The correction factor in Artin's primitive root conjecture.
Emil Artin proposed in 1927 a conjectural density for the set of primes \(p\) for which a given integer \(g\) is a primitive root modulo \(p\). However, in 1957 computer calculations by D. H. and E. Lehmer showed unexpected deviations for certain small \(g\). Artin, when informed of this, reconsidered his original statement and came up with a correction. The Lehmers, unfortunately, do not seem to have mentioned this in any of their papers. The correction is now sometimes attributed to Heilbronn. It is therefore not without historical interest that the author and H. W. Lenstra unearthed the correspondence between the Lehmers and Artin. This correspondence makes for amusing reading, as it shows an interesting clash of mathematical background and outlook. Some of the highlights of this correspondence are reprinted and discussed in the first part of this paper. Assuming the Generalized Riemann Hypothesis it was proved in 1967 by \textit{C. Hooley} [J. Reine Angew. Math. 225, 209--220 (1967; Zbl 0221.10048)] that the density for Artin's problem exists and equals a rational number times the so-called Artin constant \(A=\prod_{p}(1-{1\over p(p-1)})\approx 0.3739\), where \(p\) runs over the primes. On averaging these densities over \(g\), the average turns out to be \(A\). Thus Artin's constant can be considered to be the density for a generic number \(g\). Many variations of Artin's problem follow the same reasoning. Unfortunately, analytic number theory gives the densities involved in terms of sums involving the Möbius function. It can then be (and usually is) a challenge to derive the actual density in Euler product form. Indeed, already Hooley spoke of a `rather harder' calculation [ibid., p. 219]. \textit{H. W. Lenstra}, the author and the reviewer (in preparation) have found a way to infer the Euler product in a much more direct way. Also, for the first time the correction term is written as a product of local terms. The form of the density is then the generic positive density \(\delta\) multiplied by \(1+\prod_p E_p\), where the \(E_p\) are averages of real character sums and \(p\) runs over the primes. Note that \(-1\leq E_p\leq 1\). Thus it has become fairly straightforward to describe when the density vanishes (an odd number of them have to equal \(-1\) and the remaining ones have to equal 1, which happens only under rather restricted conditions on the parameters of the problem involved). In the second part of this well-written paper the author explains and illustrates the method of the latter three authors in case of Artin's original problem and a variation where one asks for primes \(p\equiv a\pmod f\) such that, moreover, \(g\) is a primitive root modulo \(p\). Hooley's `rather harder' calculation is rendered almost immediate by this approach. In general it is seen that the character sum method reduces the calculation of the correction factor to a number of fairly mechanical local computations. For the primes in progression variant this is amply demonstrated by comparing the author's calculation with one of the reviewer [Max-Planck-Institut preprint MPI-1998-57 (unpublished)] which evaluates this density in the classical way. The Euler product formula for the density was used by the reviewer [J. Number Theory 78, 85--98 (1999; Zbl 0931.11036)] to set up an `asymptotically exact heuristic' for these problems. This is a simple function \(w_g(p)\) satisfying \(0\leq w_g(p)\leq 1\) such that, moreover, \(\sum_{p\leq x,~p\equiv a(\mod f)}w_g(p)\) behaves asymptotically in the same way as the number of primes \(p\leq x\) such that \(g\) is a primitive root modulo \(p\) and \(p\equiv a(\mod f)\) (under the Generalized Riemann Hypothesis). The number \(w_g(p)\) can be interpreted as some sort of `probability' that \(g\) is a primitive root modulo \(p\).
scientific article; zbMATH DE number 2058875