Prime geodesic theorem of Gallagher type

0.12 MB·

by  Muharem Avdispahić

PRIME GEODESIC THEOREM OF GALLAGHER TYPE
MUHAREMAVDISPAHIC´
7
1
0 Abstract. We reduce the exponent in the error term of the prime geodesic
2 theorem for compact Riemann surfaces from 3 to 7 outside a set of finite
4 10
logarithmicmeasure.
n
a
J

9

] 1. Introduction
T
N Let Γ ⊂ PSL(2,R) be a strictly hyperbolic Fuchsian group and F = Γ\H be
the corresponding compact Riemann surface of a genus g ≥ 2, where H = {z =
.
h x+iy : y > 0} denotes the upper half-plane equipped with the hyperbolic metric
t
a ds2 = dx2+dy2. The Selberg zeta function on F is defined by the Euler product
m y2
∞
[
Z(s)=Z (s)= (1−N(P )−s−k), Re(s)>1,
Γ 0
v1 {YP0}kY=0
5 where the product is takenover all primitive conjugacy classes{P } and N(P ) is
0 0
1
the norm of a conjugacy class P . (See any of the standard references [9], [13] for
1 0
a necessary background.)
2
0 The Selberg zeta function can be continued to the whole complex plane as a
. meromorphic function of a finite order. Its zeros ρ = 1 +iγ are denumerable and
1 2
closely related to the eigenvalues λ of the Laplace-Beltrami operator on F. This
0
7 operatorbeing essentially self-adjoint, its eigenvaluesare non-negativeand tend to
1 infinity. Therefore, there are finitely many of them that are less than 1. We have
4
:
v γ =± λ− 1 for λ≥ 1 and γ =∓i 1 −λ for λ< 1. So, the zeros of Z are split
i q 4 4 q4 4
X into two parts: those lying on the critical line Res = 1 and the real zeros in the
2
r interval [0,1].
a
Compared to the Riemann zeta case, there are ”too many zeros” of Z in some
sense[9,(6.14)onp. 113]. LetN(t)denotethenumberofzerosρ= 1+iγ suchthat
2
0<γ ≤t. ThefunctionR(t),givenbyN(t)= |F|t2+R(t),growsasO t(logt)−1 ,
4π
where |F| is the volume of F. (cid:0) (cid:1)
The norm N(P ) is determined by the length of the geodesic joining two fixed
0
points,necessarilythesameonesforallrepresentativesofP . Thestatementabout
0
the number π (x) of classes {P } such that N(P )≤x, for x>0, is known as the
0 0 0
prime geodesic theorem. It has been proved by Selberg [21] and Huber [10, 11]
andsubsequentlygeneralizedtovarioussettings. Thereferences[16,5,8,18,6,19]
form an interesting range of samples.
2010 Mathematics Subject Classification. 11M36,11F72,58J50.
Keywords andphrases. Primegeodesictheorem,Selbergzetafunction,hyperbolicmanifolds.
1
2 MUHAREMAVDISPAHIC´
One should consult [9, Discussion 6.20., pp. 113-115 and Discussion 15.16., pp.
253-255] on the difficulties in improving Huber’s O x43(logx)−21 . The best esti-
(cid:16) (cid:17)
matefortheerrortermintheprimegeodesictheoremuptonowisO x34(logx)−1
(cid:16) (cid:17)
obtained by Randol [20]. Its analogue has been also established for higher di-
mensional hyperbolic manifolds [2], improving Park’s theorem [17, Th. 1.2.]. An
important ingredient, implicitly [20] or explicitly [17], is the growth rate of the
log-derivative of the Ruelle zeta vs. the Selberg zeta log-derivative [3].
