Gallagherian PGT on Some Compact Riemannian Manifolds of Negative Curvature

Abstract

The purpose of the research is to prove that some of the latest and currently most advanced results on prime geodesic theorems can be significantly improved when considered for a certain class of locally symmetric spaces of real rank one. Our special attention will be focused on the reduction of remainder terms in the Gallagherian-type theorem for a family of compact, even-dimensional spaces. The main tool to enable the process will be newly derived explicit formulas for counting functions of appropriate degree.

1 Introduction

A special place in analytic number theory is reserved for Selberg zeta functions.

While most papers, including [44], discuss zeta functions associated with Riemann surfaces, papers such as: [22] (R-rank one groups, spherical representations), [43] ($SL\left( 2,\mathbb {C}\right)$), [23] (non-compact, rank one, finite volume case), [47] (generalizations of [22, 23]), [24, 36] (hyperbolic manifolds with cusps), [35] (higher rank locally symmetric spaces), [13] (locally homogeneous manifolds of odd dimensions), etc., treat zeta functions attached to more general settings.

For some particular cases, papers like [46] ($SU\left( n,1\right)$), [38] ($SO_{0}\left( 1,n\right)$), [42] offer individually valuable information.

The natural companion of the Selberg zeta function is the Ruelle zeta function. It was named after Ruelle [40, 41], who meromorphically extended it to $\mathbb {C}$. Nevertheless, the credit for the meromorphic extension of the Selberg zeta function belongs to Fried [18] (see also, [17, 19]) who recognized the interrelationships between the two functions and discovered a way to express each of them using the other.

As for our current article, the work of Bunke and Olbrich [11] will have the greatest impact. Thus, zetas introduced in [11] will play the most significant role in the entire study.

We adopt the minimum necessary notation from [11, 25].

Throughout the paper, $\mathscr {Y}$ will denote a compact locally symmetric Riemannian manifold of strictly negative sectional curvature and of dimension n (n even). If we take G to be a connected semi-simple Lie group of real rank one and K a maximal compact subgroup of G, then there is a discrete co-compact torsion free subgroup $\varGamma$ $\subset$ G such that $\mathscr {Y}$ $=$ $\varGamma \backslash X$, where X $\cong$ G/K is the universal covering of $\mathscr {Y}$, i.e., it is either a real $H\mathbb {R}^{k}$, or a complex $H\mathbb {C}^{m}$, or a quaternionic $H\mathbb {H}^{m}$ hyperbolic space, or the hyperbolic Cayley plane $H\mathbb {C}a^{2}$.

The Cartan involution $\theta$ on the Lie algebra $\mathfrak {g}$ of G yields the decomposition $\mathfrak {g}$ $=$ $\mathfrak {k}$ $\oplus$ $\mathfrak {p}$. As usual, $\mathfrak {a}$ is a maximal abelian subspace of $\mathfrak {p}$. For the system of positive roots of $\left( \mathfrak {g},\mathfrak {a}\right)$, let $\rho$ denote the half sum of its elements, and $\mathfrak {n}$ the sum of the root spaces. Now, G $=$ KAN is the Iwasawa decomposition of G, and M is the centralizer of $\mathfrak {a}$ in K.

It is well known that every hyperbolic element g $\in$ G is conjugate in G to some element am $\in$ $\exp \left( \mathfrak {a}^{+}\right) M$ $=$ $A^{+}M$ for $\mathfrak {a}^{+}$ the positive Weyl chamber in $\mathfrak {a}$ (see, [21]).

By [11, pp. 90-91], the Selberg zeta function $Z_{S,\chi }\left( s,\sigma \right)$ is defined for s $\in$ $\mathbb {C}$, ${{\,\mathrm{Re}\,}}\left( s\right)$ > $\rho$ as

$$\begin{aligned} Z_{S,\chi }\left( s,\sigma \right) =\prod \limits _{\begin{array}{c} 1\ne \left[ g\right] \in C\varGamma \\ \text {primitive} \end{array}}\prod \limits _{k=0}^{\infty }\det \left( 1-\left( \sigma \left( m\right) \otimes \chi \left( g\right) \otimes S^{k}\left( {{\,\mathrm{Ad}\,}}\left( ma\right) _{\bar{\mathfrak {n}}}\right) \right) e^{-\left( s+\rho \right) l\left( g\right) }\right) , \end{aligned}$$

where $S^{k}$ is the k-th symmetric power of an endomorphism, $\sigma$ and $\chi$ are finite-dimensional unitary representations of M and $\varGamma$, $l\left( g\right)$ is the length of the closed geodesic on $\mathscr {Y}$ defined by g, $C\varGamma$ is the set of all conjugacy classes of $\varGamma$, and $\bar{\mathfrak {n}}$ $=$ $\theta \mathfrak {n}$.

For s $\in$ $\mathbb {C}$, ${{\,\mathrm{Re}\,}}\left( s\right)$ > $2\rho$, the Ruelle zeta function $Z_{R,\chi }\left( s,\sigma \right)$ is defined by

$$\begin{aligned} Z_{R,\chi }\left( s,\sigma \right) =\prod \limits _{\begin{array}{c} 1\ne \left[ g\right] \in C\varGamma \\ \text {primitive} \end{array}}\det \left( 1-\left( \sigma \left( m\right) \otimes \chi \left( g\right) \right) e^{-sl\left( g\right) }\right) ^{\left( -1\right) ^{n-1}}. \end{aligned}$$

In view of Fried’s result mentioned above, we have that there exist sets $I_{p}$ $=$ $\left\{ \left( \tau ,\lambda \right) \,:\,\tau \in \hat{M},\,\lambda \in \mathbb {R}\right\}$, such that

$$\begin{aligned} \begin{aligned} Z_{R,\chi }\left( s,\sigma \right) =\prod \limits _{p=0}^{n-1}\prod \limits _{\left( \tau ,\lambda \right) \in I_{p}}Z_{S,\chi }\left( s+\rho -\lambda ,\tau \otimes \sigma \right) ^{\left( -1\right) ^{p}}, \end{aligned} \end{aligned}$$

(1)

where $\hat{M}$ stands for the unitary dual of M.

In general, the prime geodesic theorem (PGT) is an assertion about the asymptotic behavior of the number of closed geodesics on a manifold selected by their length. The case of Riemann surfaces is very well investigated in the works of Huber [28, 29] and Hejhal [26, 27], as well as in the works of many other authors. It is a striking fact that despite knowing that the Riemann hypothesis is almost valid in this case, the O-term $O\left( x^{\frac{3}{4}}\left( \log x\right) ^{-\frac{1}{2}}\right)$ in PGT is much larger than the desired $O\left( x^{\frac{1}{2}+\varepsilon }\right)$. The credit for the best bound so far belongs to Randol [39] for his $O\left( x^{\frac{3}{4}}\left( \log x\right) ^{-1}\right)$. Note that $\frac{3}{4}$ appears in the sum of class numbers of orders in complex cubic fields [14] and also in a PGT for $SL_{4}$ [15].

It was Kuznetsov summation formula that allowed Iwaniec [30] to reduce $\frac{3}{4}$ to $\frac{35}{48}+\varepsilon$ for the modular group $\varGamma$ $=$ $PSL\left( 2,\mathbb {Z}\right)$. The result has gradually improved over the years. The descending sequence $\frac{7}{10}+\varepsilon$, $\frac{71}{102}+\varepsilon$, $\frac{25}{36}+\varepsilon$ is built through [12, 34, 45], respectively. With special reference to [33] (see also, [31]), we highlight the occurrence of $\frac{7}{10}$ in arbitrary congruence subgroups of $SL\left( 2,\mathbb {Z}\right)$.

In the case of the Picard group $PSL\left( 2,\mathbb {Z}\left[ {{\,\mathrm{i}\,}}\right] \right)$, the authors in [10] recently derived $\frac{3}{2}+\frac{\theta _{S}}{2}+\varepsilon$ in PGT, where $\theta _{S}$ is a subconvexity exponent for quadratic Dirichlet L-functions defined over $\mathbb {Z}\left[ {{\,\mathrm{i}\,}}\right]$.

