Pieter Moree
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.
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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.
Primitive normal progressions in finite fields
本文研究有限域中所有项本原且至少一项正规的等差数列，通过筛法、分歧界和计算改进长度4的结果，并在长度5-10及≥53时获得新界。
Given positive integers a,b,c and d such that c and d are coprime we show that the primes p=c(mod d)dividing a^k+b^k for some k>=1 have a natural density and explicitly compute this density. We demonstrate our results by considering some claims of Fermat that he made in a 1641 letter to Mersenne.
Arithmetic progression in a finite field with prescribed norms
本文研究了在有限域 $\mathbb{F}_{q^{n}}$ 中，由本原元构成且至少包含一个具有指定范数的正规元的算术级数的存在性，并证明了在特定参数下仅有10种可能例外。
Let a and f be coprime positive integers. Let g be an integer. Under the Generalized Riemann Hypothesis (GRH) it follows by a result of H.W. Lenstra that the set of primes p such that p=a(mod f) and g is a primitive root modulo p has a natural density. In this note this density is explicitly evaluated with an Euler product as result.
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.
We demonstrate, using character sum arguments, the existence of primitive polynomials of degree n over a finite field GF(q) with the coefficients for x^(n-1), x^(n-2), and x^(n-3) prescribed, so long as char(GF(q)) is at least 5 and n is at least 9. The cases n=7, 8 are also considered.
We determine primitive solutions to the equation $(x-r)^2 + x^2 + (x+r)^2 = y^n$ for $1 \le r \le 5,000$, making use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.
Normal and primitive normal elements with prescribed traces in intermediate extensions of finite fields
本文研究有限域中间扩张中具有给定迹的范数及原始范数元素，完整刻画了范数情形下的存在条件并给出显式计数，对原始范数情形提供了渐近结果。
Squares in arithmetic progression over certain non-primitive quartic number fields
本文研究了特定非原始四次数字段上等差数列中的平方问题，通过特征化二次扩张Q(√D)上的非常数平方等差数列来探讨相关数论性质。
In this article, we give a largely self-contained proof that the quartic extension $\mathbb{F}_{q^4}$ of the finite field $\mathbb{F}_q$ contains a primitive element $α$ such that the element $α+α^{-1}$ is also a primitive element of ${\mathbb{F}_{q^4}},$ and $Tr_{\mathbb{F}_{q^4}|\mathbb{F}_{q}}(α)=a$ for any prescribed $a \in \mathbb{F}_q$. The corresponding result for finite field extensions of degrees exceeding 4 has already been established by Gupta, Sharma and Cohen.
We prove the existence of primitive sets (sets of integers in which no element divides another) in which the gap between any two consecutive terms is substantially smaller than the best known upper bound for the gaps in the sequence of prime numbers. The proof uses the probabilistic method. Using the same techniques we improve the bounds obtained by He for gaps in geometric-progression-free sets.
Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in $(\mathbb Z/p\mathbb Z)^*,$ then we say that $g$ is a $t$-near primitive root modulo $p$. We point out the easy result that each primitive residue class contains a positive natural density subset of primes $p$ not having $g$ as a $t$-near primitive root and prove a more difficult variant.
Let $F=\mathbb{F}_{q^m}$, $m>6$, $n$ a positive integer, and $f=p/q$ with $p$, $q$ co-prime irreducible polynomials in $F[x]$ and deg$(p)$ $+$ deg$(q)= n$. A sufficient condition has been obtained for the existence of primitive pairs $(α, f(α))$ in $F$ such that for any prescribed $a, b$ in $E=\mathbb{F}_q$, Tr$F/E (α) = a$ and Tr$F/E (α^{-1}) = b$. Further, for every positive integer $n$, such a pair definitely exists for large enough $(q,m)$. The case $n = 2$ is dealt separately and proved that such a pair exists for all $(q,m)$ apart from at most $64$ choices.
$r$-primitive $k$-normal polynomials over finite fields with last two coefficients prescribed
本文研究有限域上最后两个系数给定的r-本原k-正规多项式的存在性，证明了其存在的充分条件，并计算了m≥7时3-本原1-正规多项式的所有例外对(q,m)。
In this paper we continue the investigations about unlike powers in arithmetic progression. We provide sharp upper bounds for the length of primitive non-constant arithmetic progressions consisting of squares/cubes and $n$-th powers.
