Relevant bibliographies by topics / Artin's primitive root conjecture

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  1. Journal articles
  2. Dissertations / Theses
  3. Book chapters

Academic literature on the topic 'Artin's primitive root conjecture'

Author: Grafiati
Published: 4 June 2021
Last updated: 13 February 2022

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Journal articles on the topic "Artin's primitive root conjecture"

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Roskam, Hans. "Artin's primitive root conjecture for quadratic fields." Journal de Théorie des Nombres de Bordeaux 14, no. 1 (2002): 287–324. http://dx.doi.org/10.5802/jtnb.360.

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Stevenhagen, Peter. "The correction factor in Artin's primitive root conjecture." Journal de Théorie des Nombres de Bordeaux 15, no. 1 (2003): 383–91. http://dx.doi.org/10.5802/jtnb.408.

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Kim, Seoyoung, and M. Ram Murty. "Artin's primitive root conjecture for function fields revisited." Finite Fields and Their Applications 67 (October 2020): 101713. http://dx.doi.org/10.1016/j.ffa.2020.101713.

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COHEN, JOSEPH. "PRIMITIVE ROOTS IN QUADRATIC FIELDS." International Journal of Number Theory 02, no. 01 (2006): 7–23. http://dx.doi.org/10.1142/s1793042106000425.

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We consider an analogue of Artin's primitive root conjecture for units in real quadratic fields. Given such a nontrivial unit, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p + 1. An extension of Artin's conjecture is that there are infinitely many such inert primes for which this order is maximal. This is known at present only under the Generalized Riemann Hypothesis. Unconditionally, we show that for any choice of 7 units in different real quadratic fields satisfying a certain simple restriction, there is at least one of the units which satisfie
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AMBROSE, CHRISTOPHER. "Artin's primitive root conjecture and a problem of Rohrlich." Mathematical Proceedings of the Cambridge Philosophical Society 157, no. 1 (2014): 79–99. http://dx.doi.org/10.1017/s0305004114000206.

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AbstractLet$\mathbb{K}$be a number field, Γ a finitely generated subgroup of$\mathbb{K}$*, for instance the unit group of$\mathbb{K}$, and κ>0. For an ideal$\mathfrak{a}$of$\mathbb{K}$let indΓ($\mathfrak{a}$]></alt-text></inline-graphic>) denote the multiplicative index of the reduction of Γ in <inline-graphic name="S0305004114000206_inline3"><alt-text><![CDATA[$(\mathcal{O}_\mathbb{K}/\mathfrak{a})$* (whenever it makes sense). For a prime ideal$\mathfrak{p}$of$\mathbb{K}$and a positive integer γ let$\mathcal{I}_\gamma^\kappa(\mathfrak{p})$be the ave
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LENSTRA, H. W., P. STEVENHAGEN, and P. MOREE. "Character sums for primitive root densities." Mathematical Proceedings of the Cambridge Philosophical Society 157, no. 3 (2014): 489–511. http://dx.doi.org/10.1017/s0305004114000450.

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AbstractIt follows from the work of Artin and Hooley that, under assumption of the generalised Riemann hypothesis, the density of the set of primes q for which a given non-zero rational number r is a primitive root modulo q can be written as an infinite product ∏p δp of local factors δp reflecting the degree of the splitting field of Xp - r at the primes p, multiplied by a somewhat complicated factor that corrects for the ‘entanglement’ of these splitting fields.We show how the correction factors arising in Artin's original primitive root problem and several of its generalisations can be inter
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Hu, Su, Min-Soo Kim, Pieter Moree, and Min Sha. "Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture." Journal of Number Theory 205 (December 2019): 59–80. http://dx.doi.org/10.1016/j.jnt.2019.03.012.

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Roskam, Hans. "Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields." Journal de Théorie des Nombres de Bordeaux 13, no. 1 (2001): 303–14. http://dx.doi.org/10.5802/jtnb.323.

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Carella, N. A. "The Generalized Artin Primitive Root Conjecture." European Journal of Pure and Applied Mathematics 11, no. 1 (2018): 23. http://dx.doi.org/10.29020/nybg.ejpam.v11i1.3182.

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Asymptotic formulas for the number of integers with the primitive root 2, and the generalized Artin conjecture for subsets of composite integers with fixed admissible primitive roots \(u\neq \pm 1,v^2\), are presented here.
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Cojocaru, Alina Carmen, and Andrew Michael Shulman. "The Distribution of the First Elementary Divisor of the Reductions of a Generic Drinfeld Module of Arbitrary Rank." Canadian Journal of Mathematics 67, no. 6 (2015): 1326–57. http://dx.doi.org/10.4153/cjm-2015-006-9.

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AbstractLet ψ be a generic Drinfeld module of rank r ≥ 2. We study the first elementary divisor d1,℘ (ψ) of the reduction of ψ modulo a prime ℘, as ℘ varies. In particular, we prove the existence of the density of the primes ℘ for which d1,℘ (ψ) is fixed. For r = 2, we also study the second elementary divisor (the exponent) of the reduction of ψ modulo ℘ and prove that, on average, it has a large norm. Our work is motivated by J.-P. Serre's study of an elliptic curve analogue of Artin's Primitive Root Conjecture, and, moreover, by refinements to Serre's study developed by the first author and
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Dissertations / Theses on the topic "Artin's primitive root conjecture"

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Pappalardi, Francesco. "On Artin's conjecture for primitive roots." Thesis, McGill University, 1993. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=41128.

