Prime geodesic theorem for the Picard manifold
Introduction
The classical Prime Geodesic Theorem states that the counting function of primitive hyperbolic classes in whose norm does not exceed X satisfies the asymptotic law where is the logarithmic integral. This theorem and its generalisations can be considered as geometric analogues of the Prime Number Theorem, while norms of primitive hyperbolic elements are sometimes called “pseudoprimes”.
In this paper we study the three dimensional version of the Prime Geodesic Theorem. Different concepts of the two dimensional theory can be extended to this case in a natural and elegant way. The role of the Poincaré upper half plane is now played by the three dimensional hyperbolic space Let be a discrete cofinite group. Prime Geodesic Theorem for the hyperbolic manifold provides an asymptotic formula for the function , which counts the number of primitive hyperbolic or loxodromic elements in Γ with norm less than or equal to X.
In the pioneering paper [26], Sarnak proved the following asymptotic formula
Further progress has been made in the case of the Picard group defined over Gaussian integers
Assuming the mean Lindelöf hypothesis for symmetric square L-functions attached to Maass forms on the Picard manifold , Koyama [14] obtained an error term of size
Finally, an unconditional improvement of Sarnak's estimate (1.2), namely was derived in [1] as a consequence of a non-trivial estimate for the second moment of symmetric-square L-functions.
Proofs of the last two results are centered around Nakasuji's explicit formula proved in [20, Theorem 4.1] and [21, Theorem 5.2]. Let us denote the remainder term in the prime geodesic theorem by . Nakasuji showed that for we have where are the eigenvalues of the hyperbolic Laplacian on . Explicit formula (1.5) brings into play the spectral exponential sum
The trivial bound follows from Weyl's law and yields Sarnak's result (1.2).
The proof of (1.4) is based on the following improvement (see [1, Theorem 3.2])
The aim of this paper is to obtain a new upper bound for the spectral exponential sum, which explicitly depends on a subconvexity exponent θ for quadratic Dirichlet L-functions defined over Gaussian integers. Theorem 1.1 For the following estimate holds
As discussed in [1, Remark 3.1], it is not obvious what is the correct order of magnitude of for all X and T. In the two dimensional case, Petridis and Risager conjectured in [24] that the spectral exponential sum exhibits square root cancellation, and Laaksonen verified this conjecture numerically.
The most important consequence of Theorem 1.1 is the new estimate on . Theorem 1.2 The error term in the Prime Geodesic Theorem for the Picard manifold can be estimated as follows
Assuming the Lindelöf hypothesis , we improve the conditional result (1.3) and establish for the first time the error estimate . Note that this is the best possible error admissible by the explicit formula of Nakasuji (1.5). Recently, the exponent 3/2 was improved conditionally by Balog-Biro-Cherubini-Laaksonen [5] under two standard conjectures for L-functions defined over Gaussian integers: the Lindelöf hypothesis for quadratic Dirichlet L-functions and the mean Lindelöf hypothesis for the second moment of symmetric square L-functions. Furthermore, the authors of [5] established a new unconditional estimate, namely .
Theorem 1.1 is a consequence of the following estimate for the mean value of Maaß Rankin-Selberg L-functions on the critical line multiplied by the oscillating factor .
Theorem 1.3 Let and . Then for we have where is a smooth characteristic function of the interval .
The standard method for investigating the left-hand side of (1.10) is to estimate everything by absolute value so that is simply replaced by 1 and the modulus of the Rankin-Selberg L-function is bounded by its square by the means of the Cauchy-Schwarz inequality. In this way, the problem is reduced to studying the second moment. This technique allows proving non-trivial results in certain ranges but the disadvantage is that the information about the behaviour of is completely lost.
The main novelty and the core idea of our approach is that absolute value estimates are replaced by the method of analytic continuation. More precisely, we prove an exact formula for the left-hand side of (1.10) which allows us to take into consideration oscillations of the weight function . This approach has already proved to be effective in the two dimensional case (see [3]), motivating us to develop the method further and to study the Prime Geodesic Theorem for the Picard manifold. We remark that in the three-dimensional setting, new technical difficulties arise (i.e. more complicated special functions in the Kuznetsov trace formula), requiring the change of methodology and more involved analysis.
