Second numbers in arithmetic progressions

On Arithmetic Progressions in Recurrences - A new characterization of the Fibonacci sequence

We show that essentially the Fibonacci sequence is the unique binary recurrence which contains infinitely many three-term arithmetic progressions. A criterion for general linear recurrences having infinitely many three-term arithmetic progressions is also given.

算术级数中素数间的巨大间隙——一种经验方法

Large gaps between primes in arithmetic progressions -- an empirical approach

本文概述了算术级数中素数间间隙的全新穷举计算结果，并给出了异常大的最小素数的新数值结果。

The ternary Goldbach problem with primes from arithmetic progressions

We establish Bombieri-Vinogradov's type result for the number of solutions of the ternary Goldbach problem with primes from arithmetic progressions.

Squarefull numbers in arithmetic progression II

In this paper, we improve the error term in a previous paper on an asymptotic formula for the number of squarefull numbers in an arithmetic progression.

On the remainder term in the circle problem in an arithmetic progression

In this paper we improve the estimate for the remainder term in the asymptotic formula concerning the circle problem in an arithmetic progression.

On a variance for twins of $k-$free numbers in arithmetic progressions

In this paper, we give a new upper bound of Barban-Davenport-Halberstam type for twins of $k-$free numbers in arithmetic progressions.

On Polignac's conjecture and arithmetic progressions

In this paper we investigate the recent advances by Zhang, Maynard and Pintz towards Polignac's conjecture and give some new results concerning the relationship between Polignac numbers and arithmetic progressions.

On an approximation by Vaughan in restricted sets of arithmetic progressions

We investigate the approximation to the number of primes in arithmetic progressions given by Vaughan. Instead of averaging the expected error term over all residue classes to modules in a given range, here we only consider subsets of arithmetic progressions that satisfy additional congruence conditions and provide asymptotic approximations.

General divisor functions in arithmetic progressions to large moduli

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.

Improved bound in Roth's theorem on arithmetic progressions

We prove that if $A\subseteq \{1,\dots,N\}$ does not contain any non-trivial three-term arithmetic progression, then $$|A|\ll \frac{(\log\log N)^{3+o(1)}}{\log N}N\,.$$

A note on small gaps between primes in arithmetic progressions

We implement the Maynard-Tao method of detecting primes in tuples to investigate small gaps between primes in arithmetic progressions, with bounds that are uniform over a range of moduli.

Arbitrarily Long Arithmetic Progressions of Primes on the Nose

This paper has been withdrawn

Exponential sums over Piatetski-Shapiro primes in arithmetic progressions

Let $γ<1<c$ and $19(c-1)+171(1-γ)<9$. In this paper, we establish an asymptotic formula for exponential sums over Piatetski-Shapiro primes $p=[n^{1/γ}]$ in arithmetic progressions.

Composite Rational Functions and Arithmetic Progressions

In this paper we deal with composite rational functions having zeros and poles forming consecutive elements of an arithmetic progression. We also correct a result published earlier related to composite rational functions having a fixed number of zeros and poles.

Are there arbitrarily long arithmetic progressions in the sequence of twin primes? II

We show that if besides the primes some other sequences (involving the Liouville function and the primes) have a common distribution level exceeding 0.7231 then for any positive even integer $h$ there are arbitrarily long arithmetic progressions of primes $p$ such that $p+h$ is also prime for each element of the progression. In case of $h=2$ this means that under some plausible unproved hypotheses about regular distribution of the primes and other sequences in arithmetic progressions we really have arbitrarily long arithmetic progressions in the sequence of twin primes.

The influence of the first term of an arithmetic progression

The goal of this article is to study the discrepancy of the distribution of arithmetic sequences in arithmetic progressions. We will fix a sequence $\A=\{\a(n)\}_{n\geq 1}$ of non-negative real numbers in a certain class of arithmetic sequences. For a fixed integer $a\neq 0$, we will be interested in the behaviour of $\A$ over the arithmetic progressions $a \bmod q$, on average over $q$. Our main result is that for certain sequences of arithmetic interest, the value of $a$ has a significant influence on this distribution, even after removing the first term of the progressions.

Arithmetic Progressions in Squarefull Numbers

We answer a number of questions of Erdős on the existence of arithmetic progressions in $k$-full numbers (i.e. integers with the property that every prime divisor necessarily occurs to at least the $k$-th power). Further, we deduce a variety of arithmetic constraints upon such progressions, under the assumption of the $abc$-conjecture of Masser and Oesterlé.

Arithmetic progressions of four squares over quadratic fields

Let d be a squarefree integer. Does there exist four squares in arithmetic progression over Q(sqrt{d})? We shall give a partial answer to this question, depending on the value of d. In the affirmative case, we construct explicit arithmetic progressions consisting of four squares over Q(sqrt{d}).

An average theorem for tuples of k-free numbers in arithmetic progressions

We obtain an asymptotic formula, in the spirit of the Montgomery-Hooley refinement of the Barban-Davenport-Halberstam Theorem, for the variance associated with tuples of k-free numbers in arithmetic progressions.

k-term Arithmetic Progressions in Sumsets

In this paper we give a very elementary proof that if A and B are subsets of {1,2,...,N}, each having at least 5N^{1 - (4(k-1))^{-1}} elements, then the sumset A+B has a k-term arithmetic progression.

