On the average number of elements in a finite field with order or index in a prescribed residue class

Moree, Pieter

doi:10.1016/j.ffa.2003.10.001

Cited by 15 publications

(24 citation statements)

References 24 publications

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“…Note that u(n) ≤ λ(n). They proved various results comparing u(n) with λ(n), see also [312,313,343].…”

Section: )mentioning

confidence: 94%

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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“…Note that u(n) ≤ λ(n). They proved various results comparing u(n) with λ(n), see also [312,313,343].…”

Section: )mentioning

confidence: 94%

Artin's Primitive Root Conjecture – A Survey

Moree

2012

Integers

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“…Euler products that contain Dirichlet characters should likewise be expressible as a product of terms of the form L(k, χ); see [10] for an example.) For most , the resulting expressions are quite unwieldy; however, when φ( ) = 2, we can obtain rather tidy forms by using manipulations that are special to these particular expressions.…”

Section: Better Convergence For Certain Cmentioning

confidence: 99%

Roots of unity and nullity modulo $n$

Finch

Martin

Sebah

2010

Proc. Amer. Math. Soc.

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Abstract. For a fixed positive integer , we consider the function of n that counts the number of elements of order in Z * n . We show that the average growth rate of this function is C (log n) d( )−1 for an explicitly given constant C , where d( ) is the number of divisors of . From this we conclude that the average growth rate of the number of primitive Dirichlet characters modulo n of order is (We also consider the number of elements of Z n whose th power equals 0, showing that its average growth rate is D (log n) −1 for another explicit constant D . Two techniques for evaluating sums of multiplicative functions, the Wirsing-Odoni and SelbergDelange methods, are illustrated by the proofs of these results.Let Z * n denote the group (under multiplication modulo n) of integers relatively prime to n, and let denote a fixed positive integer. Define a (n) to be the number of solutions of x = 1 in Z * n . The value of a (p), when p is prime, ranges over all divisors of ; however, a (n) can be much larger than if n is composite. It is therefore interesting to ask how the function a (n) behaves on average over n. We can ask the same aboutã (n), which we let denote the number of solutions of x = 1 in Z * n for which x m = 1 for all 1 ≤ m < ; such an x is said to be of order in the group Z * n . In other words, a counts the th roots of unity modulo n, whilẽ a counts the primitive th roots of unity modulo n.The following theorem, which is proved in Section 2, gives the average rate of growth for both functions a andã for every positive integer . In the statement of the theorem, we employ the usual notation p j n to mean that p j | n but p j+1 n; we also use d( ) to denote the number of divisors of . Theorem 1. For any positive integer and for any real numberas N → ∞. Furthermore, the same asymptotic formula holds for n

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“…But J P (s) is an entire function. Therefore, if one sets 22) this defines the analytic continuation of (2.20) to a meromorphic function on the complex plane.…”

Section: 34mentioning

confidence: 99%