Primes in Quadratic Progressions on Average

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.

Primes in higher-order progressions on average

In this paper, we establish some theorems on the distribution of primes in higher-order progressions on average.

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This paper has been withdrawn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.

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。

Asymptotics of sums of functions of primes located on an arithmetic progression

We investigate the problem of the distribution of sums of functions of prime numbers located on an arithmetic progression. This problem is closely related to the problem of the distribution of prime numbers on an arithmetic progression. Based on this distribution, a general formula was obtained for the asymptotic estimate of the sums of functions of primes, and also asymptotics was found for the sums of various functions of primes on a geometric progression. Several assertions about asymptotics of sums of functions of prime numbers on a geometric progression are proved. Necessary and sufficien…

Arithmetic progressions of three prime numbers with two primes of the form $\mathbf{p=x^2+y^2+1}$

In the present paper we prove that there exist infinitely many arithmetic progressions of three different primes $p_1,p_2,p_3=2p_2-p_1$ such that $p_1=x_1^2 + y_1^2 +1$, $p_2=x_2^2 + y_2^2 +1$.

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.

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.

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.

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

The main result of the paper is that assuming that the level $θ$ of distribution of primes exceeds 1/2, then there exists a positive $d\leq C(θ)$ such that there are arbitrarily long arithmetic progressions with the property that $p'=p+d$ is the next prime for each element of the progression. If $θ>0.971$, then the above holds for some $d\leq 16$.

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.

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.

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.

Large prime gaps and progressions with few primes

We show that the existence of arithmetic progressions with few primes, with a quantitative bound on "few", implies the existence of larger gaps between primes less than x than is currently known unconditionally. In particular, we derive this conclusion if there are certain types of exceptional zeros of Dirichlet L-functions.

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.

Product of an integer free of small prime factors and prime in arithmetic progression

We establish estimates for the number of ways to represent any reduced residue class as a product of a prime and an integer free of small prime factors. Our best results is conditional on the Generalised Riemann hypothesis (GRH). As a corollary, we make progress on a conjecture of Erdös, Odlyzko and Sárközy.

On Primes in Quadratic Progressions

We verify the Hardy-Littlewood conjecture on primes in quadratic progressions on average. The results in the present paper significantly improve those of a previous paper of the authors(arXiv:math.NT/0605563).

Primes in short arithmetic progressions

We give a large sieve type inequality for functions supported on primes. As application we prove a conjecture by Elliott, and give bounds for short character sums over primes. The proves uses a combination of the large sieve and the Selberg sieve.

Squares in arithmetic progressions and infinitely many primes

We give a new proof that there are infinitely many primes, relying on van der Waerden's theorem for coloring the integers, and Fermat's theorem that there cannot be four squares in an arithmetic progression. We go on to discuss where else these ideas have come together in the past.

Biases in prime factorizations and Liouville functions for arithmetic progressions

We introduce a refinement of the classical Liouville function to primes in arithmetic progressions. Using this, we discover new biases in the appearances of primes in a given arithmetic progression in the prime factorizations of integers. For example, we observe that the primes of the form $4k+1$ tend to appear an even number of times in the prime factorization of a given integer, more so than for primes of the form $4k+3$. We are led to consider variants of Pólya's conjecture, supported by extensive numerical evidence, and its relation to other conjectures.

The Eratosthenes Progression p_{k+1} = π^{-1}(p_k), k=0,1,2,..., p_0 \in N \setminus P Determines An Inner Prime Number Distribution Law

Formulas of $π(x)$-fine structure are presented.

Accurate Computations of Euler Products over Primes in Arithmetic Progressions

This note provides truncated formulae with explicit error terms to compute Euler products over primes in arithmetic progressions of rational fractions. It further provides such a formula for the product of terms of the shape $F(1/p, 1/p^s)$ when $F$ is a two-variable polynomial with coefficients in $\mathbb{C}$ and satisfying some restrictive conditions.

