On Distribution of Zeros for a Class of Meromorphic Functions

Korobeinik Yu. F.
Vladikavkaz 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.

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