The asymptotic location of the roots of a certain transcendental equation

where Pi is real and a, is a constant not zero. If v is not an integer, Xv is taken to mean exp (v log X), the principal value of the logarithm to be chosen. The symbol [a,] in (3) designates a quantity al+zE(X), where e(X) is used as a generic designation for a function which is analytic for 1X I sufficiently large and which approaches zero uniformly as IX -|) oo . The equation of the type (1) either in form (1) or form (2) has been extensively studied under hypotheses of varying degrees of restrictiveness, by Tamarkin,t Wilder,t and Polya and Schwengeler.? Of these, the two last