Strings of special primes in arithmetic progressions

The Green-Tao Theorem, one of the most celebrated theorems in modern number theory, states that there exist arbitrarily long arithmetic progressions of prime numbers. In a related but different direction, a recent theorem of Shiu proves that there exist arbitrarily long strings of consecutive primes that lie in any arithmetic progression that contains infinitely many primes. Using the techniques of Shiu and Maier, this paper generalizes Shiu's Theorem to certain subsets of the primes such as primes of the form $\lfloor πn\rfloor$ and some of arithmetic density zero such as primes of the form $…

Explicit formulae for primes in arithmetic progressions, II

We shall give an explicit version of Bombieri-Vinogradov Theorem for moduli not divisible by an exceptional modulus.

On products of primes and square-free integers in arithmetic progressions

We obtain an asymptotic formula for the number of ways to represent every reduced residue class as a product of a prime and square-free integer. This may be considered as a relaxed version of a conjecture of Erdös, Odlyzko, and Sárközy.

Divisibility in paired progressions, Goldbach's conjecture, and the infinitude of prime pairs

We investigate progressions in the set of pairs of integers $\mathbb{Z}^2$ and define a generalisation of the Jacobsthal function. For this function, we conjecture a specific upper bound and prove that this bound would be a sufficient condition for the truth of the Goldbach conjecture, the infinitude of prime twins, and more general of prime pairs with a fixed even difference.

Primes in arithmetic progressions to large moduli, and shifted primes without large prime factors

We prove the infinitude of shifted primes $p-1$ without prime factors above $p^{0.2844}$. This refines $p^{0.2961}$ from Baker and Harman in 1998. Consequently, we obtain an improved lower bound on the the distribution of Carmichael numbers. Our main technical result is a new mean value theorem for primes in arithmetic progressions to large moduli. Namely, we estimate primes of size $x$ with quadrilinear forms of moduli up to $x^{17/32}$. This extends moduli beyond $x^{11/21}$, recently obtained by Maynard, improving $x^{29/56}$ from well-known 1986 work of Bombieri, Friedlander, and Iwaniec.

Large bias for integers with prime factors in arithmetic progressions

We prove an asymptotic formula for the number of integers $\leq x$ which can be written as the product of $k ~(\geq 2)$ distinct primes $p_1\cdots p_k$ with each prime factor in an arithmetic progression $p_j\equiv a_j \bmod q$, $(a_j, q)=1$ $(q \geq 3, 1\leq j\leq k)$. For any $A>0$, our result is uniform for $2\leq k\leq A\log\log x$. Moreover, we show that, there are large biases toward certain arithmetic progressions $(a_1 \bmod q, \cdots, a_k \bmod q)$, and such biases have connections with Mertens' theorem and the least prime in arithmetic progressions.

Lucas non-Wieferich primes in arithmetic progressions and the $abc$ conjecture

We prove the lower bound for the number of Lucas non-Wieferich primes in arithmetic progressions. More precisely, for any given integer $k\geq 2$ there are $\gg \log x$ Lucas non-Wieferich primes $p\leq x$ such that $p\equiv\pm1\pmod{k}$, assuming the $abc$ conjecture for number fields. Further, we discuss some applications of Lucas sequences in Cryptography.

On the regularity of primes in arithmetic progressions

We prove that for a positive integer $k$ the primes in certain kinds of intervals can not distribute too 'uniformly' among the reduced residue classes modulo $k$. Hereby, we prove a generalization of a conjecture of Recaman and establish our results in a much more general situation, in particular for prime ideals in number fields.

Pretentious multiplicative functions and the prime number theorem for arithmetic progressions

Building on the concept of pretentious multiplicative functions, we give a new and largely elementary proof of the best result known on the counting function of primes in arithmetic progressions.

Multiplicative decomposition of arithmetic progressions in prime fields

We prove that there exists an absolute constant $c>0$ such that if an arithmetic progression $\cP$ modulo a prime number $p$ does not contain zero and has the cardinality less than $cp$, then it can not be represented as a product of two subsets of cardinality greater than 1, unless $\cP=-\cP$ or $\cP=\{-2r,r,4r\}$ for some residue $r$ modulo $p$.

