Published online by Cambridge University Press: 24 October 2008
Suppose that a, c and A are real, and that
Let n (X) be the number of zeros of
in ; then it is not difficult to prove that
for some K = K (A, a, c, θ). The problem in this paper is to determine the function K explicitly in terms of its variables.
† γ is restricted by 0≤γ<1, while η is not so restricted. Corresponding, however, to any η satisfying the conditions of the Lemma there is a γ for which either γ = η or γ = η ± 1.
‡ For if γ and z are real, then
‡ The result may be obtained more easily by an appeal to the maximum modulus principle for a complex function
and is bounded, since η, β 1 and a 1 are bounded.
† See (1.1.4).
† In the case A = A′, f(x) has a double zero at x = x 1.
‡ In the case b 2 ≠ 0, f′(x 1) = 0, f″(x 1) = 0, it may be shown that f‴(x 1)╪0. Hence f(x) has a triple zero at x = x 1.
§ Neglecting the trivial case θ = 1.
† For this gives 0≤ξ<q.
‡ By Lemma 5.