Carleman’s Theorem

Let f be holomorphic in Re z ≥ 0 and suppose f has no zeros on z = iy. Choose ρ > 0 so that ρ < (modulus of the smallest zero of f in Re z ≥ 0). Let $$ \left\{ {{{z}_{n}} = {{r}_{n}}{{e}^{{i{{\theta }_{n}}}}}} \right\} $$ be the zeros of f in Re z ≥ 0. Define the following: $$ \sum (R) = \sum (R:f) = \sum\limits_{{{{r}_{n}} \leqslant R}} {\left( {\frac{1}{{{{r}_{n}}}} - \frac{{{{r}_{n}}}}{{{{R}^{2}}}}} \right)} \cos {{\theta }_{n}} $$ proper multiplicity of the zeros taken into account); $$ I(R) = I(R:f) = \frac{1}{{2\pi }}\int_{r}^{R} {\left( {\frac{1}{{{{t}^{2}}}} - \frac{1}{{{{R}^{2}}}}} \right)\log \left| {f(it)f( - it)} \right|dt,} $$ where the integral is taken from ir to iR along the imaginary axis; and $$ J(R) = J(R:f) = \frac{1}{{\pi R}}\int_{{ - \pi /2}}^{{\pi /2}} {\log \left| {f({{{\operatorname{Re} }}^{{i\theta }}})} \right|\cos \theta d\theta ,} $$ where the integral is taken along the semicircle of radius R centered at 0. Then $$ \sum (R) = I(R) + J(R) + O(1). $$