In earlier papers of this series III and IV, poles of certain meromorphic functions involving Riemann's zeta-function at shifted arguments and Dirichlet polynomials were studied. The functions in question were quotients of products of such functions, and it was shown that they have ``many'' poles. The main result in the present paper is that the same conclusion remains valid even for finite sums of functions of this type.
Let γ denote the imaginary parts of the nontrivial zeros of the Riemann zeta-function ζ(s). For sufficiently large T and ε>0, Ivi\'c proved that ∑T<γ≤2T|ζ(12+iγ)|2<<ε(T(logT)2loglogT)3/2+ε, where the implicit constant depends only on ε. In this paper, this result is improved by (i) replacing |ζ(12+iγ)|2 by max|ζ(s)|2, where the maximum is taken over all s=σ+it in the rectangle 12−A/logT≤σ≤2,|t−γ|≤B(loglogT)/logT with some fixed positive constants A,B, and (ii) replacing the upper bound by T(logT)2loglogT. The method of proof differs completely from Ivi\'c's approach.