Artin's Primitive Root Conjecture: Unconditional Approach and Elliptic Curve Analogue

This book explores the different approaches towardsproving Artin's 'primitive root' conjectureunconditionally and the elliptic curve analogue ofthe same. The conjecture was posed by E. Artin in1927, and it still remains an open problem. In 1967, C. Hooley proved the conjecture based on theassumption of the generalized Riemann hypothesis. Thereafter, the mathematicians tried to get rid ofthe assumption and it seemed quite a daunting task. The best results we have so far are by R. Gupta andM. Ram Murty (1983), and D. R. Heath-Brown (1986). Itstates that there can be at most 12 exceptionalintegers or at most 2 exceptional primes for whichthe conjecture does not hold. But the question -'Which ones?', remains illusive. The first part ofthis book deals with these sieve theoretic results byGupta-Murty and Heath-Brown. The second halfdiscusses the elliptic curve analogue of theconjecture which was proposed by Lang and Trotter in1977 and was proved by Gupta and Murty in 1986, assuming the generalized Riemann hypothesis, forcurves with complex multiplication. These discussionswill help the reader gain a broad perspective of theconjecture and the current advancements.