The correction factor in Artin ’ s primitive root conjecture
نویسنده
چکیده
In 1927, E. Artin proposed a conjectural density for the set of primes p for which a given integer g is a primitive root modulo p. After computer calculations in 1957 by D. H. and E. Lehmer showed unexpected deviations, Artin introduced a correction factor to explain these discrepancies. The modified conjecture was proved by Hooley in 1967 under assumption of the generalized Riemann hypothesis. This paper discusses two recent developments with respect to the correction factor. The first is of historical nature, and is based on letters between Artin and the Lehmers from 1957-58 that were discovered in the Lehmer archives in Berkeley in December 2000. The second concerns a new interpretation of the correction factor in terms of local contributions by H. W. Lenstra, P. Moree and the author that is well-suited to deal with many generalizations of Artin’s original primitive root problem.
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