it would be the case in the prime number theorem once the Riemann hypothesis be proved. The obstacles in establishing an analogue of von Koch's theorem ...
arXiv:1702.01699v2 [math.NT] 13 Feb 2017
PRIME GEODESIC THEOREM FOR THE MODULAR SURFACE ´ MUHAREM AVDISPAHIC Abstract. The exponent in the error term of the prime geodesic theorem for the modular surface is reduced to 23 outside a set of finite logarithmic measure. Under the generalized Lindel¨ of hypothesis, 32 is further reduced to 58 .
1. Introduction Let Γ = P SL (2, Z) be the modular group and H the upper half-plane equipped with the hyperbolic metric. The norms N (P0 ) of primitive conjugacy classes P0 in Γ are sometimes called pseudo-primes. The length of the primitive closed geodesic on the modular surface Γ \ H joining two fixed points, which are the same for all representatives of a class P0 , equals log(N (P0 )). The statement about the number πΓ (x) of classes P0 such that N (P0 ) ≤ 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 ZΓ (s) =
∞ YY
(1 − N (P0 )−s−k ), Re(s) > 1,
{P0 }k=0
and meromorphicaly continued to the whole complex plane. The relationship between the prime geodesic theorem and the distribution of zeros of the Selberg zeta function resembles to a large extent the relationship between the prime number theorem and the zeros of the Riemann zeta. However, the function ZΓ satisfies the Riemann hypothesis. It is an outstanding 1 open problem whether the error term in the prime geodesic theorem is O(x 2 +ε ) as it would be the case in the prime number theorem once the Riemann hypothesis be proved. The obstacles in establishing an analogue of von Koch’s theorem [10, p. 84] in this setting comes from the fact that ZΓ is a meromorphic function of order 2, while the Riemann zeta is of order 1. In the case of Fuchsian groups Γ ⊂ P SL (2, R), the best estimate of the remainder 3
term in PGT is still O
x4 log x
obtained by Randol [15] (see also [4], [1] for different 3 −1 proofs). We note that its analogue O x 2 d0 (log x) is valid also for strictly
and d ≥ 3 is the hyperbolic manifolds of higher dimensions, where d0 = d−1 2 dimension of a manifold [3, Theorem 1]. The attempts to reduce the exponent 43 in PGT were successful only in special cases. The chronological list of improvements for the modular group Γ = P SL(2, Z) 2010 Mathematics Subject Classification. 11M36, 11F72, 58J50. Key words and phrases. Prime geodesic theorem, Selberg zeta function, modular group. 1
´ MUHAREM AVDISPAHIC
2
7 71 35 + ε (Iwaniec [11]), 10 + ε (Luo and Sarnak [14]), 102 + ε (Cai [5]) and includes 48 25 the present 36 + ε (Soundararajan and Young [16]). Iwaniec [11] remarked that the generalized Lindel¨ of hypothesis for Dirichlet Lfunctions would imply 23 + ε. We prove that 32 + ε is valid outside a set of finite logarithmic measure. If we assume the generalized Lindel¨ of hypothesis, 23 + ε can be replaced by 58 + ε. More precisely, the main result of this paper is the following theorem.
Theorem. Let Γ = P SL(2, Z) be the modular group, ε > 0 arbitrarily small and θ be such that 1 A θ+ε L + it, χD ≪ (1 + |t|) |D| 2
for some fixed A > 0, where D is a fundamental discriminant. There exists a set B of finite logarithmic measure such that Z x 5 θ dt + O x 8 + 4 +ε πΓ (x) = (x → ∞, x ∈ / B) . 0 log t Inserting the Conrey-Iwaniec [6] value θ =
1 6
into Theorem, we obtain
Corollary 1. 2 πΓ (x) = li (x) + O x 3 +ε (x → ∞, x ∈ / B) .
Any improvement of θ immediately results in the obvious improvement of the error term in PGT. Taking into account that the Lindel¨ of hypothesis allows θ = 0, we get Corollary 2. Under the Lindel¨ of hypothesis, 5 (x → ∞, x ∈ / B) . πΓ (x) = li (x) + O x 8 +ε
7 Remark. The obtained exponent for strictly hyperbolic Fuchsian groups is 10 +ε outside a set of finite logarithmic measure [2] and coincides with the above mentioned Luo-Sarnak unconditional result for Γ = P SL(2, Z).