Short Note: Every Large Set of Integers Contains a Three Term Arithmetic Progression

I show that a trivial modification of a standard proof of the Roth's Theorem on triples in arithmetic progression would lead to the following Theorem: If A is a "large set" that is its elements are monotone increasing integers and the sum of reciprocals of its elements diverges then the sequence contains an arithmetic progression of length three.

纵坐标成等差数列的莫德尔曲线

Mordell Curves with Ordinates in Arithmetic Progression

研究纵坐标成等差数列的莫德尔曲线，通过证明未被先前工作排除，给出特定长度的非同构曲线族，还构造适度秩椭圆曲线参数族，其子族有理点成等差数列。

Valuations, arithmetic progressions, and prime numbers

In this short note, we give two proofs of the infinitude of primes via valuation theory and give a new proof of the divergence of the sum of prime reciprocals by Roth's theorem and Euler-Legendre's theorem for arithmetic progressions.

On the second moment for primes in an arithmetic progression

Assuming the Generalized Riemann Hypothesis, we obtain a lower bound within a constant factor of the conjectured asymptotic result for the second moment for primes in an individual arithmetic progression in short intervals. Previous results were averaged over all progression of a given modulus. The method uses a short divisor sum approximation for the von Mangoldt function, together with some new results for binary correlations of this divisor sum approximation in arithmetic progressions.

Chen primes in arithmetic progressions

We find a lower bound for the number of Chen primes in the arithmetic progression $a \bmod q$, where $(a,q)=(a+2,q)=1$. Our estimate is uniform for $q \leq \log^M x$, where $M>0$ is fixed.

A note on Farey fractions with denominators in arithmetic progressions

As the conclusion of a line of investigation undertaken in two previous papers, we compute asymptotic frequencies for the values taken by numerators of differences of consecutive Farey fractions with denominators restricted to lie in arithmetic progression.

A variance for $k$-free numbers in arithmetic progressions of given modulus

An asymptotic formula for the variance of squarefree numbers in arithmetic progressions of given modulus was obtained by Nunes (see reference [3]). We improve one of the error terms.

Explicit bound for the number of primes in arithmetic progressions assuming the Generalized Riemann Hypothesis

We prove an explicit error term for the $ψ(x,χ)$ function assuming the Generalized Riemann Hypothesis. Using this estimate, we prove a conditional explicit bound for the number of primes in arithmetic progressions.

Square-free numbers in arithmetic progressions

We give asymptotics for correlation sums linked with the distribution of squarefree numbers in arithmetic progressions over a fixed modulus. As a particular case we improve a result of Blomer concerning the variance.

A mean-square value of the Riemann zeta function over an arithmetic progression

We obtain an asymptotic formula for the second discrete moment of the Riemann zeta function over the arithmetic progression $\frac{1}{2} + in$. It shows that the first main term is equal to that of the continuous mean value.

Moments of the distribution of $k$-free numbers in short intervals and arithmetic progressions

We show estimates for the distribution of $k$-free numbers in short intervals and arithmetic progressions. We argue that, at least in certain ranges, these estimates agree with a conjecture by H. L. Montgomery.

Arithmetic progressions among powerful numbers

In this paper, we study $k$-term arithmetic progressions $N, N+d, ..., N+(k-1)d$ of powerful numbers. Under the $abc$-conjecture, we obtain $d \gg_εN^{1/2 - ε}$. On the other hand, there exist infinitely many $3$-term arithmetic progressions of powerful numbers with $d \ll N^{1/2}$ unconditionally. We also prove some partial results when $k \ge 4$ and pose some open questions.

Primes in arithmetic progressions to large moduli III: Uniform residue classes

We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$, extending the Bombieri-Vinogradov theorem to moduli of size $x^{1/2+δ}$ which have conveniently sized divisors. The main feature of these estimates is that they are completely uniform with respect to the residue classes considered, unlike previous works on primes in arithmetic progressions to large moduli.

Primes in arithmetic progressions to spaced moduli.III

We obtain the analog of the Bombieri-Vinogradov theorem for square moduli up to any power of x less than 1/2.

twin素数猜想的等价形式与一个算术进展序列的对称性质相关

An Equivalent form of Twin Prime Conjecture connected with a sequence of arithmetic progressions

本文提出了一种与算术进展序列对称性质相关的 twin素数猜想等价形式。

Cyclicity of Elliptic Curves Modulo Primes in arithmetic progressions

We consider the reduction of an elliptic curve defined over the rational numbers modulo primes in a given arithmetic progression and investigate how often the subgroup of rational points of this reduced curve is cyclic as a special case of Serre's Cyclicity Conjecture.

Arithmetic progressions in binary quadratic forms and norm forms

We prove an upper bound for the length of an arithmetic progression represented by an irreducible integral binary quadratic form or a norm form, which depends only on the form and the progression's common difference. For quadratic forms, this improves significantly upon an earlier result of Dey and Thangadurai.

