Let φ(n) denote the Euler-totient function, we study the distribution of solutions of φ(n) ≤ x in arithmetic progressions, where n ≡ l(mod q) and an asymptotic formula was obtained by Perron formula.
For any x ∈ （0, 1] （except at most countably many points）, there exists a unique sequence {dn（x）}n≥1 of integers, called the digit sequence of x, such that x =∞ ∑j=1 1/d1（x）（d1（x）-1）……dj-1（x）（dj-1...For any x ∈ （0, 1] （except at most countably many points）, there exists a unique sequence {dn（x）}n≥1 of integers, called the digit sequence of x, such that x =∞ ∑j=1 1/d1（x）（d1（x）-1）……dj-1（x）（dj-1（x）-1）dj（x）. The dexter infinite series expansion is called the Liiroth expansion of x. This paper is con- cerned with the size of the set of points x whose digit sequence in its Liiroth expansion is strictly increasing and contains arbitrarily long arithmetic progressions with arbitrary com- mon difference. More precisely, we determine the Hausdorff dimension of the above set.展开更多
Let k be a positive integer.Denote by D_(1/k)the least integer d such that for every set A of nonnegative integers with the lower density 1/k,the set(k+1)A contains an infinite arithmetic progression with difference a...Let k be a positive integer.Denote by D_(1/k)the least integer d such that for every set A of nonnegative integers with the lower density 1/k,the set(k+1)A contains an infinite arithmetic progression with difference at most d,where(k+1)A is the set of all sums of k+1 elements(not necessarily distinct)of A.Chen and Li(2019)conjectured that D_(1/k)=k~2+o(k~2).The purpose of this paper is to confirm the above conjecture.We also prove that D_(1/k)is a prime for all sufficiently large integers k.展开更多
Let(λ_f(n))_(n≥1)be the Hecke eigenvalues of either a holomorphic Hecke eigencuspform or a Hecke-Maass cusp form f.We prove that,for any fixedη>0,under the Ramanujan-Petersson conjecture for GL_(2)Maass forms,th...Let(λ_f(n))_(n≥1)be the Hecke eigenvalues of either a holomorphic Hecke eigencuspform or a Hecke-Maass cusp form f.We prove that,for any fixedη>0,under the Ramanujan-Petersson conjecture for GL_(2)Maass forms,the Rankin-Selberg coefficients(λ_f(n)^(2))_(n≥1)admit a level of distributionθ=2/5+1/260-ηin arithmetic progressions.展开更多
This lecture note is mainly about arithmetic progressions,different regularity lemmas and removal lemmas.We will be very brief most of the time,trying to avoid technical details,even definitions.For most technical det...This lecture note is mainly about arithmetic progressions,different regularity lemmas and removal lemmas.We will be very brief most of the time,trying to avoid technical details,even definitions.For most technical details,we refer the reader to references.Apart from arithmetic progressions,we also discuss property testing and extremal graph theory.展开更多
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 const...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.展开更多
Consider all the arithmetic progressions of odd numbers, no term of which is of the form 2^k ＋ p, where k is a positive integer and p is an odd prime. ErdSs ever asked whether all these progressions can be obtained f...Consider all the arithmetic progressions of odd numbers, no term of which is of the form 2^k ＋ p, where k is a positive integer and p is an odd prime. ErdSs ever asked whether all these progressions can be obtained from covering congruences. In this paper, we characterize all arithmetic progressions in which there are positive proportion natural numbers that can be expressed in the form 2^k ＋ p, and give a quantitative form of Romanoff＇s theorem on arithmetic progressions. As a corollary, we prove that the answer to the above Erdos problem is affirmative.展开更多
We consider the numberπ(x,y;q,a)of primes p≤such that p≡a(mod q)and(p-a)/q is free of prime factors greater than y.Assuming a suitable form of Elliott-Halberstam conjecture,it is proved thatπ(x,y:q,a)is asymptotic...We consider the numberπ(x,y;q,a)of primes p≤such that p≡a(mod q)and(p-a)/q is free of prime factors greater than y.Assuming a suitable form of Elliott-Halberstam conjecture,it is proved thatπ(x,y:q,a)is asymptotic to p(log(x/q)/log y)π(x)/φ(q)on average,subject to certain ranges of y and q,where p is the Dickman function.Moreover,unconditional upper bounds are also obtained via sieve methods.As a typical application,we may control more effectively the number of shifted primes with large prime factors.展开更多
Abstract For relatively prime positive integers u0 and r, and for 0 〈 k ≤ n, define uk ：= u0 ＋ kr. Let Ln ：= 1cm（u0,u1,... ,un） and let a,l≥2 be any integers. In this paper, the authors show that, for integers...Abstract For relatively prime positive integers u0 and r, and for 0 〈 k ≤ n, define uk ：= u0 ＋ kr. Let Ln ：= 1cm（u0,u1,... ,un） and let a,l≥2 be any integers. In this paper, the authors show that, for integers α≥ a, r ≥max（a,l - 1） and n ≥lατ, the following inequality holds Ln≥u0r^（l-1）α＋a-l（r＋1）^n.Particularly, letting l = 3 yields an improvement on the best previous lower bound on Ln obtained by Hong and Kominers in 2010.展开更多
We prove a result on the distribution of the general divisor functions in arithmetic progressions to smooth moduli which exceed the square root of the length.