2. Preliminaries. Gallagher and the work of Soundararajan and Young. The motivation for Theorem comes from Gallagher [8] and Soundararajan and Young [16]. 5 θ 5 θ Recall that πΓ (x) = li (x)+O x 8 + 4 +ε is equivalent to ψΓ (x) = x+O x 8 + 4 +ε , P where ψΓ (x) = log N (P0 ) is the Γ analogue of the classical Chebyshev N (P0 )k ≤x
function ψ. Under the Riemann hypothesis, Gallagher improved Koch’s remainder term von 1 2 2 to ψ(x) = x + in the prime number theorem from ψ(x) = x + O x (log x) 1 2 O x 2 (log log x) outside a set of finite logarithmic measure. Following Koyama [13], we shall apply the next lemma [7] due to Gallagher to our setting.
PGT ON MODULAR SURFACE
3
Lemma A. Let A be a discrete subset of R and η ∈ (0, 1). For any sequence c(ν) ∈ C, ν ∈ A, let the series X S (u) = c (ν) e2πiνu ν∈A
be absolutely convergent. Then Z
U
|S (u)|2 du ≤
−U
πη sin πη
2 Z
+∞
−∞
2 U X c (ν) dt. η t≤ν≤t+ η U
The smoothed version of ψΓ is given by Z Y ψΓ (x, k ) = ψΓ (x + u) k (u) du, 0
where k is a smooth real valued function with the compact support on (0, Y ) such that Z ∞ Z ∞ 1 (j) k (u) du = 1 and k (u) du ≪j j for all j ≥ 0. Y −∞ −∞ i h 1 The parameter Y ∈ x 2 +ε , logx x is to be determined later. As usual, one makes use of the decomposition Z Y (ψΓ (x + u) − ψΓ (x)) k (u) du. (1) ψΓ (x) = ψΓ (x, k ) − 0
Soundararajan and Young [16] proved Z Y Z (2) (ψΓ (x + u) − ψΓ (x)) k (u) du = 0
0
Y
1 1 θ uk (u) du + O Y 2 x 4 + 2 +ε .
Furthermore, they proved (3)
ψΓ (x, k ) = x +
Z
Y
uk (u) du + E (x, k ) , 0
with (4)
E (x, k ) =
1 ρ 1+ε
X
|γ|≤ x Y
Z
0
Y
1 ρ (x + u) k (u) du + O x 2 +ε ,
where ρ = 21 + iγ denote zeros of ZΓ (see [16, relation (20)]). Combining relations (1) to (4), we obtain the theorem of Soundararajan and Young in the following form suitable for our purpose. Theorem B. Let Γ, ψΓ , Y and k be as above. Then 1 1 θ 1 X 1Z Y ρ ψΓ (x) = x + (x + u) k (u) du + O x 2 +ε + O Y 2 x 4 + 2 +ε . ρ 0 1+ε |γ|≤ x Y
´ MUHAREM AVDISPAHIC
4
3. Proof of Theorem We are interested in the logarithmic measure of the set on which X xiγ (5) > xα . ρ |γ|≤ x1+ε Y k j 1+ε 1+ε For n = log x Y , T = en and y = x Y , let iγ X 1+ε α (Y y) > (Y y) 1+ε . An = y ∈ [T, eT ) : ρ |γ|≤T Now,
(6) µ× An =
Z
dy = y
An
Z
(Y y)
dy
2α 1+ε
An
y (Y y)
2α 1+ε
≤
1 2α
(Y T ) 1+ε
Z
eT
T
2 iγ X 1+ε (Y y) dy . y ρ |γ|≤T
After a substitution y = T e2π(1+ε)u , the last integral becomes 2 iγ 1 Z 2π(1+ε) X (T Y ) 1+ε 2πiγu du. e ρ 0 |γ|≤T Translating u by
(T Y
iγ ) 1+ε
iγ e 2
ρ
Z
1 2π(1+ε)
0
1 4π ,
and applying Lemma A, with η = U =
1 4π
for |γ| ≤ T , cγ = 0 otherwise, we get 2 iγ X 1+ε (T Y ) 2πiγu e du ρ |γ|≤T
≤
≤
2 iγ X iγ 1+ε 2 (T Y ) e 2πiγu e du 1 ρ − 4π |γ|≤T 2 1 2 Z +∞ X 1 4 dt. 1 |ρ| sin
Z
1 4π
4
−∞
t
