The following theorem was proved by Paley and Wiener (4, p. 70; 1, p. 136).
Theorem 1. If f(z) is a canonical product of order 1 with real zeros, and f(0) = 1, the conditions
are equivalent. n(r) denotes the number of zeros of absolute value not exceeding r.
Instead of assuming the zeros to be all real Pfluger assumed that the zeros are close to the real axis and proved the following theorem (5 or 1, p. 143).