(i) Misprint: the second equation in equation (2.1), page 486 should read: instead of .
(ii) In Lemma 2, page 490, ξv+1 should be ξv-1.
Following a suggestion of Martin Huxley, Bob Buttsworth applied the above formula to investigate the conjectural density H(t) of primes p for which a
given positive integer t, not a perfect power, is g(p), the least primitive
root mod p. Bob attacked the explicit formula for H(t) directly and showed that
it was positive.
The proof of the general case was never published, partly on account
of its complexity. However the method involved some results concerning the
inclusion-exclusion principle and these were published.
(See R.N. Buttsworth, An Inclusion-Exclusion Transform, Ars Combinatoria 15, 279-300, 1983.
The formula for H(t), t prime, is included in this paper.)
In a letter to me dated July 1977, Hendrik Lenstra sketched a short proof of the positivity of H(t) in the general case, using the results of his paper quoted above. This proof did not use the explicit formula for H(t).
L. Murata, On the Magnitude of the Least Primitive Root, Astérisque, 198-199-200, (1991) 253-257, mentions another application of my paper to the conjectural density of primes p such that g(p) > C.
P.D.T.A. Elliot and L. Murata, On the Average of the Least Primitive Root modulo p, J. London Math. Soc. 56 (1997) 435-454, use some of my formulae to extend the results of the previous paper.
A. Paszkiewicz and A. Schinzel also use my work as a basis for their papers Numerical calculation of the density of prime numbers with a given least primitive root, Math. Comp. 71 (2002) 1781-1797 and On the least prime primitive root modulo a prime, Math. Comp. 71 (2002) 1307-1321.
R. Balasubramanian, F. Luca and R. Thangadurai, On the exact degree of ℚ(√a1,√a2 , . . . , √al ) over ℚ, Proc. Amer. Math. Soc. 138 (2010), no. 7, 2283-2288 (see review) give another proof of formula (9.2) of my Acta Arith. paper for the case k=2.
A. Schinzel, Primitive roots and quadratic non-residues, Acta Arith. 149 (2011), no. 2, 16-170, also uses my work.
Last modified 28th March 2018