Prime Geodesic Theorems for Compact Locally Symmetric Spaces of Real Rank One

: Our basic objects will be compact, even-dimensional, locally symmetric Riemannian manifolds with strictly negative sectional curvature. The goal of the present paper is to investigate the prime geodesic theorems that are associated with this class of spaces. First, following classical Randol’s appraoch in the compact Riemann surface case, we improve the error term in the corresponding result. Second, we reduce the exponent in the newly acquired remainder by using the Gallagher–Koyama techniques. In particular, we improve DeGeorge’s bound O ( x η ) , 2 ρ − ρ n ≤ η < 2 ρ up to O (cid:16) x 2 ρ − ρ n ( log x ) − 1 (cid:17) , and reduce the exponent 2 ρ − ρ n replacing it by 2 ρ − ρ 4 n + 1 4 n 2 + 1 outside a set of ﬁnite logarithmic measure. As usual, n denotes the dimension of the underlying locally symmetric space, and ρ is the half-sum of the positive roots. The obtained prime geodesic theorem coincides with the best known results proved for compact Riemann surfaces, hyperbolic three-manifolds, and real hyperbolic manifolds with cusps.

As it is known, the prime geodesic theorem gives a growth asymptotic for the function π Γ (x). Moreover, the statement regarding the number π Γ (x), as x → ∞, x / ∈ E, where E is a set of finite logarithmic measure, is known as the Gallagherian prime geodesic theorem. Usually, the Gallagherian prime geodesic theorem improves the corresponding classical result at the cost of excluding a set of finite logarithmic measure. In this research, we are interested in both kinds of theorems.
In literature, the prime geodesic theorem appear in two forms: refined and non-refined. While, in the refined form, the counting function π Γ (x) is represented as a sum of two parts: the explicit part, and some carefully derived remainder, in the non-refined form π Γ (x) is given as an asymptotic estimate without the error terms.
DeGeorge [2] obtained the best known estimate of the error term in the prime geodesic theorem in our setting in 1977. Thus, DeGeorge's result is given in the refined form and states that there is a constant η, such that 2ρ − ρ n ≤ η < 2ρ, and (see, Theorem 1 and Remark 2 in [2] (pp. 135-136)): as x → ∞. Clearly, the optimal error term in (1) is O x 2ρ− ρ n . Here, as earlier, n = k, 2m, 4m, 16, and ρ = 1 2 (k − 1), m, 2m + 1, 11, respectively. The main purpose of this research is to improve the bound O (x η ) in (1) up to O x 2ρ− ρ n (log x) −1 in the classical sense, and then prove that the exponent 2ρ − ρ n of x in the newly acquired bound O x 2ρ− ρ n (log x) −1 can be reduced in the Gallagherian sense and, hence, replaced by 2ρ − ρ 4n+1 4n 2 +1 (see, Theorems 2 and 3 below). To put our research into historical context, let us recall the following related results. Gangolli [3] (see, Theorem 4.4, and page 423), proved the non-refined prime geodesic theorem when Y is compact (also see, [4] (p. 89)): where f (x) ∼ g (x) means that lim x→∞ f (x) g(x) = 1. The relation (2) was also proved by Gangolli-Warner [5] (p. 40, Prop. 5.4) when Y is not necessarily compact but has a finite volume. In (2), n = k, 2m, 4m, 16, respectively, as before. It is easy to see that the prime geodesic theorem (1) is a refinement of (2) when Y is compact. The first refinement of the corresponding result (2) of Gangolli-Warner (hence, for Y non-compact), for k-dimensional real hyperbolic manifolds with cusps (n = k, ρ = 1 2 (k − 1)), was achieved by Park [4] (p. 91, Th. 1.2). It states that: 3 2 ρ, 2ρ , ∆ j is the Laplacian acting on the space of j-forms over Y, and π σ j ,λ i (j) is the principal series representation. The result (3) was further improved by Avdispahić-Gušić in [6] (p. 367, Th. 1), where the authors derived a variant of (3) with the error term O x 3 2 ρ (log x) −1 . As explained in [7], the correct size of the error term in [4] resp. [6] is O x The omission was present in [4] and, thus, inherited in [6] because of the missing term O x 2ρ−1 h obtained during reduction from the level of k − 1 times integrated Chebyshev function ψ 2ρ (x) to ψ 0 (x) (see, [4] (p. 101, (3.21)) and [6] (p. 370, (7))). Finally, the bound O x 4ρ 2 +ρ 2ρ+1 (log x) −1 obtained in [7] is additionally improved in [8] in the Gallagherian sense, where the authors proved that the exponent 4ρ 2 +ρ 2ρ+1 can be replaced by (k − 1) 1 − 2k+1 4k 2 +2 outside a set of finite logarithmic measure. The investigations that were conducted in [8] were undoubtedly inspired by the recent research of Koyama [9] in the case of compact hyperbolic surfaces and the generic hyperbolic surfaces of finite volume. The ingredients applied in [9] come from the results of Hejhal [10,11], Iwaniec [12], and Gallagher [13], where the author in [13] (under assuming the Rimemann hypothesis) improved the error term in the prime number theorem from ψ ( 2 outside a set of finite logarithmic measure with the Chebyshev counting function ψ (x) defined over powers of primes by ψ (x) = ∑ p k ≤x log p. Hejhal, in his comprehensive treatise [10,11], studied the Selberg zeta function over a hyperbolic Riemann surface Y (n = k = 2, ρ = 1 2 (k − 1) = 1 2 ), which is, when Γ is cocompact (Y compact) and cofinite (Y non-compact) discrete subgroup of G = PSL (2, R), respectively. His prime geodesic theorem comes with the error terms and states that (also see [14][15][16]): 3 16 of the Laplacian ∆ 0 acting on L 2 (Y). The prime geodesic theorem (4) refines the corresponding result (2) for cocompact and cofinite Γ ⊆ PSL (2, R) in the same way the prime geodesic theorem (1) refines (2) when Y is compact. The best estimate up to now of the error term in a variant of the prime geodesic theorem (4) for compact Riemann surfaces is O x 3 4 (log x) −1 , and