sums over short intervals.     

Short Kloosterman Sums with Primes

Mathematical Notes - A new estimate of the Kloosterman sum with primes modulo a prime number q is obtained, in which the number of summands can be of order q0.5+ε. This estimate refines...     

On Fractional Part Sums: A Mean-Square Asymptotics over Short Intervals

 This article is concerned with sums 𝒮(t) = ∑ _n  ψ(tf(n/t)) where ψ denotes, essentially, the fractional part minus ?, f is a C ⁴-function with f″ ≠ 0 throughout, summation being extended over an interval of order t. We establish an asymptotic formula for ∫ _T−Λ ^T+Λ (𝒮(t))²dt for any Λ = Λ(T) growing faster than log T. Received April 30, 2001; in revised form February 15, 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

素变数线性三角和的估计

设实效满足（ａ，ｑ）＝１：｜θ｜≤１．Ｎ≥３是一个整数。记ｒ＝ｌｏｇＮ，ｅ（ａｎ）＝ｅ￣（２πｉａｎ），为Ｍａｎｇｏｌｄｔ函数。在本文中，我们证明了     

A Note on Primes and Goldbach Numbers in Short Intervals

Let J(N, H) be the Selberg integral and E(x, T) the error term in Kaczorowski-Perelli's weighted form of the classical explicit formula. We prove that the estimate J(N, H) = o(H2 N) is connected with an appropriate estimate of _N ^2N| E(x,T)² dx, uniformly for H and T in some ranges. Moreover, assuming a suitable bound for _N ^2N| E(x,T)|² dx, we also obtain, for all sufficiently large N and H (log N)11/12, that every interval [N,N + H] contains H Goldbach numbers.     

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.     

The Size of Exponential Sums on Intervals of the Real Line

We prove that there is a constant c>0 depending only on M≥1 and μ≥0 such that

On the Problem of Parameter Estimation in Exponential Sums

Let I≥1 be an integer, ω ₀=0<ω ₁<⋯<ω _I ≤π, and for j=0,…,I, a _j ∈ℂ, a_-j=[`(a_j)]a_{-j}={\overline{{a_{j}}}}, ω _−j =−ω _j , and a_j 1 0a_{j}\not=0 if j 1 0j\not=0. We consider the following problem: Given finitely many noisy samples of an exponential sum of the form

Sums of Values of Nonprincipal Characters over a Sequence of Shifted Primes

For a nonprincipal character χ modulo D, we prove a nontrivial estimate of the form Σ_n≤x Λ(n)χ(n ? l) \( \ll x\exp \{ - 0.6\sqrt {\ln D} \} \) for the sum of values of χ over a sequence of shifted primes in the case when x ≥ D^1/2+ε, (l,D) = 1, and the modulus of the primitive character generated by χ is a cube-free number.     

On the Fourth Power Mean of the Character Sums Over Short Intervals

The main purpose of this paper is to study the mean value properties of the character sums over the interval [1,p/8） by using the mean value theorems of the Dirichlet L-functions, and give an interesting mean value formula for this study.     

Sums of Three or More Primes

It has long been known that, under the assumption of the Riemann Hypothesis, one can give upper and lower bounds for the error in the Prime Number Theorem, such bounds being within a factor of of each other and this fact being equivalent to the Riemann Hypothesis. In this paper we show that, provided ``Riemann Hypothesis' is replaced by ``Generalized Riemann Hypothesis', results of similar (often greater) precision hold in the case of the corresponding formula for the representation of an integer as the sum of primes for , and, in a mean square sense, for . We also sharpen, in most cases to best possible form, the original estimates of Hardy and Littlewood which were based on the assumption of a ``Quasi-Riemann Hypothesis'. We incidentally give a slight sharpening to a well-known exponential sum estimate of Vinogradov-Vaughan.

     

Character Sums in Short Intervals and the Multiplication Table Modulo a Large Prime

We obtain nontrivial estimates of character sums over short intervals for almost all moduli. These bounds and the method of Karatsuba for solving multiplicative ternary problems are used to prove that for π(X)(1 + o(1)) primes p,p ≤ X, there are p(1 + o(1)) residue classes modulo p of the form xy (mod p), where 1 ≤ x, y ≤ p^?(log p)^1,087. We also prove that for any prime p there are p(1 + o(1)) residue classes modulo p of the form xy^* (mod p), where 1 ≤ x, y ≤ p^?(log p)^1+o(1) and y^* is defined by yy^* ≡ 1 (mod p).     

Character Sums in Short Intervals and the Multiplication Table Modulo a Large Prime

We obtain nontrivial estimates of character sums over short intervals for almost all moduli. These bounds and the method of Karatsuba for solving multiplicative ternary problems are used to prove that for π(X)(1 + o(1)) primes p,p ≤ X, there are p(1 + o(1)) residue classes modulo p of the form xy (mod p), where 1 ≤ x, y ≤ p^?(log p)^1,087. We also prove that for any prime p there are p(1 + o(1)) residue classes modulo p of the form xy^* (mod p), where 1 ≤ x, y ≤ p^?(log p)^1+o(1) and y^* is defined by yy^* ≡ 1 (mod p).     

有限域上多项式指数和的p-adic估计

设f为有限域上的一个多元多项式.它的次数矩阵Df由出现在f中的次数向量构成.本文利用Df给出了多项式f的指数和的一个p-adic估计,它改进和推广了该方向许多已知的结果.     

Difference Sets and Positive Exponential Sums I. General Properties

We describe general connections between intersective properties of sets in Abelian groups and positive exponential sums. In particular, given a set A the maximal size of a set whose difference set avoids A will be related to positive exponential sums using frequencies from A.     

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.