Section snippets
Description of the problem
In this section, following the book of Elsrodt, Grunewald and Mennicke [8], we provide some background information required for understanding the Prime Geodesic Theorem in the three-dimensional case. We keep all notations of [8].
The upper half space defined by (1.1) is equipped with the hyperbolic metric
The associated Laplace-Beltrami operator is given by
According to [8, Proposition 1.6, p. 6], the hyperbolic distance between two points
Notation and preliminary results
Throughout the paper we mostly use notations of [19] that are slightly different from the standard ones (see [19, Remark p. 270]).
Let be the Gaussian number field. All sums in this paper are over Gaussian integers unless otherwise indicated.
Let be the Gamma function. According to the Stirling formula we have for . Note that instead of it is possible to write arbitrarily accurate approximations of the Gamma function by evaluating
Exact formula for the first moment of Maaß Rankin-Selberg L-functions
In this section we prove an exact formula for first moment where is an even function, holomorphic in any fixed horizontal strip and satisfying the conditions for some fixed N and .
We remark that our approach here differs from the one we used in the two dimensional case in [3]. Our method incorporates some ideas from [2] and from Motohashi's proof [17] of the exact formula for the second moment of
Proof of Theorems 1.1 and 1.2
Following the paper of Ivic and Jutila [11], let us define We will ultimately choose .
For an arbitrary and some we have (see [11]) and otherwise Using the Weyl law [8, Section 5.5], we obtain where . Following the approach of Koyama (see [14], [1]) and applying Theorem 1.3,
Proof of Theorem 1.3
For an arbitrary large integer N we define where .
For simplicity, we will write instead of . Lemma 6.1 Let and . Then for , we have Proof This follows from the definitions of functions and (see (5.1), (6.2)), the fact that
Acknowledgments
The work of Olga Balkanova was supported by the Royal Swedish Academy of Sciences (grant no. MG2018-0002).
We thank Gergely Harcos and Han Wu for bringing the paper [28] to our attention, and Giacomo Cherubini for pointing out the correct value of the subconvexity exponent. We are also grateful to the referees for careful reading of the paper and valuable suggestions and comments.
References (29)
Spectral mean values of Maass wave forms
J. Number Theory
(1992)- et al.
Prime geodesic theorem in the 3-dimensional hyperbolic space
Trans. Am. Math. Soc.
(2019) - et al.
The mean value of symmetric square L-functions
Algebra Number Theory
(2018) - et al.
Bounds for a spectral exponential sum
J. Lond. Math. Soc.
(2019) - et al.
Convolution formula for the sums of generalized Dirichlet L-functions
Rev. Mat. Iberoam.
(2019) - et al.
Bykovskii-type theorem for the Picard manifold
- et al.
Higher Transcendental Functions, vol. 1
(1953) - et al.
Sum formula for Kloosterman sums and fourth moment of the Dedekind zeta-function over the Gaussian number field
Funct. Approx. Comment. Math.
(2003) - et al.
Groups Acting on Hyperbolic Space. Harmonic Analysis and Number Theory
(1998) - et al.
Table of integrals, series, and products
Zero-density estimate for modular form L-functions in weight aspect
Acta Arith.
On the moments of Hecke series at central points II
Funct. Approx. Comment. Math.
Prime geodesic theorem
J. Reine Angew. Math.
An estimate of Hecke's L-functions of the Gaussian field on the line
Dokl. Akad. Nauk. BSSR
Cited by (10)
Ambient Prime Geodesic Theorems on Hyperbolic 3-Manifolds
2023, International Mathematics Research NoticesGallagherian PGT on Some Compact Riemannian Manifolds of Negative Curvature
2022, Bulletin of the Malaysian Mathematical Sciences Society