  In his recent research, the author improved the error term in the prime
geodesic theorem for compact, even-dimensional, rank one locally symmetric
spaces. It turned out that the obtained estimate
$O(x^{2\rho-\frac{\rho}{n}}(\log x)^{-1})$ coincides with the best known
results for compact Riemann surfaces, three manifolds, and manifolds with
cusps, where $n$ stands for the dimension of the space, and $\rho$ is the
half-sum of positive roots. The above bound was then reduced to
$O(x^{2\rho-\rho\frac{2\cdot(2n)+1}{2n\cdot(2n)+1}}(\log
x)^{\frac{n-1}{2n\cdot(2n)+1}-1}(\log\log
x)^{\frac{n-1}{2n\cdot(2n)+1}+\varepsilon})$ in the Gallagherian sense, with
$\varepsilon$ $>$ $0$, and the key role played by the counting function
$\psi_{2n}(x)$. The purpose of this research is to prove that the latter
$O$-term can be further reduced. To do so, we derive new explicit formulas for
the functions $\psi_{j}(x)$, $j$ $\geq$ $n$, and conditional formula for
$\psi_{n-1}(x)$. Applying the Gallagher-Koyama techniques, we deduce the
asymptotics for $\psi_{0}(x)$, and the Gallagherian prime geodesic theorems.
The obtained error terms $O(x^{2\rho-\rho\frac{2j+1}{2nj+1}}(\log
x)^{\frac{n-1}{2nj+1}-1}(\log\log x)^{\frac{n-1}{2nj+1}+\varepsilon})$, $n-1$
$\leq$ $j$ $<$ $2n$, improve the $O$-term given above, with the optimal
unconditional (conditional) size achieved for $j$ $=$ $n$ ($j$ $=$ $n-1$). If
$j$ $=$ $n$ $\geq$ $4$, our new bound coincides with the best known estimate in
the manifolds with cusps case. If $j$ $=$ $n-1$, the $O$-term fully agrees with
the results in the Riemann surface case ($n$ $=$ $2$, $\rho$ $=$
$\frac{1}{2}(n-1)$ $=$ $\frac{1}{2}$), and the three manifolds case ($n$ $=$
$2$, $\rho$ $=$ $1$). Finally, for $j$ $=$ $n-1$, $n$ $\geq$ $4$, our result
improves the best known bound in the manifolds with cusps case.
