The distribution of sequences in residue classes

We prove that any set of integers with lies in at least many residue classes modulo most primes . (Here is a positive constant.) This generalizes a result of Erdos and Ram Murty, who proved in connection with Artin's conjecture on primitive roots that the integers below which are multiplicatively generated by the coprime integers (i.e. whose counting function is also ) lie in at least residue classes, modulo most small primes , where as .

     

关于一个素数和一个素数的k次方和问题

设k≥2,Hk表示一个正整数n的集合,使对任意的正整数q,同余方程a+b2三n(modq)在模q的既约剩余系中有解a,b.Dk(N)表示n≤N,n∈Hk,但不能表成p1+p22=n的数的个数,其中p1,p2表示素数.则在GRH下,Dk(N)<<N1-1/k(h(k)+1)+ε,这里k=2,3;h(2)=2,h(3)=8.     

算术级数中的五素数平方和定理

设N是充分大的正整数满足N≡5mod 24,l和d是满足(l,d)=1的整数.A0,A>1是满足A0=600A+2000的正常数.本文证明对所有的整数0相似文献   

Large sieve inequalities for -forms in the conductor aspect

Duke and Kowalski in [A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, Invent. Math. 139(1) (2000) 1–39 (with an appendix by Dinakar Ramakrishnan)] derive a large sieve inequality for automorphic forms on GL(n) via the Rankin–Selberg method. We give here a partial complement to this result: using some explicit geometry of fundamental regions, we prove a large sieve inequality yielding sharp results in a region distinct to that in [Duke and Kowalski, A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, Invent. Math. 139(1) (2000) 1–39 (with an appendix by Dinakar Ramakrishnan)]. As an application, we give a generalization to GL(n) of Duke's multiplicity theorem from [Duke, The dimension of the space of cusp forms of weight one, Internat. Math. Res. Notices (2) (1995) 99–109 (electronic)]; we also establish basic estimates on Fourier coefficients of GL(n) forms by computing the ramified factors for GL(n)×GL(n) Rankin–Selberg integrals.     

On the number of empty boxes in the Bernoulli sieve II

The Bernoulli sieve is the infinite “balls-in-boxes” occupancy scheme with random frequencies _Pk=_W1?_Wk−1(1−_Wk)$P_k=W_1?W_{k-1}(1-W_k)$, where _(Wk)k∈N${\left(W_k\right)}_{k\in \mathbb{N}}$ are independent copies of a random variable W$W$ taking values in (0,1)$(0,1)$. Assuming that the number of balls equals n$n$, let _Ln$L_n$ denote the number of empty boxes within the occupancy range. In this paper, we investigate convergence in distribution of _Ln$L_n$ in the two cases which remained open after the previous studies. In particular, provided that E|logW|=E|log(1−W)|=∞$\mathbb{E}|logW|=\mathbb{E}|log(1-W)|=\infty$ and that the law of W$W$ assigns comparable masses to the neighborhoods of 0 and 1, it is shown that _Ln$L_n$ weakly converges to a geometric law. This result is derived as a corollary to a more general assertion concerning the number of zero decrements of nonincreasing Markov chains. In the case that E|logW|<∞$\mathbb{E}|logW|<\infty$ and E|log(1−W)|=∞$\mathbb{E}|log(1-W)|=\infty$, we derive several further possible modes of convergence in distribution of _Ln$L_n$. It turns out that the class of possible limiting laws for _Ln$L_n$, properly normalized and centered, includes normal laws and spectrally negative stable laws with finite mean. While investigating the second problem, we develop some general results concerning the weak convergence of renewal shot-noise processes. This allows us to answer a question asked by Mikosch and Resnick (2006) [18].     

The large sieve inequality for integer polynomial amplitudes

We obtain a close to optimal version of the large sieve inequality with amplitudes given by the values of a polynomial with integer coefficients of degree ?2.     

Sums of five almost equal prime squares II

It is proved that every large integerN≡5 (mod 24) can be written as with each primep _j satisfying , which gives a short interval version of a classical theorem of Hua. Project supported by the Trans-Century Training Programme Foundation for the Talents by the State Education Commission and the National Natural Science Foundation of China (Grant No. 19701019).     