Given $m, n, q\in \mathbb{N}$ such that $q$ is a prime power and $m\geq 3$, $a\in \mathbb{F}_q$, we establish a sufficient condition for the existence of primitive pair $(α, f(α))$ in $\mathbb{F}_{q^m}$ such that $α$ is normal over $\mathbb{F}_q$ and $\text{Tr}_{\mathbb{F}_{q^m}/\mathbb{F}_q}(α^{-1})=a$, where $f(x)\in \mathbb{F}_{q^m}(x)$ is a rational function of degree sum $n$. Further, when $n=2$ and $q=5^k$ for some $k\in \mathbb{N}$, such a pair definitely exists for all $(q, m)$ apart from at most $20$ choices.
Given $F= \mathbb{F}_{p^{t}}$, a field with $p^t$ elements, where $p $ is a prime power, $t\geq 7$, $n$ are positive integers and $f=f_1/f_2$ is a rational function, where $f_1, f_2$ are relatively prime, irreducible polynomials with $deg(f_1) + deg(f_2) = n $ in $F[x]$. We construct a sufficient condition on $(p,t)$ which guarantees primitive pairing $(ε, f(ε))$ exists in $F$ such that $Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(ε) = a$ and $Tr_{\mathbb{F}_{p^t}/\mathbb{F}_p}(f(ε)) = b$ for any prescribed $a,b \in \mathbb{F}_{p}$. Further, we demonstrate for any positive integer $n$, such a pair defi…
In this paper we address the problem of computing asymptotic formulae for the expected values and second moments of central values of primitive Dirichlet $L$-functions $L(1/2,χ_{8d}\otimesψ)$ when $ψ$ is a fixed even primitive non-quadratic character of odd modulus $q$, $χ_{8d}$ is a primitive quadratic character, $d\equiv h\pmod r$ is odd and squarefree and $r\equiv0\pmod q$ is even. Restricting to these arithmetic progressions ensures that the resulting sets of $L$-functions each form a ``family of primitive $L$-functions" in the specific sense defined by Conrey, Farmer, Keating, Rubinstein …
Primitive normal Values of rational functions with one prescribed norm and trace over finite fields
本文研究有限域扩展中本原正规对的存在性，证明带指定本原范数与非零迹的有理函数值存在的充分条件，并界定少数潜在例外域。
In this paper, we determine all primitive solutions to the equation $(x+r)^2 +(x+2r)^2 +\cdots +(x+dr)^2 = y^n$ for $2\leq d\leq 10$ and for $1\leq r\leq 10^4$. We make use of a factorization argument and the Primitive Divisors Theorem due to Bilu, Hanrot and Voutier.
$r$-primitive $k$-normal elements in arithmetic progressions over finite fields
本文研究了有限域等差数列中同时满足 $r$-本原与 $k$-正规性质的元素存在性，给出了针对一般参数的渐近结果及特定条件下的具体结果。
We prove that for any prime power $q\notin\{3,4,5\}$, the cubic extension $\mathbb{F}_{q^3}$ of the finite field $\mathbb{F}_q$ contains a primitive element $ξ$ such that $ξ+ξ^{-1}$ is also primitive, and $\textrm{Tr}_{\mathbb{F}_{q^3}/\mathbb{F}_q}(ξ)=a$ for any prescribed $a\in\mathbb{F}_q$. This completes the proof of a conjecture of Gupta, Sharma, and Cohen concerning the analogous problem over an extension of arbitrary degree $n\ge3$.
Variance of GL(2) Fourier coefficients in arithmetic progressions
改进Lau和Zhao关于SL2(Z)原始自偶模形式傅里叶系数在算术进展中方差的结果，通过使用Rankin-SelbergL函数第一矩在高度方面的界和非平凡的移位卷积和估计。
A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a metrical and a non-metrical point of view and, on the other hand, from an asymptotic and also a uniform point of view. The principal novelty is a Khintchine type theorem for uniform approximation in this context. Some applications of this theory are also discussed.