算术进展中至光滑模数的素数：一个下界版本

Primes in arithmetic progressions to smooth moduli: A minorant version

该研究提出一个函数作为素数指示函数的下界，改进了Baker--Irving和Stadlmann在算术进展中至光滑模数的素数分布结果。

关于算术级数中素数的Mertens定理中的常数

On the constants in Mertens' theorems for primes in arithmetic progressions

本文在广义黎曼猜想下，对算术级数中素数Mertens第二定理的常数给出有效描述，基于Norton1976年的渐近结果。

The prime number theorem for primes in arithmetic progressions at large values

Assuming the Riemann hypothesis, we prove the latest explicit version of the prime number theorem for short intervals. Using this result, and assuming the generalised Riemann hypothesis for Dirichlet $L$-functions is true, we then establish explicit formulae for $ψ(x,χ)$, $θ(x,χ)$, and an explicit version of the prime number theorem for primes in arithmetic progressions that hold for general moduli $q\geq 3$. Finally, we restrict our attention to $q\leq 10\,000$ and use an exact computation to refine these results.

Polignac Numbers, Conjectures of Erdös on Gaps between Primes, Arithmetic Progressions in Primes, and the Bounded Gap Conjecture

In the present work we prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent fantastic achievement of Yitang Zhang about the existence of bounded gaps between consecutive primes. Most of these results would have belonged to the category of science fiction a decade ago. However, the presented results are far from being immediate consequences of Zhang's famous theorem: they require various new ideas, other important properties of the applied sieve functi…

算术级数中的素测地定理

The Prime Geodesic Theorem in Arithmetic Progressions

本文探讨了算术级数中的素测地定理，解决了Golovchanskiı-Smotrov（1999）的猜想，并证明模曲面上闭测地线的迹在给定模数的简化剩余类中非等分布。

Primes in arithmetic progressions to large moduli II: Well-factorable estimates

We establish new mean value theorems for primes of size $x$ in arithmetic progressions to moduli as large as $x^{3/5-ε}$ when summed with suitably well-factorable weights. This extends well-known work of Bombieri, Friedlander and Iwaniec, who handled moduli of size at most $x^{4/7-ε}$. This has consequences for the level of distribution for sieve weights coming from the linear sieve.

Sifting for small primes from an arithmetic progression

In this work and its sister paper [5] we give a new proof of the famous Linnik theorem bounding the least prime in an arithmetic progression. Using sieve machinery in both papers, we are able to dipense with the log-free zero density bounds and the repulsion property of exceptional zeros, two deep innovations begun by Linnik and reelied on in earlier proofs.

On Ramanujan expansions and primes in arithmetic progressions

A celebrated theorem of Delange gives a sufficient condition for an arithmetic function to be the sum of the associated Ramanujan expansion with the coefficients provided by a previous result of Wintner. By applying the Delange theorem to the correlation of the von Mangoldt function with its incomplete form, we deduce an inequality involving the counting function of the prime numbers in arithmetic progressions. A remarkable aspect is that such an inequality is equivalent to the famous conjectural formula by Hardy and Littlewood for the twin primes.

The divisor function in arithmetic progressions modulo prime powers

We study the average value of the divisor function $τ(n)$ for $n\le x$ with $n \equiv a \bmod q$. The divisor function is known to be evenly distributed over arithmetic progressions for all $q$ that are a little smaller than $x^{2/3}$. We show how to go past this barrier when $q=p^k$ for odd primes $p$ and any fixed integer $k\ge 7$.

On the existence of products of primes in arithmetic progressions

We study the existence of products of primes in arithmetic progressions, building on the work of Ramaré and Walker. One of our main results is that if $q$ is a large modulus, then any invertible residue class mod $q$ contains a product of three primes where each prime is at most $q^{6/5+ε}$. Our arguments use results from a wide range of areas, such as sieve theory or additive combinatorics, and one of our key ingredients, which has not been used in this setting before, is a result by Heath-Brown on character sums over primes from his paper on Linnik's theorem.

