Class Numbers of Indefinite Binary Quadratic Forms and the Residual Indices of Integers Modulo p

Fomenko, O. M.

doi:10.1023/b:joth.0000035245.77356.76

Cited by 6 publications

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References 8 publications

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“…The case f (x) = log x and g = 2 was first studied by Bach et al [25]. Fomenko [159] showed that (22) holds true in case f (x) = log x, assuming GRH and Conjecture A of Hooley [239, p. 112]. Felix and Ram Murty [156] managed to prove this result without assuming Conjecture A in case f (x) = log α x, with 0 < α < 1.…”

Section: The Indicator Functionmentioning

confidence: 99%

Artin's Primitive Root Conjecture – A Survey

Moree

2012

Integers

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Abstract. 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'.

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Section: The Indicator Functionmentioning

confidence: 99%

Artin's Primitive Root Conjecture – A Survey

Moree

2012

Integers

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“…A weaker version of Theorem 1.6 using some stronger assumptions was proved by Fomenko [3]. In order to state his theorem, we first need to state a conjecture of Hooley [8, p. 112].…”

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confidence: 99%

On a Conjecture of Erdős

Felix

Murty

2012

Mathematika

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Let a be an integer different from 0, ±1, or a perfect square. We consider a conjecture of Erdős which states that #{ p : a ( p) = r } ε r ε for any ε > 0, where a ( p) is the order of a modulo p. In particular, we see what this conjecture says about Artin's primitive root conjecture and compare it to the generalized Riemann hypothesis and the ABC conjecture. We also extend work of Goldfeld related to divisors of p + a and the order of a modulo p.

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An analogue of Artin’s conjecture for multiplicative subgroups of the rationals

Pappalardi

Susa

2013

Arch. Math.

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Abstract. Given Γ ⊂ Q * a multiplicative subgroup and m ∈ N + , assuming the Generalized Riemann Hypothesis, we determine an asymptotic formula for the number of primes p ≤ x for which the indp Γ = m, where indp Γ = (p − 1)/|Γp| and Γp is the reduction of Γ modulo p. This problem is a generalization of some earlier works by Cangelmi-Pappalardi, Lenstra, Moree, Murata, Wagstaff and probably others. We prove, on GRH, that the primes with this property have a density and, in the case when Γ contains only positive numbers, we give an explicit expression for it in terms of an Euler product. We conclude with some numerical computations.Mathematics Subject Classification (2010). Primary 11R45; Secondary 11A07.

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Class Numbers of Indefinite Binary Quadratic Forms and the Residual Indices of Integers Modulo p

Cited by 6 publications

References 8 publications

Artin's Primitive Root Conjecture – A Survey

Artin's Primitive Root Conjecture – A Survey

On a Conjecture of Erdős

An analogue of Artin’s conjecture for multiplicative subgroups of the rationals

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