In this article, we establish a sufficient condition for the existence of a primitive element $α\in {\mathbb{F}_{q^n}}$ such that the element $α+α^{-1}$ is also a primitive element of ${\mathbb{F}_{q^n}},$ and $Tr_{\mathbb{F}_{q^n}|\mathbb{F}_{q}}(α)=a$ for any prescribed $a \in \mathbb{F}_q$, where $q=p^k$ for some prime $p$ and positive integer $k$. We prove that every finite field $\mathbb{F}_{q^n}~ (n \geq5),$ contains such primitive elements except for finitely many values of $q$ and $n$. Indeed, by computation, we conclude that there are no actual exceptional pairs $(q,n)$ for $n\geq5.$
Arithmetic progressions in dense subsets of primitive elements of finite fields
本文证明有限域本原元稠密子集必含非平凡k项算术级数，并构造无此类级数的大子集，结合特征和估计与相对Szemerédi定理。
By definition primitive and $2$-primitive elements of a finite field extension $\mathbb{F}_{q^n}$ have order $q^n-1$ and $(q^n-1)/2$, respectively. We have already shown that, with minor reservations, there exists a primitive element and a $2$-primitive element $ξ\in \mathbb{F}_{q^n}$ with prescribed trace in the ground field $\mathbb{F}_q$. Here we amend our previous proofs of these results, firstly, by a reduction of these problems to extensions of prime degree $n$ and, secondly, by deriving an exact expression for the number of squares in $\mathbb{F}_{q^n}$ whose trace has prescribed value …
Given a prime power $q$ and a positive integer $n$, let $\mathbb{F}_{q^{n}}$ denote the finite field with $q^n$ elements. Also let $a,b$ be arbitrary members of the ground field $\mathbb{F}_{q}$. We investigate the existence of a non-zero element $ξ\in \mathbb{F}_{q^{n}}$ such that $ξ+ ξ^{-1}$ is primitive and $T(ξ)=a, T(ξ^{-1})=b$, where $T(ξ)$ denotes the trace of $ξ$ in $\mathbb{F}_{q}$. This was a question intended to be addressed by Cao and Wang in 2014. Their work dealt instead with another problem already in the literature. Our solution deals with all values of $n \geq 5$. A related stu…
Primitive normal pairs of elements with one prescribed trace
本文研究有限域上具有指定迹的素元正规对的存在性，给出充分条件，并针对特定参数明确至多11个例外情形。
In this paper, we explore the existence of $m$-terms arithmetic progressions in $\mathbb{F}_{q^n}$ with a given common difference whose terms are all primitive elements, and at least one of them is normal. We obtain asymptotic results for $m \ge 4$ and concrete results for $m \in \{2,3\}$, where the complete list of exceptions when the common difference belongs to $\mathbb{F}_{q}^*$ is obtained. The proofs combine character sums, sieve estimates, and computational arguments using the software SageMath.
Primitive normal pairs with prescribed traces over finite fields
本文研究有限域上具有两个指定迹的本原正规对存在性，给出充分条件，并针对特定参数情形确定了至多12个可能不存在的域。
Arithmetic Progressions of Squares and Multiple Dirichlet Series
本文研究计数平方数本原三项等差数列的二元多重 Dirichlet 级数，证明其在 $\mathbb{C}^2$ 上的亚纯延拓，并借助 Tauberian 方法获得相关平方数等差数列与圆锥曲线有理点的计数。
On $r$-primitive $k$-normal polynomials with two prescribed coefficients
本文研究了有限域上具有指定次数和两个给定系数的 $r$-本原 $k$-正规多项式的存在性，建立了充分条件并利用SageMath计算验证了特定参数下的存在情况与不确定对数量。
We investigate, using the weighted linear sieve, the distribution of almost-primes among the residue classes (mod p) that generate the multiplicative group of reduced residue classes. We are concerned with finding an upper bound for the least prime or almost-prime primitive root (mod p) that holds uniformly for all p, analogous to Linnik's Theorem on a uniform upper bound for the least prime in a single arithmetic progression (mod p).
A primitive normal pair with prescribed prenorm
本文针对有限域上的本原正规对，建立了一个充分条件以确保在指定预范数下该对的存在性，并证明了在特定参数下除极少数例外该对必定存在。
A primitive normal pair in a finite field with prescribed traces and norms
本文研究了有限域中具有指定迹和范数的本原正规对的存在性问题，构造了保证多项式生成的本原正规对存在的充分条件，并证明了特定参数下仅有4种可能例外。
Primitive pairs of rational functions with prescribed traces over finite fields
本文研究有限域上有理函数本原对的构造问题，建立了在指定迹条件下存在本原对 $(f(ε), g(ε))$ 的充分条件，并显式计算了次数和为 3 时的例外情形。
Existence of primitive normal pairs over finite fields with prescribed subtrace
本文研究有限域上具有指定子迹的原始正规对的存在性，证明了在特定条件下（$m=2$，$n\geq 6$，$q=7^k$）除至多11种例外情况外，这样的对必然存在。
This paper explores the existence and distribution of primitive elements in finite field extensions with prescribed traces in several intermediate field extensions. Our main result provides an inequality-like condition to ensure the existence of such elements. We then derive concrete existence results for a special class of intermediate extensions.
