On refinements of rank one Gallagherian prime geodesic theorems

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.