The relationship between the regular behavior of the logarithmic derivative of zero-order entire function f with zeros on a finite system of curves of regular rotation for f and the existence of υ-density of zeros of f along such curves is investigated.
Let $\upsilon$ be the growth function such that $r\upsilon'(r)/\upsilon (r) \to 0$ as $r \to +\infty$, $l_\varphi^c = \{z=te^{i(\varphi+c \ln t)}, 1 \leqslant t < +\infty\}$ be the logarithmic spiral, $f$ be the entire function of zero order. The asymptotics of $\ln f(re^{i(\theta +c \ln r)})$ along ordinary logarithmic spirals $l_\theta^c$ of the function $f$ with $\upsilon$-density of zeros along $l_\varphi^c$ outside the $C_0$-set is found. The inverse statement is true just in case zeros of $f$ are placed on the finite logarithmic spirals system $\Gamma_m = \bigcup_{j=0}^m l_{\theta_j}^c$.
We study the Riemann zeta-function $\zeta(s)$ by a Fourier series method. The summation of $\log|\zeta(s)|$ with the kernel $1/|s|^{6}$ on the critical line $\mathrm{Re}\; s = \frac{1}{2}$ is the main result of our investigation. Also we obtain a new restatement of the Riemann Hypothesis.
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