A necessary and sufficient solvable condition for diagonal quadratic equation with prime variables in arithmetic progressions is given, and the best qualitative bound for small solutions of the equation is obtained,
In this paper, we extend a classical result of Hua to arithmetic progressionswith large moduli. The result implies the Linnik Theorem on the least prime in an arithmeticprogression.
Let d^(∗)_(k)(x)be the most likely common differences of arithmetic progressions of length k+1 among primes≤x.Based on the truth of Hardy–Littlewood Conjecture,we obtain that lim x→+∞d^(∗)_(k)(x)(x)=+∞uniformly i...Let d^(∗)_(k)(x)be the most likely common differences of arithmetic progressions of length k+1 among primes≤x.Based on the truth of Hardy–Littlewood Conjecture,we obtain that lim x→+∞d^(∗)_(k)(x)(x)=+∞uniformly in k,and every prime divides all sufficiently large most likely common differences.展开更多
This note establishes a pair of exponential generating functions for generalized Eulerian polynomials and Eulerian fractions, respectively. A kind of recurrence relation is obtained for the Eulerian fractions. Finally...This note establishes a pair of exponential generating functions for generalized Eulerian polynomials and Eulerian fractions, respectively. A kind of recurrence relation is obtained for the Eulerian fractions. Finally, a short proof of a certain summarion formula is given展开更多
This work is devoted to the theory of prime numbers. Firstly it introduced the concept of matrix primes, which can help to generate a sequence of prime numbers. Then it proposed a number of theorems, which together wi...This work is devoted to the theory of prime numbers. Firstly it introduced the concept of matrix primes, which can help to generate a sequence of prime numbers. Then it proposed a number of theorems, which together with theorem of Dirichlet, Siegel and Euler allow to prove the infinity of twin primes.展开更多
This paper proposes a stochastic dynamics model in which people who are endowed with different discount factors chose to buy the capital stock periodically with different periodicities and are exposed to randomness at...This paper proposes a stochastic dynamics model in which people who are endowed with different discount factors chose to buy the capital stock periodically with different periodicities and are exposed to randomness at arithmetic progression times. We prove that the realization of a stochastic equilibrium may render to the people quite unequal benefits. Its proof is based on Erdös Discrepancy Problem that an arithmetic progression sum of any sign sequence goes to infinity, which is recently solved by Terence Tao [1]. The result in this paper implies that in some cases, the sources of inequality come from pure luck.展开更多
The distribution of the Pjateckii-Sapiro prime numbers in arithmetic progressions is investigated, and a Bombier i- Vinogr adov typ e mean- value t heorem and anot her almost all resultconcerning this problem are esta...The distribution of the Pjateckii-Sapiro prime numbers in arithmetic progressions is investigated, and a Bombier i- Vinogr adov typ e mean- value t heorem and anot her almost all resultconcerning this problem are established.展开更多
Let A ■ ■_(N),and f_(A)(s)={1-|A/N,-|A|/N,for s ∈A,for s■ A.We define the pseudorandom measure of order k of the subset A as follows,P _(k)(A,N)=max D|∑n∈■_(N)|f_(A)(n+c_(1))f_(A)(n+c_(2))…f_(A)(n+c_(k))|where...Let A ■ ■_(N),and f_(A)(s)={1-|A/N,-|A|/N,for s ∈A,for s■ A.We define the pseudorandom measure of order k of the subset A as follows,P _(k)(A,N)=max D|∑n∈■_(N)|f_(A)(n+c_(1))f_(A)(n+c_(2))…f_(A)(n+c_(k))|where the maximum is taken over all D=(c_(1),c_(2),…,C_(K))∈■^(k) with 0≤c_(1)<c_(2)<…ck≤N-1.The subset A ■ ■_(N) is considered as a pseudorandom subset of degree k if P_(k)(A,N)is“small”in terms of N.We establish a link be tween the Gowers norm and our pseudorandom measure,and show that“good”pseudorandom subsets must have“small”Gowers norm.We give an example to suggest that subsets with"small" Gowers norm may have large pseudorandom measure.Finally,we prove that the pseudorandom subset of degree L(k)contains an arithmetic progression of length k,where L(k)=2·lcm(2,4,…,2|k/2|),for k≥4,and lcm(a1,a2,…,al)denotes the least common multiple of a1,a2,…,al.展开更多
It is proved constructively that there exists a thin subset S of primes, satisfying for some absolute constant c＞0, such that every sufficiently large odd integer N can beLet r be prime, and hi positive integers with...It is proved constructively that there exists a thin subset S of primes, satisfying for some absolute constant c＞0, such that every sufficiently large odd integer N can beLet r be prime, and hi positive integers with (bj, r) = 1,j = 1, 2, 3. It is also proved that, for almost all prime moduli r＜ log- N, every sufficiently large odd integer N = b1 + b2 +ba(modr) can be represented as where c ＞ 0 is an absolute constant.展开更多
基金Supported by the National Natural Science Foundation of China（11271249） Supported by the Scientific and Technological Research Program of Chongqing Municipal Education Commission（1601213） Supported by the Scientific Research Program of Yangtze Normal University（2012XJYBO31）
文摘Let φ(n) denote the Euler-totient function, we study the distribution of solutions of φ(n) ≤ x in arithmetic progressions, where n ≡ l(mod q) and an asymptotic formula was obtained by Perron formula.
