On sums of seven cubes

We show that every integer between 1290741 and is a sum of 5 nonnegative cubes, from which we deduce that every integer which is a cubic residue modulo 9 and an invertible cubic residue modulo 37 is a sum of 7 nonnegative cubes.

     

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.     

A diophantine problem concerning cubes

This paper deals with the problem of finding n integers such that their pairwise sums are cubes. We obtain eight integers, expressed in parametric terms, such that all the six pairwise sums of four of these integers are cubes, 9 of the 10 pairwise sums of five of these integers are cubes, 12 pairwise sums of six of these integers are cubes, 15 pairwise sums of seven of these integers are cubes and 18 pairwise sums of all the eight integers are cubes. This leads to infinitely many examples of four positive integers such that all of their six pairwise sums are cubes. Further, for any arbitrary positive integer n, we obtain a set of 2(n+1) integers, in parametric terms, such that 5n+1 of the pairwise sums of these integers are cubes. With a choice of parameters, we can obtain examples with 5n+2 of the pairwise sums being cubes.     

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.      

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.     

Properties of the Beurling generalized primes

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In this paper, we prove a generalization of Mertens' theorem to Beurling primes, namely that , where γ is Euler's constant and Ax is the asymptotic number of generalized integers less than x. Thus the limit exists. We also show that this limit coincides with ; for ordinary primes this claim is called Meissel's theorem. Finally, we will discuss a problem posed by Beurling, namely how small |N(x)−[x]| can be made for a Beurling prime number system Q≠P, where P is the rational primes. We prove that for each c>0 there exists a Q such that and conjecture that this is the best possible bound.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=Kw3iNo3fAbk/.     

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.     

Three cubes in one number

Separate a three-digit number into its component digits. After raising each digit to the third power and computing the sum of the cubes, determine how often the original number reappears. Modular arithmetic is used to reduce the number of potential solutions to a more manageable quantity.     

Long arithmetic progressions in critical sets

Given a density 0<σ?1, we show for all sufficiently large primes p that if S⊆Z/pZ has the least number of three-term arithmetic progressions among all sets with at least σp elements, then S contains an arithmetic progression of length at least log^1/4+o(1)p.     

Group automorphisms with prescribed growth of periodic points,and small primes in arithmetic progressions in intervals

We investigate the question of which growth rates are possible for the number of periodic points of a compact group automorphism. Our arguments involve a modification of Linnik?s Theorem, concerning small prime numbers in arithmetic progressions which lie in intervals.     

On sums of a prime and four prime squares in short intervals

In this paper, we prove that each sufficiently large integer N ≠1（mod 3） can be written as N=p＋p1^2＋p2^2＋p3^2＋p4^2, with
｜p-N/5｜≤U,｜pj-√N/5｜≤U,j=1,2,3,4,
where U=N^2/20＋c and p,pj are primes.     

On the index of fractions with square-free denominators in arithmetic progressions

We prove asymptotic formulas for the first and second moments of the index of fractions with square-free denominators of order Q streaming in a given arithmetic progression as Q→∞. A. Zaharescu was supported by NSF grant number DMS-0456615. This research was also partially supported by the CERES Program 4-147/2004 of the Romanian Ministry of Education and Research.     

Primes representable by polynomials and the lower bound of the least primes in arithmetic progressions

Letf (m) be an irreducible quadratic polynomial with integral coefficients and positive leading coefficient. Under the assumption of Extended Riemann Hypothesis, we obtain new remainder terms in the upper bounds on primes represented byf(m) orf(p) which greatly improve Bantle's recent results. As an application, we obtain, in the second part of the paper, a new result on the lower bound of the least primes in arithmetic progressions with some difference.     

Almost-all results on the p λ problem. II

Summary Suppose that 1/2 ≦ λ < 1. Balog and Harman proved that for any real θ there exist infinitely many primes p satisfying p^λ-θ < p^{-(1-λ)/2+ ε} (with an asymptotic result). In the present paper we establish that for almost all θ in the interval 0 ≦ θ < 1 there exist infinitely many primes p such that {p^λ-θ} < p^{-min{(2-λ)/6,(14-9λ)/32}+ε}. Thus we obtain a better result for almost all θ than for a single θ if λ>1/2.     

Primes of the form in short intervals

In this note, we prove that for every and , the short interval contains at least one prime number of the form with . This improves a similar result due to Huxley and Iwaniec, which requires .

     

Metrical Theorems on Prime Values of the Integer Parts of Real Sequences

Let a_n be an increasing sequence of positive reals with a_n as n . Necessary and sufficient conditions are obtained foreach of the sequences to take on infinitely many prime values for almost all > rß.For example, the sequence a_n is infinitely often prime for almostall > 0 if and only if there is a subsequence of the a_n,say b_n, with b_{n + 1} > b_n + 1 and with the series divergent. Asymptotic formulae areobtained when the sequences considered are lacunary. An earlierresult of the author reduces the problem to estimating the measureof overlaps of certain sets, and sieve methods are used to obtainthe correct order upper bounds. 1991 Mathematics Subject Classification:primary 11N05; secondary 11K99, 11N36.     

On sums and products of integers

Erdös and Szemerédi proved that if is a set of positive integers, then there must be at least integers that can be written as the sum or product of two elements of , where is a constant and . Nathanson proved that the result holds for . In this paper it is proved that the result holds for and .

     

On sums and products of integers

Erdos and Szemerédi conjectured that if is a set of positive integers, then there must be at least integers that can be written as the sum or product of two elements of . Erdos and Szemerédi proved that this number must be at least for some and . In this paper it is proved that the result holds for .

     

On the Hybrid Power Mean Involving the Two-Term Exponential Sums and Polynomial Character Sums

The main purpose of this paper is using the analytic method and the properties of trigonometric sums and character sums to study the computational problem of one kind hybrid power mean involving two-term exponential sums and polynomial character sums. Then the authors give some interesting calculating formulae for them.     

Linear Forms in Finite Sets of Integers

Let A_1, ..., A_r be finite, nonempty sets of integers, and let h_1,..., h_r be positive integers. The linear formh_1A_1 + ··· + h_rA_r is the set of all integers of the form b_1 + ··· + b_r, where b_i is an integer that can be represented as the sum ofh_i elements of the set A_i. In this paper, the structure of the linear form h_1A_1 + ··· + h_rA_r is completely determined for all sufficiently large integersh_i .