Asymptotically Exact Heuristics for (Near) Primitive Roots Original Research Article

Let gset membership, variantimage\{−1, 0, 1}. Let p be a prime. Let ordp(g) denote the exponent of p in the canonical factorization of g. If ordp(g)=0, we define rg(p)=[(image/pimage)*: left angle bracketg mod pright-pointing angle bracket], that is, rg(p) is the residual index mod p of g. For an arbitrary natural number t we consider the set Ng, t of primes p with ordp(g)=0 and rg(p)=t. Write g=±gh0, where g0 is positive and not an exact power of a rational. We introduce a function wg, t(p)set membership, variant{0, 1, 2}, for which it is proved, under the Generalized Riemann Hypothesis (GRH), that[formula]where Ng, t(x) is the counting function for Ng, t . This modifies the naive and, under GRH, false heuristic in which one takes wg, t(p)=1.