So far, the most satisfactory estimates in PGTs for large n, i.e.,

$O\left( x^{\frac{4\rho ^{2}+\rho }{2\rho +1}}\left( \log x\right) ^{-1}\right)$ resp. $O\left( x^{2\rho -\frac{\rho }{n}}\left( \log x\right) ^{-1}\right)$ are obtained by Avdispahić [3, 4] on k-dimensional manifolds with cusps resp. by Gušić [25] on $\mathscr {Y}$.

Gallagherization, a technique discovered by P. X. Gallagher [20] and initially aimed at reducing the O-term in the prime number theorem came into focus in 2016, when Koyama [32] successfully applied it to hyperbolic surfaces. From 2016 onwards, it has been extensively exploited. Adapting the method to his needs, Avdispahić, in a series of papers [1, 2, 5], managed to replace the above-mentioned $\frac{3}{4}$, $\frac{25}{36}$, $\frac{3}{2}+\frac{\theta _{S}}{2}$ with better $\frac{7}{10}$, $\frac{2}{3}$, $\frac{8}{5}$ for Riemann surfaces, $PSL\left( 2,\mathbb {Z}\right)$ and $PSL\left( 2,\mathbb {Z}\left[ {{\,\mathrm{i}\,}}\right] \right)$.

In the same context, the authors in [9] reduced $\frac{4\rho ^{2}+\rho }{2\rho +1}$. They particularly proved that for a k-dimensional manifold with cusps $X_{\varGamma }$ and $\varepsilon$ > 0, there exists a set E of finite logarithmic measure such that

$$\begin{aligned} \pi _{\varGamma }\left( x\right) =\sum \limits _{\alpha _{k}<s_{i}\left( j\right) \le 2\rho }\left( -1\right) ^{j}{{\,\mathrm{li}\,}}\left( x^{s_{i}\left( j\right) }\right) +O\left( x^{\alpha _{k}}\left( \log x\right) ^{\beta _{k}-1}\left( \log \log x\right) ^{\beta _{k}+\varepsilon }\right) \end{aligned}$$

(2)

as x $\rightarrow$ $\infty$, x $\notin$ E, where $\alpha _{k}$ $=$ $\left( k-1\right) \left( 1-\frac{2k+1}{4k^{2}+2}\right)$, $\beta _{k}$ $=$ $\frac{k-1}{2k^{2}+1}$,

$\left( s_{i}\left( j\right) -j\right) \left( 2\rho -j-s_{i}\left( j\right) \right)$ is a small eigenvalue in $\left[ 0,\frac{3}{4}\rho ^{2}\right]$ of $\varDelta _{j}$ on $\pi _{\sigma _{j},\lambda _{i}\left( j\right) }$ with $s_{i}\left( j\right)$ $=$ $\rho$ $+$ ${{\,\mathrm{i}\,}}{}\lambda _{i}\left( j\right)$ or $s_{i}\left( j\right)$ $=$ $\rho$ − ${{\,\mathrm{i}\,}}{}\lambda _{i}\left( j\right)$ in $\left( \frac{3}{2}\rho ,2\rho \right]$, $\varDelta _{j}$ is the Laplacian acting on the space of j-forms over $X_{\varGamma }$, $\pi _{\sigma _{j},\lambda _{i}\left( j\right) }$ is the principal series representation, and $\pi _{\varGamma }\left( x\right)$ is the function counting prime geodesics on $X_{\varGamma }$ of length not larger than $\log x$.

As far as the compact case $\mathscr {Y}$ is concerned, the author reduced the unconditional bound $O\left( x^{2\rho -\frac{\rho }{n}}\left( \log x\right) ^{-1}\right)$ in his most recent research [25]. By [25, p. 9, Th. 3], for $\mathscr {Y}$ as above and $\varepsilon$ > 0, there is a set E of finite logarithmic measure such that

$$\begin{aligned} \begin{aligned} \pi _{\varGamma }\left( x\right) =&\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\left( \tau ,\lambda \right) \in I_{p}}\sum \limits _{\begin{array}{c} \alpha \in S_{p,\tau ,\lambda }^{\mathbb {R}}\\ 2\rho -\rho \frac{4n+1}{4n^{2}+1}<\alpha \le 2\rho \end{array}}{{\,\mathrm{li}\,}}\left( x^{\alpha }\right) + \end{aligned} \end{aligned}$$

(3)

$$\begin{aligned}&O\left( x^{2\rho -\rho \frac{4n+1}{4n^{2}+1}}\left( \log x\right) ^{\frac{n-1}{4n^{2}+1}-1}\left( \log \log x\right) ^{\frac{n-1}{4n^{2}+1}+\varepsilon }\right) \end{aligned}$$

as x $\rightarrow$ $\infty$, x $\notin$ E, where $S_{p,\tau ,\lambda }^{\mathbb {R}}$ is the set of real singularities of $Z_{S,\chi }\left( s+\rho -\lambda ,\tau \otimes \sigma \right)$.

Note that the result (3) was obtained using the 2n times integrated Chebyshev function $\psi _{2n}\left( x\right)$ (see, Section 2). In addition, the O-term in (3) is of the form

$$\begin{aligned} O\left( x^{2\rho -\rho \frac{2\left( 2n\right) +1}{2n\left( 2n\right) +1}}\left( \log x\right) ^{\frac{n-1}{2n\left( 2n\right) +1}-1}\left( \log \log x\right) ^{\frac{n-1}{2n\left( 2n\right) +1}+\varepsilon }\right) . \end{aligned}$$

This, at first glance a coincidence, raises the question of the dependence of PGT (3) on the order of the applied counting function. Finding the answer to this question as well as the answer to the question of whether or not the O-term in (3) could be further reduced by varying $\psi _{j}\left( x\right)$, j $\in$ $\mathbb {N}$, represent two of the three main motives for this research to be conducted. The third motive comes from the knowledge that lower-order counting functions are commonly used to produce PGTs. Hejhal [26, 27] dealt with $\psi _{1}\left( x\right)$ on Riemann surfaces (n $=$ k $=$ 2), Randol [39] observed $\psi _{2}\left( x\right)$ in the compact case, Park [37] resp. Avdispahić-Gušić [6] considered $\psi _{k-1}\left( x\right)$ resp. $\psi _{k}\left( x\right)$ on manifolds with cusps (n $=$ k), etc.

The structure of the paper is as follows. In addition to motivation, Sects. 1 and 2 provide the minimum necessary notation and offer a brief historical overview of the most relevant facts. Section 3 is devoted to new results. In Sect. 3.1, we apply the classical complex analysis apparatus to derive new explicit formulas for the functions $\psi _{j}\left( x\right)$, j $\ge$ n (Theorem 1). Keeping an eye on the Gallagher’s technique, we obtain the asymptotics of $\psi _{0}\left( x\right)$ (Theorem 2) from $\psi _{j}\left( x\right)$, j $\ge$ n in Sect. 3.2. Ultimately, in Sect. 3.3 we formulate and prove the main result of the research (Theorem 3). As desired, we thus reduce the O-term in (3). Section 4 is devoted to discussion.

2 Preliminaries

Define $\psi _{j}\left( x\right)$ $=$ $\int \limits _{0}^{x}\psi _{j-1}\left( t\right) \mathrm{d}t$, j $\in$ $\mathbb {N}$, where $\psi _{0}\left( x\right)$ $=$ $\sum \limits _{\begin{array}{c} 1\ne \left[ \gamma \right] \in C\varGamma \\ l\left( \gamma \right) \le \log x \end{array}}\varLambda \left( \gamma \right)$, and $\varLambda \left( \gamma \right)$ $=$ $l\left( \gamma _{0}\right)$ for $\gamma _{0}$ the primitive element corresponding to $\gamma$.

For the sake of clarity, we recall that the singularities of meromorphically continued $Z_{S,\chi }\left( s,\sigma \right)$ are determined as follows.