Though the expected exponent on Riemann surfaces is 1 +ε, the above men-
2
tioned 3 was successfully reduced only in the case of modular surfaces Γ \ H,
4
Γ ⊂ PSL(2,Z). Iwaniec [12] obtained 35 +ε, Luo and Sarnak [15] 7 +ε, Cai [4]
48 10
71 +ε, Soundararajanand Young [22] 25 +ε.
102 36
Following Gallagher’s [7] approach to the Riemann zeta, we shall prove that 7
10
can be achieved for Γ⊂PSL(2,R) outside a set of finite logarithmic measure.
2. Main result
Let ψ(x) = Λ(P) = logN(P0) and ψ (x) = xψ(t)dt be the
1−N(P)−1 1 1
N(PP)≤x N(PP)≤x R
Chebyshev resp. integrated Chebyshev function, as usual. Recall that the error
termO x34(logx)−1 inthe primegeodesictheoremcorrespondstoO x43 inthe
(cid:16) (cid:17) (cid:16) (cid:17)
explicit formula for ψ.
Our main result is given by the following theorem that substantially improves
[14, Th. 1.]. and [1, Th. 2.]
Theorem. LetΓ⊂PSL(2,R)beastrictlyhyperbolic Fuchsiangroup. Thereexists
a set G of finite logarithmic measure such that
ψ(x)=x+ xρ +O x170(logx)15 (loglogx)51+ε (x→∞,x∈/ G),
ρ
7X<ρ<1 (cid:16) (cid:17)
10
where ε>0 is arbitrarily small.
Proof. As the starting point, we shalltake Hejhal’s explicit formula for ψ with an
1
error term [9, Th. 6.16. on p. 110]
1
ψ (x)=α x+β xlogx+α +β logx+F
1 0 0 1 1
(cid:18)x(cid:19)
x2 xρ+1 x2logx
+ + +O (x→∞),
2 ρ(ρ+1) (cid:18) T (cid:19)
Xρ
|γ|<T
∞
where F(x)=(2g−2) 2k+1 x1−k.
k(k−1)
kP=2
The asymptoticsofψ is convenientlyderivedfromthe asymptoticsofψ viathe
1
relation
x x+h
f(t)dt≤f(x)h≤ f(t)dt
Z Z
x−h x
valid for any non-decreasing function f, where h>0.
PRIME GEODESIC THEOREM 3
We have
x+h
1 xρ
ψ(x)≤ ψ(t)dt=x+ +O(logx)+O(h)
h Z ρ
x 12<Xρ<1
(1)
1 (cid:12) (x+h)ρ+1−xρ+1(cid:12) x2logx
+ (cid:12) (cid:12)+O .
(cid:12) (cid:12)
h(cid:12)(cid:12)ReX(ρ)=1 ρ(ρ+1) (cid:12)(cid:12) (cid:18) hT (cid:19)
(cid:12) |γ|≤T2 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Now, for Y <T, (cid:12) (cid:12)
(x+h)ρ+1−xρ+1 (x+h)ρ+1−xρ+1 (x+h)ρ+1−xρ+1
= + .
ρ(ρ+1) ρ(ρ+1) ρ(ρ+1)
ReX(ρ)=1 ReX(ρ)=1 ReX(ρ)=1
2 2 2
|γ|≤T |γ|≤Y Y<|γ|≤T
The trivial bound for the first sum on the right hand side is given by
1 (cid:12) (x+h)ρ+1−xρ+1(cid:12)  1 
(2) (cid:12)(cid:12) (cid:12)(cid:12)=O x21 =O x21Y .
h(cid:12)(cid:12)ReX(ρ)=1 ρ(ρ+1) (cid:12)(cid:12)  ReX(ρ)=1 |ρ| (cid:16) (cid:17)
(cid:12) |γ|≤Y2 (cid:12)  |γ|≤Y2 
(cid:12) (cid:12)  
(cid:12) (cid:12)
The second(cid:12) sum is split into (cid:12)
(x+h)ρ+1 xρ+1
− .
ρ(ρ+1) ρ(ρ+1)
ReX(ρ)=1 ReX(ρ)=1
2 2
Y<|γ|≤T Y<|γ|≤T
Let
 (cid:12) xρ+1 (cid:12)  3
DYT =x∈[T,eT):(cid:12)(cid:12)(cid:12) X ρ(ρ+1)(cid:12)(cid:12)(cid:12)>xα(logx)β(loglogx)β, 1<α< 2, β >0.
(cid:12)Re(ρ)=1 (cid:12)
(cid:12) 2 (cid:12)
Then,  (cid:12)(cid:12)(cid:12)Y<|γ|≤T (cid:12)(cid:12)(cid:12) 
dt dt
µ×DT = = t2α(logt)2β(loglogt)2β
Y Z t Z t1+2α(logt)2β(loglogt)2β
DT DT
Y Y
2
1 (cid:12) tρ+1 (cid:12) t3−2α
≤ (cid:12) (cid:12) dt
(logT)2β(loglogT)2βDZYT (cid:12)(cid:12)(cid:12)(cid:12)YR<eX(|ργ)|=≤2T1 ρ(ρ+1)(cid:12)(cid:12)(cid:12)(cid:12) t4
(cid:12) (cid:12)
(cid:12) (cid:12) 2
