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A connection between the zeta functions of zeros and poles of a meromorphicfunction has been established, and using it, a criterion for the absence ofzeros has been derived. Sufficient conditions for the existence of zeros ofsums of Cauchy kernels have been obtained, including those based on geometricconditions on the poles.

The zeta function of zeros and poles of a meromorphic function, the  criterion for the absence of zeros, and its application to sums of Cauchy  kernels