On von Koch Theorem for PSL(2, $$\mathbb {Z}$$ Z )
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On von Koch Theorem for PSL(2,Z) Muharem Avdispahi´c1 Received: 23 June 2020 / Revised: 12 October 2020 / Accepted: 10 November 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract Under a previously studied condition on the argument of the Selberg zeta function on the critical line, we reach the critical exponent 21 in the error term of the prime geodesic theorem for the modular group PSL(2,Z) outside a set of finite logarithmic measure. We also prove a conditional prime geodesic theorem of Hejhal’s type in this setting without the latter exclusion. Keywords Prime geodesic theorem · Selberg zeta function · Modular group Mathematics Subject Classification 11M36 · 11F72 · 58J50
1 Introduction Let H = {z = x + i y : y > 0} be the upper half-plane equipped with the hyperbolic 2 2 . If ⊂ P S L(2, R) is a strictly hyperbolic Fuchsian group, the metric ds 2 = d x y+dy 2 quotient space F = \ H is a compact Riemann surface of genus g ≥ 2. The length of a primitive closed geodesic on F equals log(N (P0 )), where N (P0 ) is the norm of the corresponding primitive hyperbolic conjugacy class P0 in . The Selberg zeta function Z is defined by the Euler-type product Z (s) =
∞
(1 − N (P0 )−s−k )
{P0 }k=0
in the right half-plane Re(s) > 1 and meromorphically continued to the whole complex plane.
Communicated by Rosihan M. Ali.
B 1
Muharem Avdispahi´c [email protected] Department of Mathematics, University of Sarajevo, Zmaja od Bosne 33-35, 71000 Sarajevo, Bosnia and Herzegovina
123
M. Avdispahi´c
The prime geodesic theorem, PGT, tells us that the number π (x) of classes P0 such that N (P0 ) ≤ x, x > 0, satisfies the relation π (x) ∼ li (x) =
x 2
dt (x → +∞) . log t
In the classical situation of the Riemman function ζ , von Koch (see [13, p. 84]) proved that the Riemann hypothesis is equivalent to the statement that the error term E (x) in the prime number theorem (PNT) π (x) = li (x) + E (x) (x → +∞) 1 is of the form E (x) = O x 2 log x . It has been an old dream of Hilbert and Polya that the Riemann hypothesis should be true because of a connection between the non-trivial zeros of ζ and the eigenvalues of a still unknown self-adjoint operator. Now, the non-trivial zeros of Z are solutions of the equations s(1−s) = λn , where λn ≥ 0 are the eigenvalues of the essentially self-adjoint Laplace–Beltrami operator = −y on F. The zeros sn =
1 2
+ irn =
1 2
2
∂2 ∂2 + ∂x2 ∂ y2
+ i λn −
1 4
and sn =
1 2
− irn lie on the critical
line Re (s) = for λ ≥ . There might exist at most finitely many, say M + 1, eigenvalues 0 ≤ λn < 41 , and these give rise to exceptional zeros of Z lying in the
sn ∈ 0, 21 for 1 ≤ n ≤ M. segment [0, 1]. Note that sn ∈ 21 , 1 and Since the function Z satisfies an analogue of the Riemann hypothesis, it is expected that the exponent of x in the error term E (x) of the prime geodesic theorem 1 2
1 4
π (x) = li (x) +
M
li x sn + E (x)
n=1
also equals 21 . However, the best unconditional e
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