Let $i_{1}^{*}$  :  $R\left( K\right)$ $\rightarrow$ $R\left( M\right)$ be the restriction map induced by the embedding $i_{1}$  :  M $\hookrightarrow$ K, where $R\left( K\right)$ and $R\left( M\right)$ are the representation rings over $\mathbb {Z}$ of K and M, respectively. Now, for each $\sigma$ $\in$ $\hat{M}$, there is some $\nu$ $\in$ $R\left( K\right)$, such that $i_{1}^{*}\left( \nu \right)$ $=$ $\sigma$ (see, [11, Propositions 1.1 and 1.2]).

Suppose that $E_{A}\left( .\right)$ is the family of spectral projections of a normal operator A.

Put $m_{\chi }\left( s,\nu ,\sigma \right)$ $=$ ${{\,\mathrm{Tr}\,}}E_{A_{\mathscr {Y},\chi }\left( \nu ,\sigma \right) }\left( \left\{ s\right\} \right)$, $m_{d}\left( s,\nu ,\sigma \right)$ $=$ ${{\,\mathrm{Tr}\,}}E_{A_{d}\left( \nu ,\sigma \right) }\left( \left\{ s\right\} \right)$, s $\in$ $\mathbb {C}$, where the operators $A_{\mathscr {Y},\chi }\left( \nu ,\sigma \right)$ and $A_{d}\left( \nu ,\sigma \right)$, corresponding to $\mathscr {Y}$ and $X_{d}$, are defined in [11, Subsection 1.1.3], and where $X_{d}$ denotes a compact dual space of the symmetric space X.

According to Lemma 1.18 in [11], there is a $\sigma$-admissible $\nu$ $\in$ $R\left( K\right)$ for every $\sigma$ $\in$ $\hat{M}$, where $\nu$ $\in$ $R\left( K\right)$ is called $\sigma$-admissible if $i_{1}^{*}\left( \nu \right)$ $=$ $\sigma$ and $m_{d}\left( s,\nu ,\sigma \right)$ $=$ $P_{\sigma }\left( s\right)$ for all 0 $\le$ s $\in$ $L\left( \sigma \right)$. Here, $P_{\sigma }\left( s\right)$ resp. $L\left( \sigma \right)$ denote the polynomial resp. the lattice introduced by Definition 1.13 in [11]. Additionally, $L\left( \sigma \right)$ $=$ $T\left( \epsilon _{\sigma }+\mathbb {Z}\right)$, where T and $\epsilon _{\sigma }$ are given by the same definition.

Thus, if $\nu$ is $\sigma$-admissible, the singularities (zeros and poles) of $Z_{S,\chi }\left( s,\sigma \right)$ are the following ones (see, [11, p. 107, Th. 3.15]): at $\pm {{\,\mathrm{i}\,}}{}s$ of order $m_{\chi }\left( s,\nu ,\sigma \right)$ if s $\ne$ 0 is an eigenvalue of $A_{\mathscr {Y},\chi }\left( \nu ,\sigma \right)$, at s $=$ 0 of order $2m_{\chi }\left( 0,\nu ,\sigma \right)$ if 0 is an eigenvalue of $A_{\mathscr {Y},\chi }\left( \nu ,\sigma \right)$, at $-s$, s $\in$ $T\left( \mathbb {N}-\epsilon _{\sigma }\right)$ of order $-2\left( -1\right) ^{\frac{n}{2}}\frac{\dim \left( \chi \right) {{\,\mathrm{vol}\,}}\left( \mathscr {Y}\right) }{{{\,\mathrm{vol}\,}}\left( X_{d}\right) }m_{d}\left( s,\nu ,\sigma \right)$ (in this case, s > 0 is an eigenvalue of $A_{d}\left( \nu ,\sigma \right)$). Clearly, if two singularities coincide, then the orders add up.

Let $\sigma =1$, $\chi =1$. For convenience, we omit to write them below. By (1),

$$\begin{aligned} \begin{aligned} Z_{R}\left( s\right) =\prod \limits _{p=0}^{n-1}\prod \limits _{\left( \tau ,\lambda \right) \in I_{p}}Z_{S}\left( s+\rho -\lambda ,\tau \right) ^{\left( -1\right) ^{p}}. \end{aligned} \end{aligned}$$

(4)

Denote by $\mathcal {T}$ the set of all $\tau$ $\in$ $\hat{M}$ occurring in (4).

3 Results

3.1 Explicit formulas for $\psi _{j}\left( x\right)$, j $\ge$ n

We prove the following theorem.

Theorem 1

Let $\mathscr {Y}$ be a compact locally symmetric Riemannian manifold of negative sectional curvature and of even dimension n. If j $\ge$ n, then

$$\begin{aligned} \psi _{j}\left( x\right) =\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\left( \tau ,\lambda \right) \in I_{p}} \sum \limits _{\rho <\alpha \le 2\rho }\frac{x^{\alpha +j}}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }+ \sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\begin{array}{c} \left( \tau ,\lambda \right) \in I_{p}\\ \lambda =2\rho \end{array}} \sum \limits _{{{\,\mathrm{Re}\,}}\left( \alpha \right) =\rho }\frac{x^{\alpha +j}}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }, \end{aligned}$$

(5)

where $\alpha$ is a singularity of the Selberg zeta function $Z_{S}\left( s+\rho -\lambda ,\tau \right)$.

Proof

Reasoning in the same way as in [37, p. 98], we obtain for c > $2\rho$ (see, [11, p. 91, eq. (3.4)])

$$\begin{aligned} \psi _{j}\left( x\right) =\frac{1}{2\pi {{\,\mathrm{i}\,}}{}}\int \limits _{c-{{\,\mathrm{i}\,}}{}\infty }^{c+{{\,\mathrm{i}\,}}{}\infty }-\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{s\left( s+1\right) \ldots \left( s+j\right) }ds. \end{aligned}$$

Let $T\gg 0$, $\tilde{T}=\sqrt{T^{2}-\rho ^{2}}$ and 0 < $\varepsilon$ < c − $\rho$.

By the residue theorem,

$$\begin{aligned} \begin{aligned}&\frac{1}{2\pi {{\,\mathrm{i}\,}}{}}\int \limits _{c-{{\,\mathrm{i}\,}}{}\tilde{T}}^{c+{{\,\mathrm{i}\,}}{}\tilde{T}}-\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }ds\\&\quad =\frac{-1}{2\pi {{\,\mathrm{i}\,}}{}}\left( \int \limits _{c+{{\,\mathrm{i}\,}}{}\tilde{T}}^{\rho +\varepsilon +{{\,\mathrm{i}\,}}{}\tilde{T}}+\int \limits _{\rho +\varepsilon +{{\,\mathrm{i}\,}}{}\tilde{T}}^{\rho +{{\,\mathrm{i}\,}}{}\tilde{T}}+\int \limits _{C_{T}}+ \int \limits _{\rho -{{\,\mathrm{i}\,}}{}\tilde{T}}^{\rho +\varepsilon -{{\,\mathrm{i}\,}}{}\tilde{T}}+ \int \limits _{\rho +\varepsilon -{{\,\mathrm{i}\,}}{}\tilde{T}}^{c-{{\,\mathrm{i}\,}}{}\tilde{T}}\right) \left( -\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }\right) ds\\&\qquad +\sum \limits _{\alpha \in R\left( T\right) }{{\,\mathrm{Res}\,}}_{s=\alpha }\left( -\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }\right) , \end{aligned} \end{aligned}$$

(6)

where the boundary of $R\left( T\right)$ $=$ $\left\{ s\in \mathbb {C}\,:\,\left| s\right| \le T,\,{{\,\mathrm{Re}\,}}\left( s\right) \le \rho \right\}$ $\cup$

$\left\{ s\in \mathbb {C}\,:\,\rho \le {{\,\mathrm{Re}\,}}\left( s\right) \le c,\,-\tilde{T}\le {{\,\mathrm{Im}\,}}\left( s\right) \le \tilde{T}\right\}$ is assumed not to contain any of the singularities of $-\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }$, and $C_{T}$ $=$ $\left\{ s\in \mathbb {C}\,:\,\left| s\right| =T,\,{{\,\mathrm{Re}\,}}\left( s\right) \le \rho \right\}$ is positively oriented (see, Fig. 1).

