On distribution of zeros for a class of meromorphic functions

Yu. F. Korobeǐnik

Vladikavkazskij matematičeskij žurnal, Tome 19 (2017) no. 1, pp. 41-49

Résumé

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 magnitudes: $\mathcal{P}_y$, 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 $\frac12 +i\tau$, $|\tau|$; $\mathcal{N}_y(T,\sigma)$, the sum of multiplicities of all zeros of $y$ which lies inside the rectangle with vertices $A=\frac12-\sigma - iT$, $C=\frac12+\sigma - iT$, $D=\frac12+\sigma + iT$, $F=\frac12-\sigma + iT$. Here $T$ is a $y$-regular ordinate, that is, $y(z)$ is analytic and has no zeros on the line $\operatorname{Im} z =T$, $\operatorname{Re} z \in \mathbb{R}$, $\sigma\in (h_1,h)$, $h=h(y)$, $\sigma$ is chosen in such a manner that $y(z)\ne 0$ on the segments $[F,A]$ and $[C,D]$. The problem of finding the magnitudes of $\mathcal{P}_y$, $\mathcal{N}_y(T)$ and $\mathcal{N}_y(T,\sigma)$ with the help of corresponding characteristics of the «Riemann's multiplier» $b_y(z)$ 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.