文摘For any x ∈ （0, 1] （except at most countably many points）, there exists a unique sequence {dn（x）}n≥1 of integers, called the digit sequence of x, such that x =∞ ∑j=1 1/d1（x）（d1（x）-1）……dj-1（x）（dj-1（x）-1）dj（x）. The dexter infinite series expansion is called the Liiroth expansion of x. This paper is con- cerned with the size of the set of points x whose digit sequence in its Liiroth expansion is strictly increasing and contains arbitrarily long arithmetic progressions with arbitrary com- mon difference. More precisely, we determine the Hausdorff dimension of the above set.
基金supported by National Natural Science Foundation of China(Grant Nos.12171243 and 11922113)the National Key Research and Development Program of China(Grant No.2021YFA1000700)。
文摘Let k be a positive integer.Denote by D_(1/k)the least integer d such that for every set A of nonnegative integers with the lower density 1/k,the set(k+1)A contains an infinite arithmetic progression with difference at most d,where(k+1)A is the set of all sums of k+1 elements(not necessarily distinct)of A.Chen and Li(2019)conjectured that D_(1/k)=k~2+o(k~2).The purpose of this paper is to confirm the above conjecture.We also prove that D_(1/k)is a prime for all sufficiently large integers k.
文摘Let(λ_f(n))_(n≥1)be the Hecke eigenvalues of either a holomorphic Hecke eigencuspform or a Hecke-Maass cusp form f.We prove that,for any fixedη>0,under the Ramanujan-Petersson conjecture for GL_(2)Maass forms,the Rankin-Selberg coefficients(λ_f(n)^(2))_(n≥1)admit a level of distributionθ=2/5+1/260-ηin arithmetic progressions.
文摘This lecture note is mainly about arithmetic progressions,different regularity lemmas and removal lemmas.We will be very brief most of the time,trying to avoid technical details,even definitions.For most technical details,we refer the reader to references.Apart from arithmetic progressions,we also discuss property testing and extremal graph theory.
基金Project supported partially by NNSF of China NSF of Henan Province
文摘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.
基金Supported by National Natural Science Foundation of China (Grant Nos. 10771103 and 10801075)
文摘Consider all the arithmetic progressions of odd numbers, no term of which is of the form 2^k ＋ p, where k is a positive integer and p is an odd prime. ErdSs ever asked whether all these progressions can be obtained from covering congruences. In this paper, we characterize all arithmetic progressions in which there are positive proportion natural numbers that can be expressed in the form 2^k ＋ p, and give a quantitative form of Romanoff＇s theorem on arithmetic progressions. As a corollary, we prove that the answer to the above Erdos problem is affirmative.
基金supported by the Programme de Recherche Conjoint CNRS-NSFC(Grant No.1457)supported by National Natural Science Foundation of China(Grant No.11531008)+3 种基金the Ministry of Education of China(Grant No.IRT16R43)the Taishan Scholar Project of Shandong Provincesupported by National Natural Science Foundation of China(Grant No.11601413)NSBRP of Shaanxi Province(Grant No.2017JQ1016)
文摘We consider the numberπ(x,y;q,a)of primes p≤such that p≡a(mod q)and(p-a)/q is free of prime factors greater than y.Assuming a suitable form of Elliott-Halberstam conjecture,it is proved thatπ(x,y:q,a)is asymptotic to p(log(x/q)/log y)π(x)/φ(q)on average,subject to certain ranges of y and q,where p is the Dickman function.Moreover,unconditional upper bounds are also obtained via sieve methods.As a typical application,we may control more effectively the number of shifted primes with large prime factors.