Fig. 1

Boundary of $R\left( T\right)$

First, we estimate the integrals $\int \limits _{\rho +\varepsilon +{{\,\mathrm{i}\,}}{}\tilde{T}}^{\rho +{{\,\mathrm{i}\,}}{}\tilde{T}}\cdot ds$ and $\int \limits _{\rho -{{\,\mathrm{i}\,}}{}\tilde{T}}^{\rho +\varepsilon -{{\,\mathrm{i}\,}}{}\tilde{T}}\cdot ds$ on the right-hand side of (6).

By Theorem 3.15 in [11, p. 107] and Theorem 9.1 in [16, p. 89], there is a constant $C_{1}$ such that

$$\begin{aligned} \begin{aligned} N_{S,p}^{\tau ,\lambda }\left( y\right) =C_{1}y^{n}+O\left( y^{n-1}\right) , \end{aligned} \end{aligned}$$

(7)

where $N_{S,p}^{\tau ,\lambda }\left( y\right)$ is the function counting singularities $\rho _{S,p}^{\tau ,\lambda }$ $=$ $-\rho$ $+$ $\lambda$ $+$ ${{\,\mathrm{i}\,}}{}\gamma _{S,p}^{\tau ,\lambda }$ of $Z_{S}\left( s+\rho -\lambda ,\tau \right)$ (corresponding to factor $Z_{S}\left( s+\rho -\lambda ,\tau \right) ^{\left( -1\right) ^{p}}$ in (4)) along ${{\,\mathrm{Re}\,}}\left( s\right)$ $=$ $-\rho +\lambda$.

By claim $\left( b\right)$ of Theorem 4.1 in [8,  p. 314], for $\delta$ > 0 and t $\gg$ 0 (${{\,\mathrm{i}\,}}{}t$ is not a singularity of $Z_{S}\left( s,\tau \right)$ for $\tau$ $\in$ $\mathcal {T}$)

$$\begin{aligned} \begin{aligned} \frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }=O\left( t^{n-1+\delta }\right) +\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p}\sum \limits _{\begin{array}{c} \left( \tau ,\lambda \right) \in I_{p}\\ \lambda =2\rho \end{array}} \sum \limits _{\left| t-\gamma _{S,p}^{\tau ,\lambda }\right| \le 1}\frac{1}{s-\rho _{S,p}^{\tau ,\lambda }} \end{aligned} \end{aligned}$$

(8)

for s $=$ $\sigma _{1}$ $+$ ${{\,\mathrm{i}\,}}{}t$, $\rho$ $\le$ $\sigma _{1}$ < $\frac{1}{4}t$ − $\rho$, and

$$\begin{aligned} \begin{aligned} \frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }=O\left( \frac{1}{\eta }t^{n-1+\delta }\right) \end{aligned} \end{aligned}$$

(9)

for s $=$ $\sigma _{1}$ $+$ ${{\,\mathrm{i}\,}}{}t$, $\rho$ $+$ $\eta$ $\le$ $\sigma _{1}$ < $\frac{1}{4}t$ − $\rho$, where 0 < $\eta$ $\le$ $2\rho$.

Applying the relations (8) and (7), we obtain (Cf. [37, p. 99, (3.11)] and [6, p. 368, (2)])

$$\begin{aligned} \begin{aligned} \frac{1}{2\pi {{\,\mathrm{i}\,}}{}}\int \limits _{\rho +{{\,\mathrm{i}\,}}{}\tilde{T}}^{\rho +\varepsilon +{{\,\mathrm{i}\,}}{}\tilde{T}}-\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }ds =O\left( x^{j+\rho +\varepsilon }T^{-j-2+n+\delta }\right) . \end{aligned} \end{aligned}$$

(10)

Namely, by (8), it is clear that we have

$$\begin{aligned} \begin{aligned}&\frac{1}{2\pi {{\,\mathrm{i}\,}}{}}\int \limits _{\rho +{{\,\mathrm{i}\,}}{}\tilde{T}}^{\rho +\varepsilon +{{\,\mathrm{i}\,}}{}\tilde{T}}-\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }ds =O\left( x^{j+\rho +\varepsilon }T^{-j-2+n+\delta }\right) \\&\qquad +O\left( x^{j+\rho +\varepsilon }T^{-j-1}\sum \limits _{p=0}^{n-1}\sum \limits _{\begin{array}{c} \left( \tau ,\lambda \right) \in I_{p}\\ \lambda =2\rho \end{array}} \sum \limits _{\left| \tilde{T}-\gamma _{S,p}^{\tau ,\lambda }\right| \le 1}\int \limits _{\rho +{{\,\mathrm{i}\,}}{}\tilde{T}}^{\rho +\varepsilon +{{\,\mathrm{i}\,}}{}\tilde{T}}\frac{\left| ds\right| }{\left| s-\rho _{S,p}^{\tau ,\lambda }\right| }\right) \\&\quad =O\left( x^{j+\rho +\varepsilon }T^{-j-2+n+\delta }\right) \\&\qquad +O\left( x^{j+\rho +\varepsilon }T^{-j-1}\sum \limits _{p=0}^{n-1}\sum \limits _{\begin{array}{c} \left( \tau ,\lambda \right) \in I_{p}\\ \lambda =2\rho \end{array}} \sum \limits _{\left| \tilde{T}-\gamma _{S,p}^{\tau ,\lambda }\right| \le 1}\,\,\,\int \limits _{\rho -\varepsilon +{{\,\mathrm{i}\,}}{}\tilde{T}}^{\rho +\varepsilon +{{\,\mathrm{i}\,}}{}\tilde{T}}\frac{\left| ds\right| }{\left| s-\rho _{S,p}^{\tau ,\lambda }\right| }\right) . \end{aligned} \end{aligned}$$

(11)

This is the place where we make use of the Landau trick (see, [26, p. 107]). It consists of applying the Cauchy integral theorem to the last integral. There are two possible paths, i.e., for $\gamma _{S,p}^{\tau ,\lambda }$ < $\tilde{T}$ $\left( \gamma _{S,p}^{\tau ,\lambda }>\tilde{T}\right)$ we use the upper (the lower) half of $\left| s-\rho -{{\,\mathrm{i}\,}}{}\tilde{T}\right|$ $=$ $\varepsilon$. Each semi-circular integral, and, consequently, each integral on the right-hand side of (11), is $O\left( 1\right)$. Furthermore, the number of summands in (11), corresponding to $\gamma _{S,p}^{\tau ,\lambda }$’s, is $N_{S,p}^{\tau ,\lambda }\left( \tilde{T}+1\right)$ − $N_{S,p}^{\tau ,\lambda }\left( \tilde{T}-1\right)$ $=$ $O\left( \tilde{T}^{n-1}\right)$ $=$ $O\left( T^{n-1}\right)$. Taking into account the obtained facts, the relation (11) becomes

$$\begin{aligned}&\frac{1}{2\pi {{\,\mathrm{i}\,}}{}}\int \limits _{\rho +{{\,\mathrm{i}\,}}{}\tilde{T}}^{\rho +\varepsilon +{{\,\mathrm{i}\,}}{}\tilde{T}}-\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }ds\\&\quad =O\left( x^{j+\rho +\varepsilon }T^{-j-2+n+\delta }\right) +O\left( x^{j+\rho +\varepsilon }T^{-j-2+n}\right) \\&\quad =O\left( x^{j+\rho +\varepsilon }T^{-j-2+n+\delta }\right) . \end{aligned}$$

Thus, the equality (10) holds true.