Hausdorff dimension of sets of numbers whose continued fractions contain arbitrarily long arithmetic progressions
本文研究了偏商序列严格递增且包含任意长等差数列的无理数的豪斯多夫维数。
Triple divisor type functions in arithmetic progressions to prime power moduli
本文研究非尖点GL_3自守形式Hecke特征值在素数幂模算术级数中的等分布，利用Kowalski-Lin-Michel和Milićević技术证明指数1/2+7/458-ε可取。
Let $τ$ be the primitive Dirichlet character of conductor 4, let $χ$ be the primitive even Dirichlet character of conductor 8 and let $k$ be an integer. Then the $U_2$ operator acting on cuspidal overconvergent modular forms of weight $2k+1$ and character $τ$ has slopes in the arithmetic progression ${2,4,...,2n,...}$, and the $U_2$ operator acting on cuspidal overconvergent modular forms of weight $k$ and character $χ\cdot τ^k$ has slopes in the arithmetic progression ${1,2,...,n,...}$. We then show that the characteristic polynomials of the Hecke operators $U_2$ and $T_p$ acting on the space…
Existence of normal elements with prescribed norms
本文结合组合与数论方法，探讨了在中间扩张上范数被预先指定的正规元素的存在性，并完全解决了仅考虑单一中间扩张情形下的该问题。
We prove that the equation $(x-3r)^3+(x-2r)^3 + (x-r)^3 + x^3 + (x+r)^3 + (x+2r)^3+(x+3r)^3= y^p$ only has solutions which satisfy $xy=0$ for $1\leq r\leq 10^6$ and $p\geq 5$ prime. This article complements the work on the equations $(x-r)^3 + x^3 + (x+r)^3 = y^p$ and $(x-2r)^3 + (x-r)^3 + x^3 + (x+r)^3 + (x+2r)^3= y^p$ . The methodology in this paper makes use of the Primitive Divisor Theorem due to Bilu, Hanrot and Voutier for a complete resolution of the Diophantine equation.
In this paper we derive a formula for the number of $N$-free elements over a finite field $\mathbb{F}_q$ with prescribed trace, in particular trace zero, in terms of Gaussian periods. As a consequence, we derive a simple explicit formula for the number of primitive elements, in quartic extensions of Mersenne prime fields, having absolute trace zero. We also give a simple formula in the case when $Q = (q^m-1)/(q-1)$ is prime. More generally, for a positive integer $N$ whose prime factors divide $Q$ and satisfy the so called semi-primitive condition, we give an explicit formula for the number of…
It is known that there is a one-to-one correspondence between equivalence classes of primitive indefinite binary quadratic forms and primitive hyperbolic conjugacy classes of the modular group. Due to such a correspondence, Sarnak obtained the asymptotic formula for the class number sum in order of the fundamental unit by using the prime geodesic theorem for the modular group. In the present paper, we propose asymptotic formulas of the class number sums over discriminants in arithmetic progressions. Since there are relations between the arithmetic properties of the discriminants and the conjug…
The previous paper [4] proved the existence of primitive polynomials and primitive normal polynomials of degree n with k prescribed coefficients in the finite field GF(q) for all sufficiently large q. This paper presents a loger versions of the result on primitve normal polynomials.
On pairs of $r$-primitive and $k$-normal elements with prescribed traces over finite fields
本文研究有限域中同时满足指定迹、有理函数像及本原、正规约束的元素对，给出充分条件，并在固定参数下确定特征13域中除10个例外外的所有 $(q,n)$。
The first purpose of our paper is to show how Hooley's celebrated method leading to his conditional proof of the Artin conjecture on primitive roots can be combined with the Hardy-Littlewood circle method. We do so by studying the number of representations of an odd integer as a sum of three primes, all of which have prescribed primitive roots. The second purpose is to analyse the singular series. In particular, using results of Lenstra, Stevenhagen and Moree, we provide a partial factorisation as an Euler product and prove that this does not extend to a complete factorisation.
