A note on Artin's constant

We suggest new heuristic summation formulas for the Artin constant and its analogs in higher ranks, for the density of prime numbers $p$ such that a given set of integers generates the corresponding multiplicative groups modulo powers of $p$. Depending on the choice of these integers, certain interesting rational corrections emerge, which are calculated in rank one case. The rate of convergence of these sums to the corresponding constants, connected with the generalized Riemann Hypothesis, is analyzed numerically. We also discuss the Stephens constant.