Analogues of Artin's Conjecture by Larry
نویسنده
چکیده
Artin's celebrated conjecture on primitive roots (Artin [l, p. viii], Hasse [2], Hooley [3]) suggests the following Conjecture. Let S' be a set of rational primes. For each q£-S, let Lq be an algebraic number field of degree n(q). For every square-f ree integer k, divisible only by primes of S, define Lk to be the composite of all Lq, q\ k, and denote n{k) =deg(Ljb/0). Assume that 2 * l/n(k) converges, where the sum is over those k for which Lk is defined. Then the natural density of the set P of all primes p which do not split completely in each Lq exists and has the value ]£* fx(k)/n(k), where ix is the Möbius function and the term k = l has been included with » ( 1 ) 1 . If S— {all rational primes}, Lq = Q(Çq, a ), aÇEZ, f« = a primitive qth root of 1, then the conjecture is equivalent to Artin's conjecture. If S is a finite set, then the conjecture is easily verifiable using the prime ideal theorem. For 5 = {all rational primes}, L«—Q(?/), the conjecture has been proved by Knobloch [4] (for r = 2 and only for Dirichlet densities) and by Mirsky [5]. We have proved the following theorems, whose proofs will appear elsewhere.
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