On Hilbert Cubes in Certain Sets

A set of the form {d + _{i i} a : _i = 0 or 1, _{i i} < } (where d is a non-negative integer, a₁, a₂, ... are positive integers) is called a Hilbert cube. If {a₁, a₂, ...} is a finite set of, say, k elements, then it called a k-cube, while if {a₁, a₂, ...} is infinite, then the cube is said to be an infinite cube. As a partial answer to a question of Brown, Erdös and Freedman, an upper bound is given for the size of a Hilbert cube contained in the set of the squares not exceeding n. Estimates of Gaussian sums, Gallagher's large sieves and a result of Olson play a crucial rule in the proof. Hilbert cubes in other special sets are also studied.     

A larger GL2 large sieve in the level aspect

In this paper we study the orthogonality of Fourier coefficients of holomorphic cusp forms in the sense of large sieve inequality. We investigate the family of GL ₂ cusp forms modular with respect to the congruence subgroups Γ₁(q), with additional averaging over the levels q ∼ Q. We obtain the orthogonality in the range N ≪ Q ^2−δ for any δ > 0, where N is the length of linear forms in the large sieve.     

Terence Tao的一个问题

解决了Terence Tao提出的一个问题.证明了:设K≥2,N充分大,L_N为{-KN,…,KN}的任意子集,|L_N|=K.那么在[N,(1+1/K)N]中至少存在C_K N/(log N)个素数p,使得|kp+ja~i+l|为合数,其中1≤a,|j|,k≤K,1≤i≤K log N,l∈L_N,ja~i+l≠0,常数C_K0与K有关.     

On Bombieri's asymptotic sieve

We improve the error term in the Bombieri asymptotic sieve when the summation is restricted to integers having at most two prime factors. This results in a refined bilinear decomposition for the characteristic function of the primes that enables us to get a best possible estimate for the trigonometric polynomial over primes.     

On a multiplicative function on the set of shifted primes

It is proved that if ƒ(n) is a multiplicative function taking a valueζ on the set of primes such thatζ ³ = 1,ζ ≠ 1 andƒ ³(p ^r)=1 forr≥2, then there exists aθ ∈ (0, 1), for which

A Barban-Davenport-Halberstam theorem for integers with fixed number of prime divisors

The purpose of this paper is to study the distribution of integers with a given number prime divisors over arithmetic progressions, via using the large-sieve inequality, Huxley-Hooley contour and the zero-density estimate, and present a Barban-Davenport-Halberstam type theorem for it.     

On the distribution of

Suppose that α is an irrational number and β is a real number. It is proved that there are infinitely many prime numbers p such that ‖ αp-β‖ <p ^-9/28.     

On Diophantine approximation with one prime and three squares of primes

Let λ₁, λ₂, λ₃, λ₄ be non-zero real numbers, not all of the same sign, w real. Suppose that the ratios λ₁/λ₂, λ₁/λ₃ are irrational and algebraic. Then there are in.nitely many solutions in primes p_j, j =1, 2, 3, 4, to the inequality . This improves the earlier result.     

On the large sieve in algebraic number fields

In the present note Bombieri's central theorem concerning the average distribution of the prime numbers in arithmetic progressions is generalized to arbitrary algebraic number fields.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 673–680, December, 1967.Finally, I express my profound gratitude to B. V. Levin for setting the problem and the help he rendered and to A. I. Vinogradov for valuable suggestions.     

On the ternary goldbach problem with primes in independent arithmetic progressions

We show that for every fixed A > 0 and θ > 0 there is a ϑ = ϑ(A, θ) > 0 with the following property. Let n be odd and sufficiently large, and let Q ₁ = Q ₂:= n ^1/2(log n)^−ϑ and Q ₃:= (log n) ^θ . Then for all q ₃ ≦ Q ₃, all reduced residues a ₃ mod q ₃, almost all q ₂ ≦ Q ₂, all admissible residues a ₂ mod q ₂, almost all q ₁ ≦ Q ₁ and all admissible residues a ₁ mod q ₁, there exists a representation n = p ₁ + p ₂ + p ₃ with primes p _i ≡ a _i (q _i ), i = 1, 2, 3.      

On twin primes associated with the Hawkins random sieve

We establish an asymptotic formula for the number of k-difference twin primes associated with the Hawkins random sieve, which is a probabilistic model of the Eratosthenes sieve. The formula for k=1 was obtained by M.C. Wunderlich [A probabilistic setting for prime number theory, Acta Arith. 26 (1974) 59-81]. We here extend this to k?2 and generalize it to all l-tuples of Hawkins primes.