No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Higher-degree Artin conjecture
Abstract
For an algebraic number α we consider the orders of the reductions of α in finite fields. In the case where α is an integer, it is known by the work on Artin’s primitive root conjecture that the order is ‘almost always almost maximal’ assuming the Generalized Riemann Hypothesis (GRH), but unconditional results remain modest. We consider higher-degree variants under GRH. First, we modify an argument of Roskam to settle the case where α and the reduction have degree two. Second, we give a positive lower density result when α is of degree three and the reduction is of degree two. Third, we give higher-rank results in situations where the reductions are of degree two, three, four or six. As an application we give an almost equidistribution result for linear recurrences modulo primes. Finally, we present a general result conditional to GRH and a hypothesis on smooth values of polynomials at prime arguments.
ResearchGate has not been able to resolve any citations for this publication.
Let be an integer and let be a polynomial of degree d whose Galois group is . Let be a non-degenerate linearly recursive sequence of integers which has P as its characteristic polynomial. We prove, under the generalised Riemann hypothesis, that the lower density of the set of primes which divide at least one non-zero element of the sequence is positive.
The topic of exponential sums entered firmly into the mainstream of number theory with the work [We] of H. Weyl. Although today very general sums
We studied the distribution of units of an algebraic number field modulo prime ideals. Here we study the distribution of units of a cubic abelian field modulo rational prime numbers. For a decomposable prime number p , is an upper bound of the order of the unit group modulo p , and we show that the conjectural density of primes which attain it is really positive.
We determine the order of magnitude of H(x,y,z), the number of integers n\le x having a divisor in (y,z], for all x,y and z. We also study H_r(x,y,z), the number of integers n\le x having exactly r divisors in (y,z]. When r=1 we establish the order of magnitude of H_1(x,y,z) for all x,y,z satisfying z\le x^{0.49}. For every r\ge 2, and , we determine the the order of magnitude of H_r(x,y,z) when y is large and y+y/(\log y)^{\log 4 -1 - \epsilon} \le z \le \min(y^{C},x^{1/2-\epsilon}). As a consequence of these bounds, we settle a 1960 conjecture of Erdos and several related conjectures. One key element of the proofs is a new result on the distribution of uniform order statistics.
Fix a∈Z, a∉{0,±1}. A simple argument shows that for each ϵ>0, and almost all (asymptotically 100% of) primes p, the multiplicative order of a modulo p exceeds p12−ϵ. It is an open problem to show the same result with 12 replaced by any larger constant. We show that if a,b are multiplicatively independent, then for almost all primes p, one of a,b,ab,a2b,ab2 has order exceeding p12+130. The same method allows one to produce, for each ϵ>0, explicit finite sets A with the property that for almost all primes p, some element of A has order exceeding p1−ϵ. Similar results hold for orders modulo general integers n rather than primes p.
For all number fields the failure of maximality for the Kummer extensions is bounded in a very strong sense. We give a direct proof (without relying on the Bashmakov-Ribet method) of the fact that if G is a finitely generated and torsion-free multiplicative subgroup of a number field K having rank r, then the ratio between nr and the Kummer degree [K(ζn,Gn): K(ζn)] is bounded independently of n. We then apply this result to generalize to higher rank a theorem of Ziegler from 2006 about the multiplicative order of the reductions of algebraic integers (the multiplicative order must be in a given arithmetic progression, and an additional Frobenius condition may be considered).
In this note, we study conjectures of Artin and Langlands and derive the automorphy of all solvable groups of order at most 200, three groups excepted.
The classical theorem of Bombieri and Vinogradov is generalized to a non-abelian, non-Galois setting. This leads to a prime number theorem of "mixed-type" for arithmetic progressions "twisted" by splitting conditions in number fields. One can view this as an extension of earlier work of M. R. Murty and V. K. Murty on a variant of the Bombieri-Vinogradov theorem. We develop this theory with a view to applications in the study of the Euclidean algorithm in number fields and arithmetic orbifolds.
There exist infinitely many integers n such that the greatest prime factor of is at least . The proof is a combination of Hooley’s method – for reducing the problem to the evaluation of Kloosterman sums – and the majorization of Kloosterman sums on average due to the authors.
One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the literature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contributions in the survey on `elliptic Artin' are due to Alina Cojocaru. Wojciec Gajda wrote a section on `Artin for K-theory of number fields,' and Hester Graves (together with me) on `Artin's conjecture and Euclidean domains.'
Let F be a cubic number field with negative discriminant. Taking into account the extension degree of the ray class field modulo a prime ideal, we study the residual index I(p) of residue classes represented by units in the multiplicative group of the residue field modulo a prime ideal p. The possible minimal value ℓ(p) is given, and we give the density of prime ideals p with I(p)=ℓ(p) if the degree of p is one, and a conjecture if the degree is 2 or 3.
Let ε be a fundamental unit in a real quadratic field and let S be the set of rational primes p for which ε has maximal order modulo p. Under the assumption of the generalized Riemann hypothesis, we show that S has a density δ(S)=c·A in the set of all rational primes, where A is Artin's constant and c is a positive rational number.
In this note we prove that as the modulus runs through the set of prime numbers, the roots of a quadratic congruence are uniformly
distributed in a natural sense. It is well known that this implies uniform distribution of the angles of certain exponential
sums introduced by Salie. The proof is based on previous work by Duke, Friedlander, and Iwaniec.
Fix an element in a quadratic eld K. Dene S as the set of rational primes p, for which has maximal order modulo p. Under the assumption of the generalized Riemann hypothesis, we show that S has a density. Moreover, we give necessary and sucient conditions for the density of S to be positive.
Let S be a linear recurrent integer sequence of order k 3, and dene PS as the set of primes that divide at least one term of S. We give a heuristic approach to the problem whether PS has a natural density, and prove that part of our heuristics is correct. Under the assumption of a generalization of Artin's primitive root conjecture, we nd that PS has positive lower density for `generic' sequences S. Some numerical examples are included. 1.