Let $q$ be a prime power and $n, r$ integers such that $r\mid q^n-1$. An element of $\mathbb{F}_{q^n}$ of multiplicative order $(q^n-1)/r$ is called \emph{$r$-primitive}. For any odd prime power $q$, we show that there exists a $2$-primitive element of $\mathbb{F}_{q^n}$ with arbitrarily prescribed $\mathbb{F}_q$ trace when $n\geq 3$. Also we explicitly describe the values that the trace of such elements may have when $n=2$. A feature of this amended version is the reduction of the discussion to extensions of prime degree $n$.
We describe a straightforward method to generate a random prime q such that the multiplicative group GF(q)* also has a random large prime-order subgroup. The described algorithm also yields this order p as well as a p'th primitive root of unity. The methods here are efficient asymptotically, but due to large constants may not be very useful in practical settings.
Let $p>2$ be prime and $g$ a primitive root modulo $p$. We present an argument for the fact that discrete logarithms of the numbers in any arithmetic progression are uniformly distributed in $[1,p]$ and raise some questions on the subject.
Let GF(q), q=p^r, be a finite field with a primitive element g. In this paper we use exponential sums and Jacobi sums to compute the number of the irreducible polynomials of degree m over GF(q) with trace fixed and norm restricted to a coset of a subgroup <g^s>, s|(q-1). We give the number explicitly for s=2, 3, 4 when q=p, and for s|(p^e+1) when r=2en. Finally, we give explicit formulae for the number when both trace and norm are fixed, p=2 and m =< 30.
In this paper, we introduce and study a variant of Kummer's notion of (ir)regularity of primes which we call G-irregularity. It is based on Genocchi numbers $G_n$, rather than Bernoulli number $B_n.$ We say that an odd prime $p$ is G-irregular if it divides at least one of the integers $G_2,G_4,\ldots, G_{p-3}$, and G-regular otherwise. We show that, as in Kummer's case, G-irregularity is related to the divisibility of some class number. Furthermore, we obtain some results on the distribution of G-irregular primes. In particular, we show that each primitive residue class contains infinitely ma…
In this paper, using a deep result on the existence of primitive divisors of Lehmer numbers due to Y. Bilu, G. Hanrot and P. M. Voutier, we first give an explicit formula for all positive integer solutions of the Diophantine equation $(x-d)^2+x^2+(x+d)^2=y^n$ (*) when $n$ is an odd prime and $d=p^r$, $p>3$ a prime. So this improves the results on the papers of A. Koutsianas and V. Patel \cite{KP} and A. Koutsianas \cite{Kou}. Secondly, under the assumption of our first result, we prove that (*) has at most one solution $(x,y)$. Next, for a general $d$, we prove the following two results: (i) i…
We consider the family of polynomials $f_{d,c}(x)=x^d+c$ over the rational field $\Q$. Fixing integers $d, n\ge 2$, we show that the density of primes that can appear as primitive prime divisors of $f_{d,c}^n(0)$ for some $c\in\Q$ is positive. In fact, under certain assumptions, we explicitly calculate the latter density when $d=2$. Furthermore, fixing $d,n\ge 2$, we show that for a given integer $N>0$, there is $c\in \Q$ such that $\f^n(0)$ has at least $N$ primitive prime divisors each of which is appearing up to any predetermined power. This shows that there is no uniform upper bound on the…
Let $Q(n)$ denote the count of the primitive subsets of the integers $\{1,2\ldots n\}$. We give a new proof that $Q(n) = α^{(1+o(1))n}$ which allows us to give a good error term and to improve upon the lower bound for the value of this constant $α$. We also show that the method developed can be applied to many similar problems that can be stated in terms of the divisor graph, including other questions about primitive sets, geometric-progression-free sets, and the divisor graph path-cover problem.
Euler constants from primes in arithmetic progression
本文系统研究由算术级数中素数定义的生成级数的欧拉常数γ(d,a)，给出其与一般狄利克雷级数欧拉常数的关系，并进行数值计算与实例讨论。
A famous conjecture of Artin states that there are infinitely many prime numbers for which a fixed integer $g$ is a primitive root, provided $g \neq -1$ and $g$ is not a perfect square. Thanks to work of Hooley, we know that this conjecture is true, conditional on the truth of the Generalized Riemann Hypothesis. Using a combination of Hooley's analysis and the techniques of Maynard-Tao used to prove the existence of bounded gaps between primes, Pollack has shown that (conditional on GRH) there are bounded gaps between primes with a prescribed primitive root. In the present article, we provide …
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