Similarly,

$$\begin{aligned} \begin{aligned} \frac{1}{2\pi {{\,\mathrm{i}\,}}{}}\int \limits _{\rho +\varepsilon -{{\,\mathrm{i}\,}}{}\tilde{T}}^{\rho -{{\,\mathrm{i}\,}}{}\tilde{T}}-\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }ds =O\left( x^{j+\rho +\varepsilon }T^{-j-2+n+\delta }\right) . \end{aligned} \end{aligned}$$

(12)

To estimate the integrals $\int \limits _{c+{{\,\mathrm{i}\,}}{}\tilde{T}}^{\rho +\varepsilon +{{\,\mathrm{i}\,}}{}\tilde{T}}\cdot ds$ and $\int \limits _{\rho +\varepsilon -{{\,\mathrm{i}\,}}{}\tilde{T}}^{c-{{\,\mathrm{i}\,}}{}\tilde{T}}\cdot ds$ appearing in (6), we make use of the asymptotics (9).

It immediately follows that

$$\begin{aligned} \begin{aligned} \frac{1}{2\pi {{\,\mathrm{i}\,}}{}}\int \limits _{\rho +\varepsilon +{{\,\mathrm{i}\,}}{}\tilde{T}}^{c+{{\,\mathrm{i}\,}}{}\tilde{T}}-\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }ds =O\left( \varepsilon ^{-1}x^{c+j}T^{-j-2+n+\delta }\right) \end{aligned} \end{aligned}$$

(13)

and

$$\begin{aligned} \begin{aligned} \frac{1}{2\pi {{\,\mathrm{i}\,}}{}}\int \limits _{c-{{\,\mathrm{i}\,}}{}\tilde{T}}^{\rho +\varepsilon -{{\,\mathrm{i}\,}}{}\tilde{T}}-\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }ds =O\left( \varepsilon ^{-1}x^{c+j}T^{-j-2+n+\delta }\right) . \end{aligned} \end{aligned}$$

(14)

The remaining integral on the right-hand side of (6) is $\int \limits _{C_{T}}\cdot ds$.

Recall the following facts.

By Corollary 4.2 in [7,  p. 530], a meromorphic extension over $\mathbb {C}$ of the Ruelle zeta function $Z_{R}\left( s\right)$ can be expressed as

$$\begin{aligned} \begin{aligned} Z_{R}\left( s\right) =\frac{Z_{R}^{1}\left( s\right) }{Z_{R}^{2}\left( s\right) }, \end{aligned} \end{aligned}$$

(15)

where $Z_{R}^{1}\left( s\right)$ and $Z_{R}^{2}\left( s\right)$ are entire functions of order at most n over $\mathbb {C}$.

Moreover, according to Fried’s known result [18,  p. 509, Prop. 7], there is a constant C > 0 such that for arbitrarily large choices of r

$$\begin{aligned} \begin{aligned} \int \limits _{\left| s\right| =r}\left| \frac{Z^{'}\left( s\right) }{Z\left( s\right) }\right| \left| ds\right| \le Cr^{n}\log r \end{aligned} \end{aligned}$$

(16)

for $Z\left( s\right)$ the ratio of two nonzero entire functions of order not larger than n.

Following Park’s reasoning in manifolds with cusps case [37, p. 99], i.e., using the assertions (15) and (16), we estimate

$$\begin{aligned} \frac{1}{2\pi {{\,\mathrm{i}\,}}{}}\int \limits _{C_{T}}-\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }ds =&O\left( x^{\rho +j}T^{-j-1}\int \limits _{C_{T}}\left| \frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\right| \left| ds\right| \right) \nonumber \\ =&O\left( x^{\rho +j}T^{-j-1}\int \limits _{\left| s\right| =T}\left| \frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\right| \left| ds\right| \right) \nonumber \\ =&O\left( x^{\rho +j}T^{-j-1+n}\log T\right) . \end{aligned}$$

(17)

Finally, we deal with the residues in (6).

Having in mind the positions of the singularities of $Z_{S}\left( s,\tau \right)$, $\tau$ $\in$ $\mathcal {T}$ [11, p. 107, Th. 3.15], we write

$$\begin{aligned} \begin{aligned}&\sum \limits _{\alpha \in R\left( T\right) }{{\,\mathrm{Res}\,}}_{s=\alpha }\left( -\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }\right) \\&\quad =\sum \limits _{\rho<\alpha \le 2\rho }{{\,\mathrm{Res}\,}}_{s=\alpha }\left( -\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }\right) + \sum \limits _{\begin{array}{c} \alpha \in R\left( T\right) \\ {{\,\mathrm{Re}\,}}\left( \alpha \right) =\rho \end{array}}{{\,\mathrm{Res}\,}}_{s=\alpha }\left( .\right) + \sum \limits _{\begin{array}{c} \alpha \in R\left( T\right) \\ {{\,\mathrm{Re}\,}}\left( \alpha \right)<\rho \end{array}}{{\,\mathrm{Res}\,}}_{s=\alpha }\left( .\right) \\&\quad =\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\left( \tau ,\lambda \right) \in I_{p}}\sum \limits _{\rho<\alpha \le 2\rho } {{\,\mathrm{Res}\,}}_{s=\alpha }\left( \frac{Z_{S}^{'}\left( s+\rho -\lambda ,\tau \right) }{Z_{S}\left( s+\rho -\lambda ,\tau \right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }\right) \\&\qquad +\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\begin{array}{c} \left( \tau ,\lambda \right) \in I_{p}\\ \lambda =2\rho \end{array}}\sum \limits _{\begin{array}{c} \alpha \in R\left( T\right) \\ {{\,\mathrm{Re}\,}}\left( \alpha \right) =\rho \end{array}} {{\,\mathrm{Res}\,}}_{s=\alpha }\left( \frac{Z_{S}^{'}\left( s+\rho -\lambda ,\tau \right) }{Z_{S}\left( s+\rho -\lambda ,\tau \right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }\right) \\&\qquad +\sum \limits _{\begin{array}{c} \alpha \in R\left( T\right) \\ {{\,\mathrm{Re}\,}}\left( \alpha \right) <\rho \end{array}}{{\,\mathrm{Res}\,}}_{s=\alpha }\left( -\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }\right) . \end{aligned} \end{aligned}$$

(18)

Consider the first sum on the right-hand side of (18).

For arbitrarily selected and then fixed p $\in$ $\left\{ 0,1,\ldots ,n-1\right\}$, $\left( \tau ,\lambda \right)$ $\in$ $I_{p}$, let us take a closer look at ${{\,\mathrm{Res}\,}}_{s=\alpha }\left( \frac{Z_{S}^{'}\left( s+\rho -\lambda ,\tau \right) }{Z_{S}\left( s+\rho -\lambda ,\tau \right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }\right)$, where $\rho$ < $\alpha$ $\le$ $2\rho$. Clearly, all such $\alpha$’s are singularities of $Z_{S}\left( s+\rho -\lambda ,\tau \right)$.

Let $\alpha$ be one such singularity.

It is immediately seen that (Cf. [26, p. 88])

$$\begin{aligned} {{\,\mathrm{Res}\,}}_{s=\alpha }\left( \frac{Z_{S}^{'}\left( s+\rho -\lambda ,\tau \right) }{Z_{S}\left( s+\rho -\lambda ,\tau \right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }\right) =-o_{\alpha }\frac{x^{\alpha +j}}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }, \end{aligned}$$

where $o_{\alpha }$ is the order of $\alpha$.

The argument for the second sum on the right-hand side of (18) is similar.

