A Note on Artin’s Constant

P p(pi−1) , where the summation is over first N prime pi, k ∈ Z+. The classical summation formula is as follows: A = limN→∞Σ(N), where Σ(N) = 1 N ∑ φ(pi−1) pi−1 . The changes needed for arbitrary g are addressed in Theorem 1, a good exercise in basic analytic and algebraic number theory. The same procedure can be applied to other number-theoretic constants like A (see, e.g., [Ni]). In Theorem 2, we demonstrate how it works for the Stephens constant and for Artin’s constants of higher ranks (for the density of prime p such that a given set of “generic” integers generates Zp ). The following three features of this approach vs. the summation formulas are worth noticing. 1) The restricted summation suggested by P. Moore to make the Σ– formula matching the right heuristic density for arbitrary g ∈ Z gives the desired answer in our approach only when g is not a pure (odd) power in Z. Otherwise, nontrivial rational multiplicative corrections occur; they are calculated in Theorem 1, (ii). 2) The p–terms in the denominator and numerator of Rk(N) do not influence the limit, which can be heuristically associated with switching from primitive roots in Zp to those in (Z/(p )). The extra p–factors disappear (cancel) in the corresponding summation formula. When k = 0, our R–formula forA (without restricting the summation) follows from [Pi]. 3) The R–formulas oscillate significantly around A (and the other constants). The magnitude of oscillations increases as k grows; see Figure 1. Representing Rk(N) = ∑N i=1 wi φ(pi−1) pi−1 , the weights wi change from O( logN Nk+2 ) for small i to O( (logN) k+1