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DOI: 10.23671/VNC.2017.1.5820

# On Distribution of Zeros for a Class of Meromorphic Functions

Korobeinik Yu. F.
Vladikakazian Mathematical Journal 2017. Vol. 19. Issue 1.
Abstract:
In this article some class $$\mathcal{K}_0$$ of meromorphic functions is introduced. Each function $$y(z)$$ from $$\mathcal{K}_0$$ satisfies the functional equation $$y(z)=b_y(z)y(1-z)$$ with its own "Riemann's multiplier" $$b_y(z)$$ which is a meromorphic function with real zeros and poles. All poles of an arbitrary function from $$\mathcal{K}_0$$ are real and belong to the interval $$(\frac12,\frac12+h_1]$$, $$h_1=h_1(y)$$. Using the theory of residues we prove some relation connecting the following mag\-ni\-tu\-des: $$\mathcal{P}_y$$, the sum of all orders of poles of $$y \in \mathcal{K}_0$$; $$\mathcal{N}_y(T)$$, the sum of multiplicities of all zeros of $$y$$ having the form $$\frac12 +i\tau$$, $$|\tau|> \(b_y(z)$$ is posed. This problem is solved in the paper for~$$\mathcal{P}_y$$. Moreover, the obtained equality enables one to deduce a definite relation the left part of which contains the number $$2\alpha_{T_0}+ 4\beta_{T_0}$$ where $$T_0$$ is arbitrary $$y$$-nonregular ordinate, $$\alpha_{T_0}$$ is the multiplicities of all possible zero of $$y$$ of the form $$\frac12+iT_0$$, $$\beta_{T_0}$$ is the sum of multiplicities of all possible zeros of $$y$$ belonging to $$\frac12+iT_0,+\infty +iT_0$$. It is proved that the class $$\mathcal{K}_0$$ contains the Riemann's Zeta-Function.
Keywords: zeros of meromorphic functions, functional equation.