Number Theory ON HOOLEY’S THEOREM WITH WEIGHTS.pdf

Rend. Sem. Mat. Univ. Pol. Torino Vol. 53, 4 (1995) Number Theory ON HOOLEY’S THEOREM WITH WEIGHTS F. Pappalardi Abstract. We adapt Hooley’s proof that the Generalized Riemann Hypothesis implies the Artin Conjecture for primitive roots to various other problems. We consider the sum ( ) where is the index of 2 modulo and is a given function. In various cases we establish asymptotic formulas for such a sum and analyse the constants. While we claim no originality, we outline the method to approach this problem in a fairly general case. 1. Introduction For a fixed prime number , we denote by the index of (mod ). For a function , we consider the sum We will establish various estimates for such a sum. If and for , then the famous Artin Conjecture for primitive roots states that where is the Artin constant, prime In 1967, C. Hooley (see [5]) proved the Artin Conjecture as a consequence of the Generalized Riemann Hypothesis. The weaker form of the Artin Conjecture states that any fixed integer that is not a perfect square is a primitive root for infinitely many primes. Heath-Brown [4], Gupta and Murty [3] (see also [9]) solved this form of the Artin conjecture for a very large class of Supported by Human Capital and Mobility Program of the European Community, under contract ERBCHBICT930706 376 F. Pappalardi numbers but the asymptotic formula in (1) is still to be proven