Hence,

$$\begin{aligned} \begin{aligned}&\sum \limits _{\alpha \in R\left( T\right) }{{\,\mathrm{Res}\,}}_{s=\alpha }\left( -\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }\right) \\&\quad =\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\left( \tau ,\lambda \right) \in I_{p}}\sum \limits _{\rho<\alpha \le 2\rho }\frac{x^{\alpha +j}}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }+ \sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\begin{array}{c} \left( \tau ,\lambda \right) \in I_{p}\\ \lambda =2\rho \end{array}}\sum \limits _{\begin{array}{c} \alpha \in R\left( T\right) \\ {{\,\mathrm{Re}\,}}\left( \alpha \right) =\rho \end{array}}\frac{x^{\alpha +j}}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }\\&\qquad +\sum \limits _{\begin{array}{c} \alpha \in R\left( T\right) \\ {{\,\mathrm{Re}\,}}\left( \alpha \right) <\rho \end{array}}{{\,\mathrm{Res}\,}}_{s=\alpha }\left( -\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }\right) . \end{aligned} \end{aligned}$$

(19)

To complete the story with the residues, we proceed in exactly the same way as in [37, pp. 100-101]. The use of the canonical product to $Z_{R}\left( s\right)$ leads to a function $Z\left( s\right)$ with no singularities in ${{\,\mathrm{Re}\,}}\left( s\right)$ $\ge$ $\rho$. Remembering (17), we obtain

$$\begin{aligned} \begin{aligned}&\sum \limits _{\begin{array}{c} \alpha \in R\left( T\right) \\ {{\,\mathrm{Re}\,}}\left( \alpha \right) <\rho \end{array}}{{\,\mathrm{Res}\,}}_{s=\alpha }\left( -\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }\right) \\&\quad =\frac{1}{2\pi {{\,\mathrm{i}\,}}{}}\int \limits _{C^{'}}-\frac{Z^{'}\left( s\right) }{Z\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }ds =O\left( x^{\rho +j}T^{-j-1+n}\log T\right) , \end{aligned} \end{aligned}$$

(20)

where $C^{'}$ is the positively oriented boundary of $R\left( T\right) \cap \left\{ s\in \mathbb {C}\,:\,{{\,\mathrm{Re}\,}}\left( s\right) <\rho \right\}$.

It is clear that

$$\begin{aligned} \frac{1}{2\pi {{\,\mathrm{i}\,}}{}}\int \limits _{c+{{\,\mathrm{i}\,}}{}\tilde{T}}^{c+{{\,\mathrm{i}\,}}{}\infty }-\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }ds =O\left( x^{c+j}T^{-j}\right) \end{aligned}$$

and

$$\begin{aligned} \frac{1}{2\pi {{\,\mathrm{i}\,}}{}}\int \limits _{c-{{\,\mathrm{i}\,}}{}\infty }^{c-{{\,\mathrm{i}\,}}{}\tilde{T}}-\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }ds =O\left( x^{c+j}T^{-j}\right) . \end{aligned}$$

Therefore,

$$\begin{aligned} \begin{aligned} \psi _{j}\left( x\right) =\frac{1}{2\pi {{\,\mathrm{i}\,}}{}}\int \limits _{c-{{\,\mathrm{i}\,}}{}\tilde{T}}^{c+{{\,\mathrm{i}\,}}{}\tilde{T}}-\frac{Z_{R}^{'}\left( s\right) }{Z_{R}\left( s\right) }\frac{x^{s+j}}{\prod \limits _{k=0}^{j}\left( s+k\right) }ds +O\left( x^{c+j}T^{-j}\right) . \end{aligned} \end{aligned}$$

(21)

Combining (10)–(14), (17), (19)–(21), and letting T $\rightarrow$ $\infty$ in (6), we end with

$$\begin{aligned} \psi _{j}\left( x\right) =&\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\left( \tau ,\lambda \right) \in I_{p}}\sum \limits _{\rho <\alpha \le 2\rho }\frac{x^{\alpha +j}}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }+\\&\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\begin{array}{c} \left( \tau ,\lambda \right) \in I_{p}\\ \lambda =2\rho \end{array}}\sum \limits _{{{\,\mathrm{Re}\,}}\left( \alpha \right) =\rho }\frac{x^{\alpha +j}}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }. \end{aligned}$$

This completes the proof. $\square$

3.2 Explicit formulas for $\psi _{0}\left( x\right)$

Theorem 2

Let $\mathscr {Y}$ be a compact locally symmetric Riemannian manifold of negative sectional curvature and of even dimension n. Suppose that j $\ge$ n and $\varepsilon$ > 0. There exists a set E of finite logarithmic measure such that

$$\begin{aligned} \psi _{0}\left( x\right) =&\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\left( \tau ,\lambda \right) \in I_{p}} \sum \limits _{2\rho -\rho \frac{2j+1}{2nj+1}<\alpha \le 2\rho }\frac{x^{\alpha }}{\alpha }\\&\quad +O\left( x^{2\rho -\rho \frac{2j+1}{2nj+1}}\left( \log x\right) ^{\frac{n-1}{2nj+1}}\left( \log \log x\right) ^{\frac{n-1}{2nj+1}+\varepsilon }\right) \end{aligned}$$

as x $\rightarrow$ $\infty$, x $\notin$ E, where $\alpha$ is a singularity of the Selberg zeta function

$Z_{S}\left( s+\rho -\lambda ,\tau \right)$.

Proof

By (5),

$$\begin{aligned} \psi _{j}\left( x\right) =&\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\left( \tau ,\lambda \right) \in I_{p}} \sum \limits _{\rho<\alpha \le 2\rho }\frac{x^{\alpha +j}}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }\nonumber \\&\quad +\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\begin{array}{c} \left( \tau ,\lambda \right) \in I_{p}\\ \lambda =2\rho \end{array}} \sum \limits _{\begin{array}{c} {{\,\mathrm{Re}\,}}\left( \alpha \right) =\rho \nonumber \\ \left| {{\,\mathrm{Im}\,}}\left( \alpha \right) \right| \le Y \end{array}}\frac{x^{\alpha +j}}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }\nonumber \\&\quad +\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\begin{array}{c} \left( \tau ,\lambda \right) \in I_{p}\\ \lambda =2\rho \end{array}} \sum \limits _{\begin{array}{c} {{\,\mathrm{Re}\,}}\left( \alpha \right) =\rho \nonumber \\ Y<\left| {{\,\mathrm{Im}\,}}\left( \alpha \right) \right| \le W \end{array}}\frac{x^{\alpha +j}}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }\nonumber \\&\quad +\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\begin{array}{c} \left( \tau ,\lambda \right) \in I_{p}\\ \lambda =2\rho \end{array}} \sum \limits _{\begin{array}{c} {{\,\mathrm{Re}\,}}\left( \alpha \right) =\rho \\ \left| {{\,\mathrm{Im}\,}}\left( \alpha \right) \right| >W \end{array}}\frac{x^{\alpha +j}}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }. \end{aligned}$$

(22)

Inspired by the third sum, we introduce the set $E_{p,\tau }^{\ell }$ of those $x\in \left[ e^{\ell },e^{\ell +1}\right)$ for which

$$\begin{aligned} \left| \sum \limits _{\begin{array}{c} {{\,\mathrm{Re}\,}}\left( \alpha \right) =\rho \\ Y<\left| {{\,\mathrm{Im}\,}}\left( \alpha \right) \right| \le W \end{array}}\frac{x^{\alpha +j}}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }\right| >x^{\gamma }\left( \log x\right) ^{\beta }\left( \log \log x\right) ^{\beta +\varepsilon }, \end{aligned}$$

where $\gamma$ and $\beta$ are two positive numbers whose formulas will be determined during the course of the proof of this theorem.

For the sake of clarity, let us emphasize once again that $\alpha$ appearing in the definition of $E_{p,\tau }^{\ell }$ is a singularity of the Selberg zeta $Z_{S}\left( s+\rho -\lambda ,\tau \right)$ corresponding to p $\in$ $\left\{ 0,1,\ldots ,n-1\right\}$ and $\left( \tau ,\lambda \right)$ $\in$ $I_{p}$, $\lambda$ $=$ $2\rho$.

