the present study we restrict ourself to the GCE calculations within the Boltzmann statistics approximation. The paper is organized as follows. In Section II the ideal HG model is presented. In Section III the one-component VDW gas is used to calculate the average multiplicity and the scaled variance. These results are extended to the multi-component VDW gas in Section IV. Both primordial and final state (after resonance decay) hadron yields and their fluctuations are considered. In Section V the scaled variance of negative, positive and all charged hadrons is calculated along the chemical freeze-out line. This gives the VDW HG predictions for fluctuations of hadron multiplicities in central A+A collisions at different collision energies from SIS to LHC. A summary, presented in Section VI, closes the paper.

Ideal Hadron-resonance Gas
==========================

The partition function of the ideal Boltzmann HG in the GCE has the form: [$$\begin{aligned}
 \label{Z-id} Z_{id}~=~\exp\left[\sum_j V~\phi_j ~
\right]~,
 \end{aligned}$$]{} where: $$\begin{aligned}
\label{phi-j}
\phi_j~  \equiv
~\lambda_j~\frac{g_j}{2\pi^2}~~\int_0^{\infty}k^2dk~
\exp\left[-~\frac{(k^2~+~m_j^2)^{1/2}}{T} \right]
=~\lambda_j~\frac{g_j}{2\pi^2}~m_j^2~T~K_2\left(\frac{m_j}{T}\right)~.\end{aligned}$$ In Eqs. (\[Z-id\],\[phi-j\]), $\lambda_j=\exp\left(\mu_j/T\right)$ is the particle specific fugacity, $g_j$ the degeneracy of $j$th particle species, $m_j$ the particle mass, and $\mu_j = b_j\mu_B + s_j\mu_S  + q_j\mu_Q $ the chemical potential due to the $j$th particle internal quantum 