  The signal strength of the $gg \to H \to \gamma \gamma$ reaction in $pp$
collisions at the LHC is studied within the context of the SM with UED. The
impact of an arbitrary number $n$ of UED on both the $gg\to H$ and $H\to \gamma
\gamma$ subprocesses is studied. The 1-loop contributions of Kaluza-Klein
excitations to these subprocesses are proportional to discrete and continuous
sums, which can diverge. By implementing dimensional regularization, it is
shown that discrete regularized sums can naturally be expressed as
multidimensional Epstein functions, and that divergences, if exist, emerge
through the poles of these functions. It is found that continuous sums
converge, but the discrete ones diverge, with the exception of the $n=1$ case,
in which the 1-dimensional Epstein function converges. It is argued that
divergences that arise from discrete sums for $n\geq 2$ are genuine UV
divergences, since they correspond to short-distance effects in the compact
manifold. Then, the amplitudes are renormalized in a modern sense by
incorporating interactions of canonical dimension higher than four that allow
us to generate the required counterterms, which are determined using a
$\overline{\rm MS}$-like renormalization scheme. We find that the $gg\to H$
subprocess is quite sensitive to both the size and the dimension of the compact
manifold, but the SM prediction for $H\to \gamma \gamma$ subprocess is
practically unchanged. In the $n=1$ case, it is found that the experimental
constraint on the compactification scale $R^{-1}\geq 1.5$ TeV allow us to
reproduce the experimental limit on the signal strength $1.01\leq
\mu^{(1)}_{\gamma \gamma}\leq 1.2$. In the $n\geq 2$ cases, it is found that
the experimental limit on $\mu^{(n)}_{\gamma \gamma}$ leads to stronger lower
bounds for the compactification scale given by $R^{-1}\geq 1.55, 2.45, 3.57,
5.10, 7.25$ TeVs for $n=2, 4, 6, 8, 10$, respectively.
