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On the average number of elements in a finite field with order or index in a prescribed residue class. (English) Zbl 1061.11050

For a prime \(p\), let \(\mathbb F^*_p\) denote the multiplicative group of the finite field having \(p\) elements. Let \(\delta(p;a,d)\) and \(\rho(p;a,d)\) denote the density of elements of \(\mathbb F^*_p\) having order, respectively index, congruent to \(a\pmod d\). Let \(N(a,d)(x) = \sum_{p\leq x} \delta(p;a,d)\) and \(N'(a,d)(x)=\sum_{p\leq x} \rho(p;a,d)\). The author proves that the limits \[ \delta(a,d)=\lim_{x\to\infty} N(a,d)(x)/\pi(x) \quad\text{and}\quad \rho(a,d)=\lim_{x\to\infty} N'(a,d)(x)/\pi(x) \] both exist. He also gives several formulas for computing theses densities, and he elucidates some connections between the two densities.
The densities also have connection to corresponding characteristic 0 densities. Let \(g\) be a rational number not equal to \(-1,0,\) or \(1\). Let \(\nu_p(g)\) denote the index of \(p\) in the canonical factorization of \(g\). If \(\nu_p(g)=0\), then \(g\) may be considered an element of \(\mathbb F^*_p\).
Let \(N_g(a,d)\) and \(N_g'(a,d)\) be the number of primes \(p\) with \(\nu_p(g)=0\) such that the order, respectively index, of \(g\pmod p\) is \(\equiv a \pmod d\). Let \(\delta_g(a,d)=\lim_{x\to\infty} N_g(a,d)(x)/\pi(x)\) and \(\rho_g(a,d)=\lim_{x\to\infty} N_g'(a,d)(x)/\pi(x)\) To avoid trivialities, assume that \(g\in G\), where \(G\) is the set of rational numbers \(g\) that cannot be written in the form \(\pm g_0^h\) for some rational \(g_0\). In this paper, the author proves, conditionally on GRH, that if \(D(g)\to\infty\) with \(g\in G\), then \(\rho_g(a,d)\) tends to \(\rho(a,d)\).
In a forthcoming paper [On the distribution of the order and index of \(g\pmod p\) over residue classes, preprint, arxiv.org/abs/math.NT/0211259, J. Number Theory (in press) http://dx.doi.org/10.1016/j.jnt.2004.09.004, see also the author’s reviews of L. Murata and K. Chinen, J. Number Theory 105, No. 1, 60–81, 82–100 (2004) in Zbl 1045.11066 and Zbl 1045.11067], he will prove the more difficult analogue for \(\delta\), again conditionally on GRH.

MSC:

11N60 Distribution functions associated with additive and positive multiplicative functions
11N69 Distribution of integers in special residue classes
11A07 Congruences; primitive roots; residue systems
11N37 Asymptotic results on arithmetic functions
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References:

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