aa r X i v : . [ m a t h . N T ] M a r PRIME GEODESIC THEOREM FOR THE MODULAR SURFACE
MUHAREM AVDISPAHI´C
Abstract.
Under the generalized Lindel¨of hypothesis, the exponent in theerror term of the prime geodesic theorem for the modular surface is reducedto + ε outside a set of finite logarithmic measure. Introduction
Let Γ =
P SL (2 , Z ) be the modular group and H the upper half-plane equippedwith the hyperbolic metric. The norms N ( P ) of primitive conjugacy classes P inΓ are sometimes called pseudo-primes. The length of the primitive closed geodesicon the modular surface Γ \ H joining two fixed points, which are the same for allrepresentatives of P , equals log( N ( P )). The statement about the number π Γ ( x ) ofclasses P such that N ( P ) ≤ x , for x >
0, is known as the prime geodesic theorem,PGT.The main tool in the proof of PGT is the Selberg zeta function, defined by Z Γ ( s ) = Y { P } ∞ Y k =0 (1 − N ( P ) − s − k ), Re( s ) > Z Γ satisfies the Riemann hypothesis. It is an outstandingopen problem whether the error term in the prime geodesic theorem is O ( x + ε ) asit would be the case in the prime number theorem once the Riemann hypothesis beproved.The obstacles in establishing an analogue of von Koch’s theorem [13, p. 84] inthis setting comes from the fact that Z Γ is a meromorphic function of order 2, whilethe Riemann zeta is of order 1.In the case of Fuchsian groups Γ ⊂ P SL (2 , R ), the best estimate of the remainderterm in PGT is still O (cid:18) x log x (cid:19) obtained by Randol [18] (see also [7], [1] for differentproofs). We note that its analogue O (cid:16) x d (log x ) − (cid:17) is valid also for strictlyhyperbolic manifolds of higher dimensions, where d = d − and d ≥ in PGT were successful only in specialcases. The chronological list of improvements for the modular group Γ = P SL (2 , Z ) Mathematics Subject Classification.
Key words and phrases.
Prime geodesic theorem, Selberg zeta function, modular group. includes + ε (Iwaniec [15]), + ε (Luo and Sarnak [17]), + ε (Cai [8]) andthe present + ε (Soundararajan and Young [19]).Iwaniec [14] remarked that the generalized Lindel¨of hypothesis for Dirichlet L -functions would imply + ε .We proved [2] that + ε is valid outside a set of finite logarithmic measure. Inthe present note, we relate the error term in the Gallagherian P GT on P SL (2 , Z )to the subconvexity bound for Dirichlet L - functions. This enables us to replace + ε by + ε under the generalized Lindel¨of hypothesis. More precisely, the mainresult of this paper is the following theorem. Theorem.
Let
Γ =
P SL (2 , Z ) be the modular group, ε > arbitrarily small and θ be such that L (cid:18)
12 + it, χ D (cid:19) ≪ (1 + | t | ) A | D | θ + ε for some fixed A > , where D is a fundamental discriminant. There exists a set B of finite logarithmic measure such that π Γ ( x ) = Z x dt log t + O (cid:16) x + θ + ε (cid:17) ( x → ∞ , x / ∈ B ) . Inserting the Conrey-Iwaniec [9] value θ = into Theorem, we obtain Corollary 1. π Γ ( x ) = li ( x ) + O (cid:16) x + ε (cid:17) ( x → ∞ , x / ∈ B ) . Any improvement of θ immediately results in the obvious improvement of theerror term in PGT. Taking into account that the Lindel¨of hypothesis allows θ = 0,we get Corollary 2.
Under the Lindel¨of hypothesis, π Γ ( x ) = li ( x ) + O (cid:16) x + ε (cid:17) ( x → ∞ , x / ∈ B ) . Remark 1.
The obtained exponent for strictly hyperbolic Fuchsian groups is + ε outside a set of finite logarithmic measure [3] and coincides with the above men-tioned Luo-Sarnak unconditional result for Γ =
P SL (2 , Z ) . In the case of a co-compact Kleinian group or a noncompact congruence group for some imaginaryquadratic number field, the respective Gallagherian bound is + ε [4] . Preliminaries.
The motivation for Theorem comes from several sources, including Gallagher[11], Iwaniec [15] and Balkanova and Frolenkov [6].Recall that π Γ ( x ) = li ( x )+ O (cid:16) x + θ + ε (cid:17) is equivalent to ψ Γ ( x ) = x + O (cid:16) x + θ + ε (cid:17) ,where ψ Γ ( x ) = P N ( P ) k ≤ x log N ( P ) is the Γ analogue of the classical Chebyshevfunction ψ .Under the Riemann hypothesis, Gallagher improved von Koch’s remainder termin the prime number theorem from ψ ( x ) = x + O (cid:16) x (log x ) (cid:17) to ψ ( x ) = x + O (cid:16) x (log log x ) (cid:17) outside a set of finite logarithmic measure.Following Koyama [16], we shall apply the next lemma [10] due to Gallagher toour setting. GT ON MODULAR SURFACE 3
Lemma A.
