The “Pits Effect” for the Integral Function$f\left( z \right) = \sum {\exp \left\{ { - {\vartheta ^{ - 1}}\left( {n\log n - n} \right) + \pi i\alpha {n^2}} \right\}{z^n},\alpha = \tfrac{1}{2}\left( {\sqrt 5 - 1} \right)}$

Abstract

If f ₀(z) = Σc _n z ⁿ is any integral function of finite non-zero order ϱ, consider the class of functions where r ⁿ(t) are Rademacher’s functions, representing a ‘random’ factor of the form ± 1. Littlewood and Offord [1] have shown that ‘most’ f(z) behave with great crudity and violence. If we erect an ordinate |f(z)| at the point z of the z-plane, then the resulting surface is an exponentially rapidly rising bowl, approximately of revolution, with exponentially small ‘pits’ going down to the bottom. The zeros of f, more generally the w-points where f = w, all lie in the pits for |z| > R(w). Finally the pits are very uniformly distributed in direction, and as uniformly distributed in distance as is compatible with the order ϱ.