In this thesis, I have explored the different approaches towards proving Artin;;s`primitive root;; conjecture unconditionally and the elliptic curve analogue of thesame. This conjecture was posed by E. Artin in the year 1927, and it still remains anopen problem. In 1967, C. Hooley proved the conjecture based on the assumptionof the generalized Riemann hypothesis. Thereafter, the mathematicians tried to getrid of the assumption and it seemed quite a daunting task. In 1983, the pioneeringattempt was made by R. Gupta and M. Ram Murty, who proved unconditionallythat there exists a specific set of 13 distinct numbers such that for at least oneof them, the conjecture is true. Along the same line, using sieve theory, D. R.Heath-Brown reduced this set down to 3 distinct primes in the year 1986. This isthe best unconditional result we have so far. In the first part of this thesis, we will review the sieve theoretic approach taken by Gupta-Murty and Heath-Brown. Thesecond half of the thesis will deal with the elliptic curve analogue of the Artin;;sconjecture, which is also known as the Lang-Trotter conjecture. Lang and Trotterproposed the elliptic curve analogue in 1977, including the higher rank version, andalso proceeded to set up the mathematical formulation to prove the same. Theanalogue conjecture was proved by Gupta and Murty in the year 1986, assumingthe generalized Riemann hypothesis, for curves with complex multiplication. Theyalso proved the higher rank version of the same. We will discuss their proof indetails, involving the sieve theoretic approach in the elliptic curve setup. Finally, I will conclude the thesis with a refinement proposed by Gupta and Murty to find out a finite set of points on the curve such that at least one satisfies the conjecture.
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Artin's Conjecture: Unconditional Approach and Elliptic Analogue
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