On the distribution of the order and index of g(mod p) over residue classes III
نویسنده
چکیده
For a fixed rational number g 6∈ {−1, 0, 1} and integers a and d we consider the sets Ng(a, d), respectively Rg(a, d), of primes p for which the order, respectively the index of g(mod p) is congruent to a(mod d). Under the Generalized Riemann Hypothesis (GRH), it is known that these sets have a natural density δg(a, d), respectively ρg(a, d). It is shown that these densities can be expressed as linear combinations of certain constants introduced by Pappalardi. Furthermore it is proved that δg(a, d) and ρg(a, d) equal their g-averages for almost all g.
منابع مشابه
On the distribution of the order and index of g(mod p) over residue classes II
For a fixed rational number g 6∈ {−1, 0, 1} and integers a and d we consider the set Ng(a, d) of primes p for which the order of g(mod p) is congruent to a(mod d). It is shown, assuming the Generalized Riemann Hypothesis (GRH), that this set has a natural density δg(a, d). Moreover, δg(a, d) is computed in terms of degrees of certain Kummer extensions. Several properties of δg(a, d) are establi...
متن کاملOn the distribution of the order and index of g(mod p) over residue classes
For a fixed rational number g 6∈ {−1, 0, 1} and integers a and d we consider the set Ng(a, d) of primes p for which the order of g(mod p) is congruent to a(mod d). For d = 4 and d = 3 we show that, under the Generalized Riemann Hypothesis (GRH), these sets have a natural density δg(a, d) and compute it. The results for d = 4 generalise earlier work by Chinen and Murata. The case d = 3 was appar...
متن کاملRing structures of mod p equivariant cohomology rings and ring homomorphisms between them
In this paper, we consider a class of connected oriented (with respect to Z/p) closed G-manifolds with a non-empty finite fixed point set, each of which is G-equivariantly formal, where G = Z/p and p is an odd prime. Using localization theorem and equivariant index, we give an explicit description of the mod p equivariant cohomology ring of such a G-manifold in terms of algebra. This makes ...
متن کاملON THE SZEGED INDEX OF NON-COMMUTATIVE GRAPH OF GENERAL LINEAR GROUP
Let $G$ be a non-abelian group and let $Z(G)$ be the center of $G$. Associate with $G$ there is agraph $Gamma_G$ as follows: Take $Gsetminus Z(G)$ as vertices of$Gamma_G$ and joint two distinct vertices $x$ and $y$ whenever$yxneq yx$. $Gamma_G$ is called the non-commuting graph of $G$. In recent years many interesting works have been done in non-commutative graph of groups. Computing the clique...
متن کاملOn the Distribution of the Order over Residue Classes
For a fixed rational number g ∈ {−1, 0, 1} and integers a and d we consider the set Ng(a, d) of primes p such that the order of g modulo p is congruent to a (mod d). Under the Generalized Riemann Hypothesis (GRH), it can be shown that the set Ng(a, d) has a natural density δg(a, d). Arithmetical properties of δg(a, d) are described, and δg(a, d) is compared with δ(a, d): the average density of ...
متن کامل