On the Distribution of the Zeros and α-Values of a Random Integral Function (I)

Abstract

1.1. The present paper is the outcome of an original aim, to establish, as fully as might be, the conjecture that almost integral functions of finite non-zero order p behave in essentials, and with generalization on obvious principles from p = 2 to general p, like the elliptic function o-(z) = o-(z; W1 X W2). This is an integral function of z of order p = 2, with a behavior of the greatest simplicity. It is exponentially large except in regions of exponentially small area round the zeros, and these regions are distributed uniformly over the z-plane. If we map the values of log+ I o-(z) I on the z-plane, we obtain a surface like a bowl (the height at distance R from the origin being of order R2), with uniformly distributed pits, each exponentially small in area. The results we have obtained can fairly be said to fulfil the original aim. The conceptions most and almost integral functions imply a basis of probability. Some such bases are more restrictive than others: at one extreme both the moduli and arguments of the coefficients are subject to variations; at the other, one set, say the moduli, are arbitrarily given and only the arguments are random. Since the assertions of the conjecture have a far wider scope in the second case we have made that our principal study. We suppose a set of coefficients a. to be given, subject only to Eanz?' being an integral function of finite non-zero order p; a random factor ?i1 is then superimposed, and we have a family a of functions f(z) = E i a Wez. We find now that there is a sub-class a*, containing almost the f of a, such that all f of a* show the behavior; f is exponentially large except in pits of exponentially small area. If now D is any bounded set of a-values containing a = 0, it follows from Rouch6's theorem that each pit of an f of A*, except for a finite number, contains the same number of a-values as it does zeros; thus all a-values behave essentially alike. As for the distribution of the pits in the z-plane, we are able to assert that for all f of A* it is approximately the same. And finally this common distribution has considerable regularity in respect of direction, and as much regularity in respect of distance (from the origin) as is compatible with the nature of the case. The reservation in the last clause arises as follows. Suppose the coefficients an are at the same time highly lacunary (zero except for widely spaced n) and