算术级数中Goldbach估计的平均阶

Average orders of Goldbach Estimates in Arithmetic Progressions

本文研究了算术级数中Goldbach表示的平均数问题，获得了渐近结果，并证明了该结果本质上是最佳的。

On Character Sums and Exponential Sums over Generalized Arithmetic Progressions

We study upper bounds for sums of Dirichlet characters. We prove a uniform upper bound of the character sum over all proper generalized arithmetic progressions, which generalizes the classical Polya and Vinogradov inequality. Our argument is based on getting an upper bound for the l1 norm of the Fourier coefficients of a generalized arithmetic progression. Our method also applies to give upper bounds for polynomial exponential sums.

Arithmetic progressions of Carmichael numbers in a reduced residue class

Fix coprime natural numbers $a,q$. Assuming the Prime $k$-tuple Conjecture, we show that there exist arbitrarily long arithmetic progressions of Carmichael numbers, each of which lies in the reduced residue class $a$ mod $q$ and is a product of three distinct prime numbers.

关于形如 ${3^x+2^y}$ 的等差数列的长度

On the length of an arithmetic progression of the form ${3^x+2^y}$

研究数论中形如 ${3^x+2^y}$ 的等差数列长度问题，并证明其长度至多为六。

Explicit Short Intervals for Primes in Arithmetic Progressions on GRH

We prove explicit versions of Cramér's theorem for primes in arithmetic progressions, on the assumption of the generalized Riemann hypothesis.

Smooth-supported multiplicative functions in arithmetic progressions beyond the $x^{1/2}$-barrier

We show that smooth-supported multiplicative functions $f$ are well-distributed in arithmetic progressions $a_1a_2^{-1} \pmod q$ on average over moduli $q\leq x^{3/5-\varepsilon}$ with $(q,a_1a_2)=1$.

The divisor function in arithmetic progressions to smooth moduli

By using the $q$-analogue of van der Corput's method we study the divisor function in an arithmetic progression to modulus $q$. We show that the expected asymptotic formula holds for a larger range of $q$ than was previously known, provided that $q$ has a certain factorisation.

Integers without divisors from a fixed arithmetic progression

Let a be an integer and q a prime number. In this paper, we find an asymptotic formula for the number of positive integers n < x with the property that no divisor d > 1 of n lies in the arithmetic progression a modulo q.

Arithmetic progressions consisting of unlike powers

In this paper we present some new results about unlike powers in arithmetic progression. We prove among other things that for given $k\geq 4$ and $L\geq 3$ there are only finitely many arithmetic progressions of the form $(x_0^{l_0},x_1^{l_1},...,x_{k-1}^{l_{k-1}})$ with $x_i\in{\Bbb Z},$ gcd$(x_0,x_1)=1$ and $2\leq l_i\leq L$ for $i=0,1,...,k-1.$ Furthermore, we show that, for L=3, the progression $(1,1,...,1)$ is the only such progression up to sign.

When the Nontrivial, Small Divisors of a Natural Number are in Arithmetic Progression

Iannucci considered the positive divisors of a natural number $n$ that do not exceed $\sqrt{n}$ and found all forms of numbers whose such divisors are in arithmetic progression. In this paper, we generalize Iannucci's result by excluding the trivial divisors $1$ and $\sqrt{n}$ (when $n$ is a square). Surprisingly, the length of our arithmetic progression cannot exceed $5$.

Distance between arithmetic progressions and perfect squares

In this paper, we study how close the terms of a finite arithmetic progression can get to a perfect square. The answer depends on the initial term, the common difference and the number of terms in the arithmetic progression.

有限域上椭圆曲线的偏差猜想与算术级数中的Hurwitz类数

The bias conjecture for elliptic curves over finite fields and Hurwitz class numbers in arithmetic progressions

本文研究了Frobenius迹位于算术级数的有限域椭圆曲线的二阶矩偏差猜想，证明偏差对正密度级数为正、对正密度级数为负，与经典猜想相反，并给出了相关矩量的显式公式。

On products of disjoint blocks of arithmetic progressions and related equations

In this paper we deal with Diophantine equations involving products of consecutive integers, inspired by a question of Erdős and Graham.

Cesàro averages for Goldbach representations with summands in arithmetic progressions

We consider weighted averages of the number of representations of an even integer as a sum of two prime numbers, where each summand lies in a given arithmetic progression modulo a common integer $q$. Our result is uniform in a suitable range for $q$.

Searching for simultaneous arithmetic progressions on elliptic curves

We look for elliptic curves featuring rational points whose coordinates form two arithmetic progressions, one for each coordinate. A constructive method for creating such curves is shown, for lengths up to 5.

The distribution of r-free numbers in arithmetic progressions

A positive integer n is called r-free if n is not divisible by the r-th power of a prime. Generalizing earlier work of Orr, we provide an upper bound of Bombieri-Vinogradov type for the r-free numbers in arithmetic progressions.

等差数列在几何数列的和集中的表现

Arithmetic progressions in sumsets of geometric progressions

本文研究了几何数列和集中的等差数列结构，结合线性形式对数界和模ularity 性质，给出完整刻画方法。