Estimating the logarithmic measure $\mu ^{\times }E_{p,\tau }^{\ell }$ $=$ $\int \limits _{E_{p,\tau }^{\ell }}\frac{\mathrm{d}x}{x}$ of the set $E_{p,\tau }^{\ell }$ in the same way as in [25, pp. 10-11], we find that $\mu ^{\times }E_{p,\tau }^{\ell }$ is

$$\begin{aligned}&O\left( \frac{e^{2\left( \rho +j-\gamma \right) \ell }}{\ell ^{2\beta }\left( \log \ell \right) ^{2\beta +2\varepsilon }} \int \limits _{e^{\ell }}^{e^{\ell +1}} \left| \sum \limits _{\begin{array}{c} {{\,\mathrm{Re}\,}}\left( \alpha \right) =\rho \\ Y<\left| {{\,\mathrm{Im}\,}}\left( \alpha \right) \right| \le W \end{array}} \frac{x^{{{\,\mathrm{i}\,}}{{\,\mathrm{Im}\,}}\left( \alpha \right) }}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }\right| ^{2}\frac{\mathrm{d}x}{x}\right) \\&\quad =O\left( \frac{e^{2\left( \rho +j-\gamma \right) \ell }}{\ell ^{2\beta }\left( \log \ell \right) ^{2\beta +2\varepsilon }} \int \limits _{-\frac{1}{4\pi }}^{\frac{1}{4\pi }} \left| \sum \limits _{\begin{array}{c} {{\,\mathrm{Re}\,}}\left( \alpha \right) =\rho \\ Y<\left| {{\,\mathrm{Im}\,}}\left( \alpha \right) \right| \le W \end{array}} \frac{e^{{{\,\mathrm{i}\,}}{{\,\mathrm{Im}\,}}\left( \alpha \right) \left( \ell +\frac{1}{2}\right) }}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }e^{2\pi {{\,\mathrm{i}\,}}{{\,\mathrm{Im}\,}}\left( \alpha \right) u}\right| ^{2}\mathrm{d}u\right) . \end{aligned}$$

The application of Lemma 1 [32,  p. 78] yields

$$\begin{aligned}&\int \limits _{-\frac{1}{4\pi }}^{\frac{1}{4\pi }} \left| \sum \limits _{\begin{array}{c} {{\,\mathrm{Re}\,}}\left( \alpha \right) =\rho \\ Y<\left| {{\,\mathrm{Im}\,}}\left( \alpha \right) \right| \le W \end{array}} \frac{e^{{{\,\mathrm{i}\,}}{{\,\mathrm{Im}\,}}\left( \alpha \right) \left( \ell +\frac{1}{2}\right) }}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }e^{2\pi {{\,\mathrm{i}\,}}{{\,\mathrm{Im}\,}}\left( \alpha \right) u}\right| ^{2}\mathrm{d}u\nonumber \\&\quad =O\left( \int \limits _{-\infty }^{+\infty }\left( \sum \limits _{\begin{array}{c} t\le \left| {{\,\mathrm{Im}\,}}\left( \alpha \right) \right| \le t+1\\ Y<\left| {{\,\mathrm{Im}\,}}\left( \alpha \right) \right| \le W \end{array}}\frac{1}{\prod \limits _{k=0}^{j}\left| \alpha +k\right| }\right) ^{2}\mathrm{d}t\right) . \end{aligned}$$

(23)

The O-term in (23) is $O\left( \frac{1}{Y^{2j+3-2n}}\right)$ according to the asymptotic law (7).

We obtain, $\mu ^{\times }E_{p,\tau }^{\ell }$ $=$ $O\left( \frac{e^{2\left( \rho +j-\gamma \right) \ell }}{Y^{2j+3-2n}\ell ^{2\beta }\left( \log \ell \right) ^{2\beta +2\varepsilon }}\right)$.

For Y $\sim$ $e^{\frac{\left( 2\rho +2j-2\gamma \right) \ell }{2j+3-2n}}\ell ^{\frac{1-2\beta }{2j+3-2n}}\left( \log \ell \right) ^{\frac{1-2\beta }{2j+3-2n}}$, it follows that $\mu ^{\times }E_{p,\tau }^{\ell }$ $=$ $O\left( \frac{1}{\ell \left( \log \ell \right) ^{1+2\varepsilon }}\right)$ and $\mu ^{\times }\bigcup \limits _{p}\bigcup \limits _{\begin{array}{c} \left( \tau ,\lambda \right) \in I_{p} \lambda =2\rho \end{array}}E_{p,\tau }^{\ell }$ $=$ $O\left( \frac{1}{\ell \left( \log \ell \right) ^{1+2\varepsilon }}\right)$.

Thus, the convergence of the series $\sum \limits _{\ell }\frac{1}{\ell \left( \log \ell \right) ^{1+2\varepsilon }}$ implies that $\mu ^{\times }E<\infty$ for E $=$ $\bigcup \limits _{\ell }\bigcup \limits _{p}\bigcup \limits _{\begin{array}{c} \left( \tau ,\lambda \right) \in I_{p} \lambda =2\rho \end{array}}E_{p,\tau }^{\ell }$.

Obviously, for x $\notin$ E

$$\begin{aligned} \sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\begin{array}{c} \left( \tau ,\lambda \right) \in I_{p}\\ \lambda =2\rho \end{array}}\sum \limits _{\begin{array}{c} {{\,\mathrm{Re}\,}}\left( \alpha \right) =\rho \\ Y<\left| {{\,\mathrm{Im}\,}}\left( \alpha \right) \right| \le W \end{array}}\frac{x^{\alpha +j}}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }=O\left( x^{\gamma }\left( \log x\right) ^{\beta }\left( \log \log x\right) ^{\beta +\varepsilon }\right) . \end{aligned}$$

Hence, for x $\notin$ E

$$\begin{aligned} \begin{aligned}&d^{-j}\varDelta _{j}^{+}\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\begin{array}{c} \left( \tau ,\lambda \right) \in I_{p}\\ \lambda =2\rho \end{array}}\sum \limits _{\begin{array}{c} {{\,\mathrm{Re}\,}}\left( \alpha \right) =\rho \\ Y<\left| {{\,\mathrm{Im}\,}}\left( \alpha \right) \right| \le W \end{array}}\frac{x^{\alpha +j}}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }\\&\quad =O\left( \frac{x^{\gamma }\left( \log x\right) ^{\beta }\left( \log \log x\right) ^{\beta +\varepsilon }}{d^{j}}\right) , \end{aligned} \end{aligned}$$

(24)

where $\varDelta _{j}^{+}f\left( x\right) =\int \limits _{x}^{x+d}\int \limits _{t_{j}}^{t_{j}+d}\ldots \int \limits _{t_{2}}^{t_{2}+d}f^{\left( j\right) }\left( t_{1}\right) \mathrm{d}t_{1}\ldots \mathrm{d}t_{j}$ and d is a constant.

In particular, we require d $=$ $O\left( x\right)$.

By the mean value theorem,

$$\begin{aligned} d^{-j}\varDelta _{j}^{+}\frac{x^{\alpha +j}}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }=O\left( \left| \alpha \right| ^{-1}x^{\rho }\right) \end{aligned}$$

for $\alpha$ appearing in (22) with ${{\,\mathrm{Re}\,}}\left( \alpha \right) =\rho$. Therefore, by (7) (Cf. [39, p. 246])

$$\begin{aligned} \begin{aligned} d^{-j}\varDelta _{j}^{+}\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\begin{array}{c} \left( \tau ,\lambda \right) \in I_{p}\\ \lambda =2\rho \end{array}}\sum \limits _{\begin{array}{c} {{\,\mathrm{Re}\,}}\left( \alpha \right) =\rho \\ \left| {{\,\mathrm{Im}\,}}\left( \alpha \right) \right| \le Y \end{array}}\frac{x^{\alpha +j}}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }=O\left( x^{\rho }Y^{n-1}\right) . \end{aligned} \end{aligned}$$

(25)

It is clear that

$$\begin{aligned} \begin{aligned} d^{-j}\varDelta _{j}^{+}\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\begin{array}{c} \left( \tau ,\lambda \right) \in I_{p}\\ \lambda =2\rho \end{array}}\sum \limits _{\begin{array}{c} {{\,\mathrm{Re}\,}}\left( \alpha \right) =\rho \\ \left| {{\,\mathrm{Im}\,}}\left( \alpha \right) \right| > W \end{array}}\frac{x^{\alpha +j}}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }= O\left( \frac{x^{\rho +j}}{d^{j}W^{j+1-n}}\right) . \end{aligned} \end{aligned}$$

(26)

