  We study a quantum Hamiltonian that is given by the (negative) Laplacian and
an infinite chain of $\delta$-like potentials with strength $\kappa>0$ on the
half line $\rz_{\geq0}$ and which is equivalent to a one-parameter family of
Laplacians on an infinite metric graph. This graph consists of an infinite
chain of edges with the metric structure defined by assigning an interval
$I_n=[0,l_n]$, $n\in\nz$, to each edge with length $l_n=\frac{\pi}{n}$. We show
that the one-parameter family of quantum graphs possesses a purely discrete and
strictly positive spectrum for each $\kappa>0$ and prove that the Dirichlet
Laplacian is the limit of the one-parameter family in the strong resolvent
sense. The spectrum of the resulting Dirichlet quantum graph is also purely
discrete. The eigenvalues are given by $\lambda_n=n^2$, $n\in\nz$, with
multiplicities $d(n)$, where $d(n)$ denotes the divisor function. We thus can
relate the spectral problem of this infinite quantum graph to Dirichlet's
famous divisor problem and infer the non-standard Weyl asymptotics
$\mathcal{N}(\lambda)=\frac{\sqrt{\lambda}}{2}\ln\lambda +\Or(\sqrt{\lambda})$
for the eigenvalue counting function. Based on an exact trace formula, the
Vorono\"i summation formula, we derive explicit formulae for the trace of the
wave group, the heat kernel, the resolvent and for various spectral zeta
functions. These results enable us to establish a well-defined (renormalized)
secular equation and a Selberg-like zeta function defined in terms of the
classical periodic orbits of the graph, for which we derive an exact functional
equation and prove that the analogue of the Riemann hypothesis is true.