基金supported by the National Natural Science Foundation of China(No.10971145)the Ph.D.Programs Foundation of Ministry of Education of China(No.20100181110073)the Science&Technology Program of Sichuan Province(No.2013JY0125)
文摘Abstract For relatively prime positive integers u0 and r, and for 0 〈 k ≤ n, define uk ：= u0 ＋ kr. Let Ln ：= 1cm（u0,u1,... ,un） and let a,l≥2 be any integers. In this paper, the authors show that, for integers α≥ a, r ≥max（a,l - 1） and n ≥lατ, the following inequality holds Ln≥u0r^（l-1）α＋a-l（r＋1）^n.Particularly, letting l = 3 yields an improvement on the best previous lower bound on Ln obtained by Hong and Kominers in 2010.
文摘We prove a result on the distribution of the general divisor functions in arithmetic progressions to smooth moduli which exceed the square root of the length.
基金supported partially by the National Natural Science Foundation of China (Grant No.10671056)
文摘A necessary and sufficient solvable condition for diagonal quadratic equation with prime variables in arithmetic progressions is given, and the best qualitative bound for small solutions of the equation is obtained,
基金Project supported by National Natural Science Foundation(No.10171027,60373039)of ChinaResearch Foundation(No.XK01071)of Henan University
文摘In this paper, we extend a classical result of Hua to arithmetic progressionswith large moduli. The result implies the Linnik Theorem on the least prime in an arithmeticprogression.
文摘Let d^(∗)_(k)(x)be the most likely common differences of arithmetic progressions of length k+1 among primes≤x.Based on the truth of Hardy–Littlewood Conjecture,we obtain that lim x→+∞d^(∗)_(k)(x)(x)=+∞uniformly in k,and every prime divides all sufficiently large most likely common differences.
文摘This note establishes a pair of exponential generating functions for generalized Eulerian polynomials and Eulerian fractions, respectively. A kind of recurrence relation is obtained for the Eulerian fractions. Finally, a short proof of a certain summarion formula is given
文摘This work is devoted to the theory of prime numbers. Firstly it introduced the concept of matrix primes, which can help to generate a sequence of prime numbers. Then it proposed a number of theorems, which together with theorem of Dirichlet, Siegel and Euler allow to prove the infinity of twin primes.
文摘This paper proposes a stochastic dynamics model in which people who are endowed with different discount factors chose to buy the capital stock periodically with different periodicities and are exposed to randomness at arithmetic progression times. We prove that the realization of a stochastic equilibrium may render to the people quite unequal benefits. Its proof is based on Erdös Discrepancy Problem that an arithmetic progression sum of any sign sequence goes to infinity, which is recently solved by Terence Tao [1]. The result in this paper implies that in some cases, the sources of inequality come from pure luck.
文摘The distribution of the Pjateckii-Sapiro prime numbers in arithmetic progressions is investigated, and a Bombier i- Vinogr adov typ e mean- value t heorem and anot her almost all resultconcerning this problem are established.
基金supported in part by the National Natural Science Foundation of China(Grant No.11571277).
文摘Let A ■ ■_(N),and f_(A)(s)={1-|A/N,-|A|/N,for s ∈A,for s■ A.We define the pseudorandom measure of order k of the subset A as follows,P _(k)(A,N)=max D|∑n∈■_(N)|f_(A)(n+c_(1))f_(A)(n+c_(2))…f_(A)(n+c_(k))|where the maximum is taken over all D=(c_(1),c_(2),…,C_(K))∈■^(k) with 0≤c_(1)<c_(2)<…ck≤N-1.The subset A ■ ■_(N) is considered as a pseudorandom subset of degree k if P_(k)(A,N)is“small”in terms of N.We establish a link be tween the Gowers norm and our pseudorandom measure,and show that“good”pseudorandom subsets must have“small”Gowers norm.We give an example to suggest that subsets with"small" Gowers norm may have large pseudorandom measure.Finally,we prove that the pseudorandom subset of degree L(k)contains an arithmetic progression of length k,where L(k)=2·lcm(2,4,…,2|k/2|),for k≥4,and lcm(a1,a2,…,al)denotes the least common multiple of a1,a2,…,al.
文摘It is proved constructively that there exists a thin subset S of primes, satisfying for some absolute constant c＞0, such that every sufficiently large odd integer N can beLet r be prime, and hi positive integers with (bj, r) = 1,j = 1, 2, 3. It is also proved that, for almost all prime moduli r＜ log- N, every sufficiently large odd integer N = b1 + b2 +ba(modr) can be represented as where c ＞ 0 is an absolute constant.