(cid:12) (cid:12)
eT
T3−2α (cid:12) tρ+1 (cid:12) dt
= O (cid:12) (cid:12) .
(cid:18)(logT)2β(loglogT)2β(cid:19)ZT (cid:12)(cid:12)(cid:12)(cid:12)YR<eX(|ργ)|=≤2T1 ρ(ρ+1)(cid:12)(cid:12)(cid:12)(cid:12) t4
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
4 MUHAREMAVDISPAHIC´
According to Koyama [14, p. 79],
2
eT
(cid:12) tρ+1 (cid:12) dt 1
(cid:12) (cid:12) =O .
ZT (cid:12)(cid:12)(cid:12)(cid:12)YR<eX(|ργ)|=≤1T2 ρ(ρ+1)(cid:12)(cid:12)(cid:12)(cid:12) t4 (cid:18)Y (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
Taking Y =T3−2α(logT)(cid:12)1−2β(loglogT)1−2(cid:12)β+ε, we obtain
1
µ×DT ≪ .
Y logT(loglogT)1+ε
For n = ⌊logx⌋, T = en, denote E = DT. Then, µ×E ≪ 1 and
n Y n n(logn)1+ε
µ×∪E ≪ 1 <∞.
n n(logn)1+ε
If x∈ enP,en+1 \E , one gets
n
(cid:2) (cid:1)
(cid:12) xρ+1 (cid:12)
(3) (cid:12) (cid:12)≤xα(logx)β(loglogx)β.
(cid:12) (cid:12)
(cid:12) X ρ(ρ+1)(cid:12)
(cid:12)Re(ρ)=1 (cid:12)
(cid:12) 2 (cid:12)
(cid:12)Y<|γ|≤T (cid:12)
(cid:12) (cid:12)
We are interested in(cid:12)achieving h<x43(cid:12). If it happens that x+h∈ en,en+1 \En,
one shall have (cid:2) (cid:1)
(x+h)ρ+1
=O xα(logx)β(loglogx)β
ρ(ρ+1)
X (cid:0) (cid:1)
Re(ρ)=1
2
Y<|γ|≤T
as well, since x+h<2x.
The other possibility is that x+h ∈ en+1,en+2 . In that case, we proceed as
follows (cid:2) (cid:1)
(x+h)ρ+1 (x+h)ρ+1 (x+h)ρ+1
= − .
ρ(ρ+1) ρ(ρ+1) ρ(ρ+1)
X X X
Re(ρ)=1 Re(ρ)=1 Re(ρ)=1
2 2 2
Y<|γ|≤T Y<|γ|≤eT T<|γ|≤eT
For x+h∈ en+1,en+2 \E , we get
n+1
(cid:2) (cid:1)
(x+h)ρ+1
(4) =O xα(logx)β(loglogx)β .
ρ(ρ+1)
X (cid:0) (cid:1)
Re(ρ)=1
2
Y<|γ|≤eT
To estimate (x+h)ρ+1, we shall consider
ρ(ρ+1)
Re(Pρ)=1
2
T<|γ|≤eT
 (cid:12) xρ+1 (cid:12) 
DTeT =x∈(cid:2)eT,e2T(cid:1):(cid:12)(cid:12)(cid:12)(cid:12)ReX(ρ)=1 ρ(ρ+1)(cid:12)(cid:12)(cid:12)(cid:12)>xα(logx)β(loglogx)β.
(cid:12) 2 (cid:12)
 (cid:12)(cid:12)(cid:12)T<|γ|≤eT (cid:12)(cid:12)(cid:12) 
PRIME GEODESIC THEOREM 5
By the argumentation above leading to the estimate of µ×DT, we get
Y
T3−2α 1 1
µ×DeT ≪ · ≪ .
T (logT)2β(loglogT)2β T T2α−2
Recall that n = ⌊logx⌋, T = en and denote F = DeT. Notice that µ×F <
n+1 T n+1
1 and the series 1 converges since α > 1. Thus, if we additionally
e(2α−2)n e(2α−2)n
assume x+h∈ en+1,ePn+2 \F , we get
n+1
(cid:2) (x(cid:1)+h)ρ+1
(5) =O xα(logx)β(loglogx)β .
ρ(ρ+1)
X (cid:0) (cid:1)
Re(ρ)=1
2
T<|γ|≤eT
Looking back at (1), and taking into account the relations (2), (3), (4) and (5), we
are left to optimize
h, xlohgx, x12Y and xα(logx)βh(loglogx)β, i.e.,
h, x21 ·x3−2α(logx)1−2β(loglogx)1−2β and xα(logx)βh(loglogx)β
since α>1, T ≈x and Y =O x3−2α(logx)1−2β(loglogx)1−2β+ε .
Choosing h ≈ xα2(logx)β2(lo(cid:0)glogx)β2, we get 1 +3−2α = α (cid:1)and 1−2β = β.
2 2 2
Hence, α = 7 and β = 2. This completes the proof since the sets E = ∪E and
5 5 n
F =∪F have finite logarithmic measure.
n
(cid:3)
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University of Sarajevo, Department of Mathematics, Zmaja od Bosne 33-35, 71000
Sarajevo,Bosnia andHerzegovina
E-mail address: [email protected]