Let A be a discrete subset of R and η ∈ (0 , . For any sequence c ( ν ) ∈ C , ν ∈ A , let the series S ( u ) = X ν ∈ A c ( ν ) e πiνu be absolutely convergent. Then Z U − U | S ( u ) | du ≤ (cid:18) πη sin πη (cid:19) Z + ∞−∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Uη X t ≤ ν ≤ t + ηU c ( ν ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt . Iwaniec [15] established the following explicit formula with an error term for ψ Γ on Γ = P SL (2 , Z ). Lemma B.
For ≤ T ≤ x (log x ) , one has ψ Γ ( x ) = x + X | γ |≤ T x ρ ρ + O (cid:16) xT (log x ) (cid:17) ,where ρ = + iγ denote zeros of Z Γ . Recently, O. Balkanova and D. Frolenkov have proved the following estimate.
Lemma C. X | γ |≤ Y x iγ ≪ max (cid:16) x + θ Y , x θ Y (cid:17) log Y , X | γ |≤ Y x iγ ≪ Y log Y if Y > x + θ κ ( x ) ,where ρ = + iγ are the zeros of Z Γ , θ is the subconvexity exponent for Dirichlet L − functions, and κ ( x ) is the distance from √ x + √ x to the nearest integer. Proof of Theorem.
Inserting T = x (log x ) into Lemma B, we obtain(1) ψ Γ ( x ) = x + X | γ |≤ T x ρ ρ + O (cid:16) x (log x ) (cid:17) .We would like to bound the expression P | γ |≤ T x ρ ρ , where Y ∈ (0 , T ) is a parameterto be determined later on.Let n = ⌊ log x ⌋ and B n = ( x ∈ (cid:2) e n , e n +1 (cid:1) : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P | γ |≤ T x iγ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > x ε Y ) . Looking atthe logarithmic measure of B n , we get µ ∗ B n = Z B n dxx = Z A n x ε Y dxx ε Y ≤ e n +1 Z e n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | γ |≤ Y x iγ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dxx ε Y (2) ≤ e nε Y e n +1 Z e n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | γ |≤ Y x iγ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dxx . MUHAREM AVDISPAHI´C
After substitution x = e n · e π ( u + π ), the last integral becomes2 π π Z − π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | γ |≤ T e ( n + ) iγ ρ e πiγu (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) du .Applying Lemma A, with η = U = π and c γ = e ( n + 12 ) iγ ρ for | γ | ≤ T , c γ = 0otherwise, we get(3) π Z − π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | γ |≤ T e ( n + ) iγ ρ e πiγu (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) du ≤ (cid:18) sin (cid:19) ∞ Z −∞ + ∞ X t<γ ≤ t +1 | γ |≤ Y | ρ | dt .Note that P t<γ ≤ t +1 1 | ρ | = O (1) since { γ : t < | γ | ≤ t + 1 } = O ( t ) by the Weyllaw.Thus,(4) + ∞ Z −∞ + ∞ X t<γ ≤ t +1 | γ |≤ Y | ρ | dt = O Y Z dt = O ( Y ) .The relations (2), (3) and (4) imply µ ∗ B n ≪ Ye nε Y = e nε . Hence, the set B = ∪ B n has a finite logarithmic measure.For x / ∈ B , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P | γ |≤ Y x iγ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ x ε Y , i.e.(5) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | γ |≤ Y x ρ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ x + ε Y .Now, we rely on Lemma C to estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P Y < | γ |≤ T x ρ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Let us put S ( x, T ) = P | γ |≤ T x iγ . By Abel’s partial summation, we have X Y < | γ |≤ T x iγ ρ = S ( x, T ) + iT − S ( x, Y ) + iY + i T Z Y S ( x, u ) (cid:0) + iu (cid:1) du .Multiplying the last relation by x and recalling that Lemma C yields P | γ |≤ Y x iγ ≪ x + θ + ε Y for Y < T = x (log x ) , we get(6) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X Y < | γ |≤ T x ρ ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≪ x + θ + ε T + x + θ + ε Y + T Z Y x + θ + ε u u du ≪ x + θ + ε Y .Combining (5) and (6), we see that the optimal choice for the parameter Y is Y ≈ x + θ . Then, P | γ |≤ T x ρ ρ = O (cid:16) x + ε Y (cid:17) = O (cid:16) x + θ + ε (cid:17) for x / ∈ B . GT ON MODULAR SURFACE 5
The relation (1) becomes ψ Γ ( x ) = x + O (cid:16) x + θ + ε (cid:17) ( x → ∞ , x / ∈ B ) ,as asserted. References [1] Avdispahi´c, M. “On Koyama’s refinement of the prime geodesic theorem.”
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University of Sarajevo, Department of Mathematics, Zmaja od Bosne 33-35, 71000Sarajevo, Bosnia and Herzegovina
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