算术等差数列中最小素数的条件界

A conditional bound for the least prime in an arithmetic progression

在假设狄利克雷L函数广义 Lindelöf 假设下，研究了算术等差数列中最小素数的界，其结果接近广义黎曼假设所暗示的经典估计。

Uniformly counting primes with a given primitive root and in an arithmetic progression

We study the number of primes with a given primitive root and in an arithmetic progression under the assumption of a suitable form of the generalized Riemann Hypothesis. Previous work of Lenstra, Moree and Stevenhagen has given asymptotics without an explicit error term, we provide an explicit error term by combining their work with the method of Hooley regarding Artin's primitive root conjecture. We give an application to a Diophantine problem involving primes with a given primitive root.

Divisor problem in arithmetic progressions modulo a prime power

We obtain an asymptotic formula for the average value of the divisor function over the integers $n \le x$ in an arithmetic progression $n \equiv a \pmod q$, where $q=p^k$ for a prime $p\ge 3$ and a sufficiently large integer $k$. In particular, we break the classical barrier $q \le x^{2/3}$ for such formulas, and generalise a recent result of R.~Khan (2015), making it uniform in $k$.

A mean value of the representation function for the sum of two primes in arithmetic progressions

In this note, assuming a variant of the Generalized Riemann Hypothesis, which does not exclude the existence of real zeros, we prove an asymptotic formula for the mean value of the representation function for the sum of two primes in arithmetic progressions. This is an improvement of the result of F. Rüppel in 2009, and the generalization of the result of A. Languasco and A. Zaccagnini concerning the ordinary Goldbach problem in 2012.

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.

算术级数中的最小素数

The Least Prime in Arithmetic an Progression

本文利用现代硬件更新Wagstaff关于算术级数中最小素数的计算，提出并验证了形如 $3ϕ(n)\log(n)\log(ϕ(n))$ 的改进上界猜想。

On the density of primes in arithmetic progression having a prescribed primitive root

Let a,f and g be integers, with a and f coprime. Under the generalized Riemann hypothesis it follows from work of Hooley and Lenstra that the set of primes p such that p=a(mod f) and g is primitive root mod p has a natural density. In this note we explicitly evaluate this density and give some applications of this result.

整数等差数列与其数字和互质的研究

Arithmetic progressions of integers that are relatively prime to their digital sums

本文研究了$b$-反尼文数等差数列的最大长度，探讨了其在不同基底下的性质和特征。

等差数列素数定理的改进

Refinements to the prime number theorem for arithmetic progressions

本文证明了等差数列素数定理的一致版本，并由此推出多个经典结果，证明依赖于Vinogradov-Korobov无零点区域、无对数零点密度估计和Deuring-Heilbronn零点排斥现象。

Primes in Geometric-Arithmetic Progression

A geometric-arithmetic progression of primes is a set of $k$ primes (denoted by GAP-$k$) of the form $p_1 r^j + j d$ for fixed $p_1$, $r$ and $d$ and consecutive $j$, {\it i.e}, $\{p_1, \, p_1 r + d, \, p_1 r^2 + 2 d, \, p_1 r^3 + 3 d, \,...}$. We study the conditions under which, for $k \ge 2$, a GAP-$k$ is a set of $k$ primes in geometric-arithmetic progression. Computational data (along with the MATHEMATICA codes) containing progressions up to GAP-13 is presented. Integer sequences for the sets of differences $d$ corresponding to the GAPs of orders up to 11 are also presented.

Goldbach's like conjectures arising from arithmetic progressions whose first two terms are primes

For two odd primes $p$ and $q$ such that $p<q$, let $A(p,q):=(a_k)_{k=1}^{\infty}$ be the arithmetic progression whose $k$th term is given by $a_k=(k-1)(q-p)+p$ (i.e., with $a_1=p$ and $a_2=q$). Here we conjecture that for every positive integer $a>1$ there exist a positive integer $n$ and two odd primes $p$ and $q$ such that $a$ can be expressed as a sum of the first $2n$ terms of the arithmetic progression $A(p,q)$. Notice that in the case of even $a$, this conjecture immediately follows from Goldbach's conjecture. We also propose the analogous conjecture for odd positive integers $a>1$ as w…