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

Abstract

For any prime p we consider the density of elements in the multiplicative group of the finite field F_p having order, respectively index, congruent to a(mod d). We compute these densities on average, where the average is taken over all finite fields of prime order. Some connections between the two densities are established. It is also shown how to compute these densities with high numerical accuracy.

25 pages, 4 tables. A conjecture made in the previous version is now resolved. Tables are also improved, thanks to a C++ program written by Yves Gallot

References in corpus (2)

• On the distribution of the order and index of g(mod p) over residue classes II
• On a distribution property of the residual order of a (mod p)

Cited by in corpus (7)

• Convoluted convolved Fibonacci numbers
• On the distribution of the order and index of g(mod p) over residue classes II
• The formal series Witt transform
• Homological Algebra and Divergent Series
• On the distribution of the order and index of g(mod p) over residue classes III
• The distribution of the multiplicative index of algebraic numbers over residue classes
• Asymptotically exact heuristics for prime divisors of a^k+b^k