On primes p for which d divides ord_p(g)

TL;DR

This paper investigates the distribution of primes p for which a fixed integer d divides the order of g modulo p, providing a new explicit formula for their density and numerical validation.

Contribution

It introduces a simple identity for counting such primes and derives a more concise expression for their natural density, improving upon previous results.

Findings

Derived a simple identity for N_g(d)(x)

Obtained a more compact formula for the natural density

Numerical demonstration confirms theoretical results

Abstract

Let N_g(d) be the set of primes p such that the order of g modulo p is divisible by a prescribed integer d. Wiertelak showed that this set has a natural density and gave a rather involved explicit expression for it. Let N_g(d)(x) be the number of primes p<=x that are in N_g(d). A simple identity for N_g(d)(x) is established. It is used to derive a more compact expression for the natural density than known hitherto. A numerical demonstration, using a program of Y. Gallot, is presented.