it is achieved by Randol [17] (see also, [18]). An important ingredient, which is implicitly applied in [17] and explicitly in [4], is the Ruelle zeta function. The bound O x 3 4 (log x) −1 is also achieved in the case of prime geodesic theorem derived for compact symmetric spaces formed as quotients of the Lie group SL 4 (R), which is, when Y is locally symmetric space Γ\G/K, where G = SL 4 (R), K is the maximal compact subgroup of G, and Γ is a discrete cocompact subgroup of G (see, [19]). By Theorem 4.4.1 in [19] (p. 197): is the first higher Euler characteristic of the centraliser Γ γ of γ in Γ, l γ is the length of γ, P is a parabolic subgroup of G, and E p P (Γ) ⊂ E P (Γ) is the subset of primitive classes, where E P (Γ) is the set of all conjugacy classes [γ] in Γ, such that γ is conjugate in G to an element of A − B, with A − the negative Weyl chamber in Deitmar obtained an analogous result of the result (5) in [20] in the case of complex cubic fields, extending, in that way, the work of Sarnak [21] in the real quadratic case. The research has been extended to a noncompact situation in [22], while the full higher rank case has been explored in [23,24] (see also, [25]). As it is known, the Selberg zeta function for compact or generic hyperbolic surfaces satisfies an analogue of the Riemann hypothesis. This fact raises the expectation that the exponent 3 4 of x in (4) could be decreased to 1 2 . However, the quantity of zeros of the Selberg zeta function causes major obstacles in achieving such result. Thus, the aforementioned 3 4 was only successfully reduced in the case of modular surfaces. In particular, Iwaniec [12] obtained 35 48 + ε, Luo and Sarnak [26] 7 10 + ε, Cai [27] 71 102 + ε, and Soundararajan and Young [28] 25 36 poles of the corresponding scattering determinant, then the following prime geodesic theorem for hyperbolic 3-manifolds holds true (see, [7] (p. 691, Th. 1.1)): .., s M are the real zeros of the attached Selberg zeta function lying in the interval (1, 2). The motivation to work within the described setting, i.e., with compact, even-dimensional, locally symmetric Riemannian manifolds of strictly negative sectional curvature, stems from the author's desire to improve the best known error term O x 2ρ− ρ n in the corresponding prime geodesic theorem (1), dating back to 1977 up to O x 2ρ− ρ n (log x) −1 , and the wish to further reduce the exponent 2ρ − ρ n of x in the Gallagherian sense, which is, the wish to replace 2ρ − ρ n with better, smaller one 2ρ − ρ 4n+1 4n 2 +1 outside a set of finite logarithmic measure. Additionally, the fact that it was an improvement of a more than forty-year-old result was quite motivating for the author.
Regarding the techniques that were applied in the proofs of our results, we want to point out that the proofs of Theorems 1 and 2 are inspired by Randol's 1977 approach in the case of compact Riemann surfaces [17] (pp. 245-246), and that the proof of Theorem 3 relies on the 1980 method developed by Gallagher and applied in the classical case on prime number theorem [13]. Hence, the mentioned techniques are not new, and are already known in literature. However, it must be noted that new techniques are nothey t invented so often in this area of research, and that the ones given above have been exploited many times since 1977 and 1980. In particular, in addition to 1977, Randol's method was successfully applied in 2002 in the proof of prime geodesic theorem for complex cubic fields [20] (p. 165), and then again, in 2006 and 2008 for the same reason, in the case of compact symmetric spaces formed as quotients of the Lie group SL 4 (R) (see, [29] (pp. 62-65), [19] (p. 197)). Furthermore, it was applied in 2012 in the proof of prime geodesic theorem for real hyperbolic manifolds with cusps [6] (p. 370) (also see, [22]), etc. On the other hand, besides 1980, Gallagher's technique was re-updated by Koyama in 2016 in the proof of the corresponding Gallagherian-style prime geodesic theorem derived for compact hyperbolic surfaces and generic hyperbolic surfaces of finite volume [9] (p. 78, Th. 2). Thereafter, the technique was re-applied to obtain the following improved results: first, in 2018, in Koyama's own setting [30], then, in 2018, in the case of PSL (2, Z) [31], once again in 2018 in the case of hyperbolic 3-manifolds [7] (p. 691, Th. 1.2), one more time in 2020 in the case of real hyperbolic manifolds with cusps [8] (p. 3021, Th. 2), etc. Summarizing what is said above, we may emphasize that the methods that were applied in this research, although old, are still not obsolete, and represent valuable and unavoidable tool in achieving more refined error terms in prime geodesic theorems for various types of underlying locally symmetric spaces. Accordingly, once again, the techniques are not new, but the results are, and the results are all that we are interested in. Like most of similar pure mathematics researches: [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]27,28,[30][31][32][33][34][35][36], etc., the present research has no direct application. In fact, it is a typical example of research in the field of pure, theoretical mathematics. So, one could hardly expect to obtain some immediate application. Finally, regarding the author's additional motivation to consider this subject, we recall that, in the Concluding Remark of [17] (p. 246), Randol noted that it would be interesting to determine the extent to which his methods are applicable for more general spaces. Accordingly, in the same way Theorem 1 in [6] (see, pages 367 and 371) represents the answer to this query in the case of real hyperbolic manifolds with cusps, now, Theorems 1 and 2 can be interpreted as the answer to the same query in the case at hand.