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Various generalizations of the Artin's Conjecture for primitive roots are considered. It is proven that for at least half of the primes p, the first log p primes generate a primitive root. A uniform version of the Chebotarev Density Theorem for the field ${ cal Q}( zeta sb{l},2 sp{1/l})$ valid for the range $l < { rm log} x$ is proven. A uniform asymptotic formula for the number of primes up to x for which there exists a primitive root less than s is established. Lower bounds for the exponent of the class group of imaginary quadratic fields valid for density one sets of discriminants are deter
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Ambrose, Christopher Daniel [Verfasser], Valentin [Akademischer Betreuer] Blomer, and Preda [Akademischer Betreuer] Mihăilescu. "On Artin's primitive root conjecture / Christopher Daniel Ambrose. Gutachter: Valentin Blomer ; Preda Mihailescu. Betreuer: Valentin Blomer." Göttingen : Niedersächsische Staats- und Universitätsbibliothek Göttingen, 2014. http://d-nb.info/1054191484/34.

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Ambrose, Christopher Daniel. "On Artin's primitive root conjecture." Doctoral thesis, 2014. http://hdl.handle.net/11858/00-1735-0000-0022-5F1A-F.

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Artins Vermutung über Primitivwurzeln besagt, dass es zu jeder ganzen Zahl a, die weder 0, ±1 noch eine Quadratzahl ist, unendlich viele Primzahlen p gibt, sodass a eine Primitivwurzel modulo p ist, d.h. a erzeugt eine multiplikative Untergruppe von Q*, dessen Reduktion modulo p Index 1 in (Z/pZ)* hat. Dies wirft die Frage nach Verteilung von Index und Ordnung dieser Reduktion in (Z/pZ)* auf, wenn man p variiert. Diese Arbeit widmet sich verallgemeinerten Fragestellungen in Zahlkörpern: Ist K ein Zahlkörper und Gamma eine endlich erzeugte unendliche Untergruppe von K*, so werden Momente von In
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FELIX, ADAM TYLER. "Variations on Artin's Primitive Root Conjecture." Thesis, 2011. http://hdl.handle.net/1974/6635.

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Let $a \in \mathbb{Z}$ be a non-zero integer. Let $p$ be a prime such that $p \nmid a$. Define the index of $a$ modulo $p$, denoted $i_{a}(p)$, to be the integer $i_{a}(p) := [(\mathbb{Z}/p\mathbb{Z})^{\ast}:\langle a \bmod{p} \rangle]$. Let $N_{a}(x) := \#\{p \le x:i_{a}(p)=1\}$. In 1927, Emil Artin conjectured that \begin{equation*} N_{a}(x) \sim A(a)\pi(x) \end{equation*} where $A(a)>0$ is a constant dependent only on $a$ and $\pi(x):=\{p \le x: p\text{ prime}\}$. Rewrite $N_{a}(x)$ as follows: \begin{equation*} N_{a}(x) = \sum_{p \le x} f(i_{a}(p)), \end{equation*} where $f:\mathbb{N}
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Camire, Patrice. "Artin's Primitive Root Conjecture and its Extension to Compositie Moduli." Thesis, 2008. http://hdl.handle.net/10012/3844.

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If we fix an integer a not equal to -1 and which is not a perfect square, we are interested in estimating the quantity N_{a}(x) representing the number of prime integers p up to x such that a is a generator of the cyclic group (Z/pZ)*. We will first show how to obtain an aymptotic formula for N_{a}(x) under the assumption of the generalized Riemann hypothesis. We then investigate the average behaviour of N_{a}(x) and we provide an unconditional result. Finally, we discuss how to generalize the problem over (Z/mZ)*, where m > 0 is not necessarily a prime integer. We present an average result in
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Sen, Gupta Sourav. "Artin's Conjecture: Unconditional Approach and Elliptic Analogue." Thesis, 2008. http://hdl.handle.net/10012/3845.

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In this thesis, I have explored the different approaches towards proving Artin's `primitive root' conjecture unconditionally and the elliptic curve analogue of the same. This conjecture was posed by E. Artin in the year 1927, and it still remains an open problem. In 1967, C. Hooley proved the conjecture based on the assumption of the generalized Riemann hypothesis. Thereafter, the mathematicians tried to get rid of the assumption and it seemed quite a daunting task. In 1983, the pioneering attempt was made by R. Gupta and M. Ram Murty, who proved unconditionally that there exists a specific se
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Book chapters on the topic "Artin's primitive root conjecture"

1

Rosen, Michael. "Artin’s Primitive Root Conjecture." In Graduate Texts in Mathematics. Springer New York, 2002. http://dx.doi.org/10.1007/978-1-4757-6046-0_10.

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Ram Murty, M. "Artin’s Conjecture for Primitive Roots." In Mathematical Conversations. Springer New York, 2001. http://dx.doi.org/10.1007/978-1-4613-0195-0_11.

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