A~derivation of the Cartwright--Levinson theorem from the theorem of Kolmogorov
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XIII, Tome 135 (1984), pp. 76-86
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An entire function $f$ of exponential type is said to
belong to the Cartwright class $C$ if
$$
\int_{-\infty}^{+\infty}\frac{\log^+|f(x)|}{1+x^2}\,dx+\infty.
$$
Let $N_+(r)(N_-(r))$ denote the number of zeros of $f$ in $|z|\leqslant R$ with $\operatorname{Re}z\geqslant0$ ($\operatorname{Re}z0$ respectively). A
simple proof, based on the weak type (1.1) Kolmogorov inequality,
of the following important result is given.
Theorem. Let $f\in C$ и $\displaystyle\varlimsup_{y\to+\infty}\frac{\log |f(iy)|}y=\varlimsup_{y\to-\infty}\frac{\log |f(iy)|}{|y|}=a$. Then
$$
\lim_{r\to+\infty}\frac{N_+(r)}r=\lim_{r\to+\infty}\frac{N_-(r)}r=\frac a\pi.
$$
@article{ZNSL_1984_135_a6,
author = {P. Kusis},
title = {A~derivation of the {Cartwright--Levinson} theorem from the theorem of {Kolmogorov}},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {76--86},
publisher = {mathdoc},
volume = {135},
year = {1984},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_1984_135_a6/}
}
P. Kusis. A~derivation of the Cartwright--Levinson theorem from the theorem of Kolmogorov. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XIII, Tome 135 (1984), pp. 76-86. http://geodesic.mathdoc.fr/item/ZNSL_1984_135_a6/