On the Distributions of the Zeros of Certain Analytic Functions.

where J and N are positive integers, and the X's are real or complex constants. It is assumed that we do not have XON = XiN = .. = XJN, and that we do not have XOn = Xln = . . . = )kXj * Ofor any n < N. It is also assumed that no two terms in the right-hand member of (1) are identical or differ only by a constant factor. The work reported here is essentially a generalization, in one direction, of certain well-known work by Wilder, Tamarkin, P61ya, Schwengler and others.1'2'3'4'5'6 We first arrange f(z) in a certain normal form. The function can be written M f(z) = E fm(z) exp [gmn(z) + xm ZN], m = 0