Preliminary Material
We introduce the notation following [1] (see also, [37,38]). Because Γ ⊂ G is cocompact and torsion-free, there are only two types of conjugacy classes: the class of the identity 1 ∈ Γ and classes of hyperbolic elements. Let CΓ be the set of all conjugacy classes [γ] in Γ. To simplify the notation, we shall write γ for an element of CΓ, and γ 0 for a primitive element. Thus, if γ and γ 0 occur in the same formula, it is understood that γ 0 will be the primitive element underlying γ.
Denote, by M, the centralizer of a in K with the Lie algebra m. Let i * : R (K) → R (M) be the restriction map that is induced by the embedding i : M → K, where R (K) and R (M) are the representation rings over Z of K and M, respectively (see, [1] (p. 19)).
Suppose that σ ∈M, whereM is the unitary dual of the Lie group M.
There are the Iwasawa decompositions g = k ⊕ a ⊕ n and G = KAN. If g ∈ G is a hyperbolic element, then g is conjugated to some element am ∈ A + M (see, e.g., [3,5]), where A + = exp (a + ), and a + is the positive Weyl chamber in a. Following [1] (p. 59), we put l (g) = |log (a)|.
For the sake of simplicity, we fix some χ ∈Γ, σ ∈M, and reduce the notation by omitting to write them in the sequel unless necessary.
Finally, we introduce the functions ψ j (x), j ∈ N recursively by ψ j (x) =

Prime Geodesic Theorem
We prove the following theorem: Theorem 1. Let Y be as above. Subsequently: where S R p,τ,λ denotes the set of real singularities of Z S (s + ρ − λ, τ).
Proof. As the starting point, we take the following explicit formula for ψ k (x), k ∈ N, k ≥ 2n: where S k is the set of poles of − Z R (s) Z R (s) x s+k s(s+1)...(s+k) , and c k (α) is the residue at α. Note that Formula (7) is easily obtained, as in the case of compact Riemann surfaces [17] (p. 245) and the compact symmetric spaces formed as quotients of the Lie group SL 4 (R) [29] (p. 63).

Gallagherian Prime Geodesic Theorem
Theorem 3. Let Y be as above. For ε > 0, there exists a set E of finite logarithmic measure, such that: Proof. As the starting point, we take the explicit formula for ψ 2n (x) given by the relation (9).
It follows that: Since N (t) = At n + O t n−1 , the number of |Im (α)| participants in the last sum is O t n−1 . Moreover, Combining the relations (15)- (17), we conclude that: Taking: Consequently, for x / ∈ E: Consider the first sum in (14). Reasoning in the same way as in the derivation of (11), we conclude that: Similarly, for the third sum in (14), we have: Now, the relations (9), (10), (12)- (14), and (18)- (20) give us for x / ∈ E: Clearly, Additionally, By our selection of Y: Combining the relations (22) and (24), and comparing the exponents of x and log x with the corresponding exponents in (23), we arrive at: Thus, α = 8n 3 +2n+ρ−6nρ , β = 2n 2 −n−1 4n 2 +1 . Substituting the obtained d and Y into (21), we end up with: +ε as x → ∞, x / ∈ E. Now, the assertion of theorem follows the same argumentation as in the proof of Theorem 2. This completes the proof.
Funding: This research received no external funding.