The mean value theorem also yields that

$$\begin{aligned} d^{-j}\varDelta _{j}^{+}\frac{x^{\alpha +j}}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }=\alpha ^{-1}\left( x+\theta _{\alpha }\right) ^{\alpha } \end{aligned}$$

for some $\theta _{\alpha }$ $\in$ $\left[ 0,jd\right]$, where $\alpha$ is a singularity of $Z_{S}\left( s+\rho -\lambda ,\tau \right)$ which occurs in the first sum on the right-hand side of (5). We obtain,

$$\begin{aligned} \begin{aligned}&d^{-j}\varDelta _{j}^{+}\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\left( \tau ,\lambda \right) \in I_{p}}\sum \limits _{\rho<\alpha \le 2\rho } \frac{x^{\alpha +j}}{\prod \limits _{k=0}^{j}\left( \alpha +k\right) }\\&\quad =\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\left( \tau ,\lambda \right) \in I_{p}}\sum \limits _{\rho <\alpha \le 2\rho } \frac{x^{\alpha }}{\alpha }+O\left( x^{2\rho -1}d\right) . \end{aligned} \end{aligned}$$

(27)

Combining , (24)–(27) with $\psi _{0}\left( x\right)$ $\le$ $d^{-j}\varDelta _{j}^{+}\psi _{j}\left( x\right)$, we conclude that for x $\notin$ E

$$\begin{aligned} \begin{aligned} \psi _{0}\left( x\right) \le&\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\left( \tau ,\lambda \right) \in I_{p}}\sum \limits _{\rho <\alpha \le 2\rho }\frac{x^{\alpha }}{\alpha }+O\left( x^{2\rho -1}d\right) + O\left( x^{\rho }Y^{n-1}\right) +\\&O\left( \frac{x^{\gamma }\left( \log x\right) ^{\beta }\left( \log \log x\right) ^{\beta +\varepsilon }}{d^{j}}\right) +O\left( \frac{x^{\rho +j}}{d^{j}W^{j+1-n}}\right) . \end{aligned} \end{aligned}$$

(28)

As in [25, p. 12], we have $x^{2\rho -1}d$ $=$ $x^{\rho }Y^{n-1}$ if d $=$ $x^{1-\rho }Y^{n-1}$ and

$x^{2\rho -1}d$ $\le$ $\frac{x^{\gamma }\left( \log x\right) ^{\beta }\left( \log \log x\right) ^{\beta +\varepsilon }}{d^{j}}$ if d $=$ $x^{\frac{\gamma -2\rho +1}{j+1}}\left( \log x\right) ^{\frac{\beta }{j+1}}\left( \log \log x\right) ^{\frac{\beta }{j+1}}$.

Furthermore, we require the following asymptotic:

Y $\sim$ $x^{\frac{2\rho +2j-2\gamma }{2j+3-2n}}\left( \log x\right) ^{\frac{1-2\beta }{2j+3-2n}}\left( \log \log x\right) ^{\frac{1-2\beta }{2j+3-2n}}$.

The solution of the implied system of equations:

$$\begin{aligned} \frac{\gamma -2\rho +1}{j+1}=&1-\rho +\left( n-1\right) \frac{2\rho +2j-2\gamma }{2j+3-2n},\\ \frac{\beta }{j+1}=&\left( n-1\right) \frac{1-2\beta }{2j+3-2n} \end{aligned}$$

is:

$$\begin{aligned} \gamma =&\frac{2\left( n-\rho \right) j^{2}+\left( 4n-3\right) \rho j+\rho +j}{2nj+1},\\ \beta =&\frac{\left( n-1\right) \left( j+1\right) }{2nj+1}. \end{aligned}$$

Thus, we can write

$$\begin{aligned} d=&\,x^{1-\rho \frac{2j+1}{2nj+1}}\left( \log x\right) ^{\frac{n-1}{2nj+1}}\left( \log \log x\right) ^{\frac{n-1}{2nj+1}},\\ Y\sim&x^{\frac{2\rho j}{2nj+1}}\left( \log x\right) ^{\frac{1}{2nj+1}}\left( \log \log x\right) ^{\frac{1}{2nj+1}}. \end{aligned}$$

These two, and (28) give

$$\begin{aligned} \psi _{0}\left( x\right) \le&\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\left( \tau ,\lambda \right) \in I_{p}}\sum \limits _{2\rho -\rho \frac{2j+1}{2nj+1}<\alpha \le 2\rho }\frac{x^{\alpha }}{\alpha }+\\&O\left( x^{2\rho -\rho \frac{2j+1}{2nj+1}}\left( \log x\right) ^{\frac{n-1}{2nj+1}}\left( \log \log x\right) ^{\frac{n-1}{2nj+1}+\varepsilon }\right) \end{aligned}$$

as x $\rightarrow$ $\infty$, x $\notin$ E.

Similarly,

$$\begin{aligned} \psi _{0}\left( x\right) \ge&\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\left( \tau ,\lambda \right) \in I_{p}}\sum \limits _{2\rho -\rho \frac{2j+1}{2nj+1}<\alpha \le 2\rho }\frac{x^{\alpha }}{\alpha }+\\&O\left( x^{2\rho -\rho \frac{2j+1}{2nj+1}}\left( \log x\right) ^{\frac{n-1}{2nj+1}}\left( \log \log x\right) ^{\frac{n-1}{2nj+1}+\varepsilon }\right) \end{aligned}$$

as x $\rightarrow$ $\infty$, x $\notin$ E.

This completes the proof. $\square$

3.3 Gallagherian prime geodesic theorem

The following theorem is the main theorem of the research.

Theorem 3

(Gallagherian Prime Geodesic Theorem) Let $\mathscr {Y}$ be a compact locally symmetric Riemannian manifold of negative sectional curvature and of even dimension n. Suppose that j $\ge$ n and $\varepsilon$ > 0. There exists a set E of finite logarithmic measure such that

$$\begin{aligned} \begin{aligned} \pi _{\varGamma }\left( x\right) =&\sum \limits _{p=0}^{n-1}\left( -1\right) ^{p+1}\sum \limits _{\left( \tau ,\lambda \right) \in I_{p}} \sum \limits _{2\rho -\rho \frac{2j+1}{2nj+1}<\alpha \le 2\rho }{{\,\mathrm{li}\,}}\left( x^{\alpha }\right) \\&\quad +O\left( x^{2\rho -\rho \frac{2j+1}{2nj+1}}\left( \log x\right) ^{\frac{n-1}{2nj+1}-1}\left( \log \log x\right) ^{\frac{n-1}{2nj+1}+\varepsilon }\right) \end{aligned} \end{aligned}$$

(29)

as x $\rightarrow$ $\infty$, x $\notin$ E, where $\alpha$ is a singularity of the Selberg zeta function

$Z_{S}\left( s+\rho -\lambda ,\tau \right)$.

Proof

Follows immediately from Theorem 2 (see, e.g., [37, p. 102]). $\square$

4 Discussion

Consider the sequence $\left\{ a_{j}\right\}$, where

$$\begin{aligned} a_{j}=2\rho -\rho \frac{2j+1}{2nj+1}. \end{aligned}$$

Since n > 1, the sequence $\left\{ a_{j}\right\}$ is increasing.

According to Theorems 1–3, our interest is in j $\ge$ n. Hence,

$$\begin{aligned} 2\rho -\rho \frac{2n+1}{2n^{2}+1}=a_{n}<a_{n+1}<\cdots<a_{2n-1}<a_{2n}=2\rho -\rho \frac{4n+1}{4n^{2}+1}. \end{aligned}$$

The obtained inequalities show that PGT (29) improves (3) for n $\le$ j < 2n. The optimal size of the error term in (29) is achieved for j $=$ n.

Theorem 3 provides the best bound in this category of spaces for n $\ge$ 4.

Note that the O-term in (29) fully agrees with the O-term in (2) for n $=$ k.

A formula for $\psi _{n-1}\left( x\right)$, analogous to formula (5), would definitely further reduce the remainder in (29). Unfortunately, no such formula has been established in this study. Namely, the method applied in the Proof of Theorem 1 is not applicable to $j=n-1$ since the estimates (17) and (20) do not vanish as T $\rightarrow$ $\infty$ in that case. A possible way to the variant of $\psi _{n-1}\left( x\right)$ would be via [26, p. 110, Th. 6.16].