On the Distribution of Values of Euler＇s Function over Integers in Arithmetic Progressions

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.

ON POINTS CONTAIN ARITHMETIC PROGRESSIONS IN THEIR LROTH EXPANSION

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.

On infinite arithmetic progressions in sumsets

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.

Rankin-Selberg coefficients in large arithmetic progressions

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.

The Ternary Goldbach Problem with Primes in Arithmetic Progressions

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.

On the Density of Integers of the Form 2~k + p in Arithmetic Progressions

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.

Primes in arithmetic progressions with friable indices

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.

An Additive Problem with Primes in Arithmetic Progressions

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.

Small Solutions of Quadratic Equations with Prime Variables in Arithmetic Progressions

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,

The Most Likely Common Difference of Arithmetic Progressions Among Primes

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.

Arithmetic Progressions,Different Regularity Lemmas and Removal Lemmas

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.

Infinite Number of Twin Primes

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.

Inequality of Realization of a Stochastic Dynamics Based on the Erdos Discrepancy Problem

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.

An Investigation on Generalized Eulerian Polynomials and Fractions

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

Gowers norms and pseudorandom measures of subsets

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.