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On Koyama's refinement of the prime geodesic theorem.

1. Introduction. Let [GAMMA] [subset] c PSL(2, R) be a strictly hyperbolic Fuchsian group acting on the upper half-plane H equipped with the hyperbolic metric. The quotient space T \ H can be identified with a compact Riemann surface F of a genus g [greater than or equal to] 2. The object of our attention is the asymptotic behaviour of the summatory von Mangoldt function

[mathematical expression not reproducible]

where the sum is taken over primitive hyperbolic conjugacy classes P in [GAMMA] (prime geodesics on F), N(P) = exp(length(P)) is the norm of a class P and k runs through positive integers.

In the recent paper [6] published in this journal, Shin-ya Koyama studied the existence of a subset E in [R.sub.[greater than or equal to]2] with finite logarithmic measure such that

[mathematical expression not reproducible].

Here and in the sequel, p denotes zeros of the Selberg zeta function [Z.sub.[GAMMA]]. It is known that the complex zeros of [Z.sub.[GAMMA]] are of the form p = 1/2 [+ or -] i[gamma] and that [Z.sub.[GAMMA]] has finitely many real zeros, all lying in the interval [0,1]. Koyama was motivated by Gallagher's [4] approach to the prime number theorem under the Riemann hypothesis.

We give a new proof of the following sharper result (cf. [7], [3]).

Theorem 1.

[[not upsilon].sub.[GAMMA]] = x + [[summation over].sub.3/4 < p < 1] [x.sup.p]/p + O ([x.sup.3/4]) (x [right arrow] [infinity]).

We observe that the analogue is also valid for higher dimensional hyperbolic manifolds with cusps. Applying the Gallagher-Koyama method, we further reduce the error term outside a set of finite logarithmic measure.

Theorem 2. For [alpha] > 0, there exists a set H of finite logarithmic measure such that

[mathematical expression not reproducible]

where [epsilon] > 0 is arbitrarily small.

2. From Hejhal to Randol.

Proof of Theorem 1. We shall take the same starting point as in [6], i.e., Hejhal's explicit formula with an error term for the function [mathematical expression not reproducible] (cf. [5, Theorem 6.16. on p. 110]):

(1) [mathematical expression not reproducible]

Recall that [mathematical expression not reproducible].

The novelty of our approach consists in integrating (1) at this point and then temporarily getting rid of Hejhal's error term. Indeed, the integration of (1) firstly yields the explicit formula with an error term for [mathematical expression not reproducible]. Now, letting T [right arrow] [infinity] in the obtained formula, we end up with

[mathematical expression not reproducible]

Usually, to derive the asymptotics of [[not upsilon].sub.[GAMMA]](x) from the asymptotics of [[not upsilon].sub.2[GAMMA]] (x), one introduces the second-difference operators:

[[DELTA].sup.+.sub.2] f(x) = f (x + 2h) - 2f (x + h) + f(x) and [[DELTA].sup.-.sub.2] f(x) = f (x + 2h) - 2f (x + h) + f(x)

where h > 0 is to be determined later.

Since [[not upsilon].sub.[GAMMA]] is a non-decreasing function, we have

[mathematical expression not reproducible]

We apply [[DELTA].sup.+.sub.2] to all summands in the explicit formula for [[not upsilon].sub.2,[GAMMA]] (x). E.g., [[DELTA].sup.+.sub.2] ([x.sup.3]/6)= [xh.sup.2] + [h.sup.3], which gives us 1/[h.sup.2] [[DELTA].sup.+.sub.2] ([x.sup.3]/6) = x + h, etc.

Applying 1/[h.sup.2] [[DELTA].sup.+.sub.2] to the sum [mathematical expression not reproducible], we end up with

[[summation over. (1/2<p<1)] [x.sup.p]/p + O(h).

When dealing with the absolutely convergent series [mathematical expression not reproducible], we take into account that

[mathematical expression not reproducible]

Thus,

[mathematical expression not reproducible]

We are left to optimize the terms O(h), O ([x.sup.1/2] M), O([x.sup.5/2]/[h.sup.2]M). This is achieved by choosing h = [x.sup.3/4], M = [x.sup.1/4]. All other ingredients are dominated by O([x.sup.3/4]).

The same procedure works in case of [[DELTA].sup.-.sub.2] [[not upsilon].sub.2,[GAMMA]]) (x), i.e., for estimating [[not upsilon].sub.2,[GAMMA]] (x) from below.

So,

[[not upsilon].sub.[GAMMA]] (x) = x + [[summation over].sub.3/4 < p < 1] [x.sup.p]/p + O ([x.sup.3/4])

Remark 1. The error term O([x.sup.3/4]) in Theorem 1 yields O [x.sup.3/4]/log x in the prime geodesic theorem. Concerning the explicit formula for [[not upsilon].sub.1,[GAMMA]], one can consult [2], where a better estimate for the logarithmic derivative of the Selberg zeta function is established.

Remark 2. The full analogue is valid for higher dimensional hyperbolic manifolds with cusps. Namely, the error term in the prime geodesic theorem in that setting reads [O([x.sup.3do/2][(logx).sup.-1]) where [d.sub.o] = d-1/2 and d is the dimension of a manifold [1, Theorem 1].

3. An application of the Gallagher-Koyama method.

Proof of Theorem 2. In estimating [[not upsilon].sub.[GAMMA]](x) we shall use explicit formula (1) and the relation 1/h [[DELTA].sup.-.sub.1] [[not upsilon].sub.[GAMMA]](x) [less than or equal to] [[not upsilon].sub.[GAMMA]] (x) [less than or equal to] 1/h [[DELTA].sup.+.sub.1] [[not upsilon].sub.1,[GAMMA] (x), where 0 < h < x/2 is to be determined later on. Here, [[DELTA].sup.+.sub.1] f (x) = f (x + h)- f (x) and [[DELTA].sup.-.sub.1]f (x) = f (x)- f (x - h).

Let [beta] > 4[alpha] +1. According to (1) and the relation above, we have

(2) [mathematical expression not reproducible].

Now,

[mathematical expression not reproducible]

For the first sum on the right-hand side, we have

[mathematical expression not reproducible]

The second sum is to be split into

[mathematical expression not reproducible]

By Koyama's argument [6, p. 80],

[mathematical expression not reproducible]

Hence,

[[mu].sup.x] [D.sup.T.sub.Y] [much greater than] [(1 + log T).sup.4[alpha]]/Y.

For x [member of] [[e.sup.n], [e.sup.n+1]), let T = [e.sup.n]. The error term in (2) becomes O(x log x/h). Let Y take values [Y.sub.1] = [(log T).sup.[beta]] = [n.sup.[beta]], [Y.sub.2] = [(n - 1).sup.[beta]], [Y.sub.3] = [e.sup.n-1]. Denote

[mathematical expression not reproducible], respectively. We have

[mathematical expression not reproducible]

Put H = E [union] F [union] G. Obviously, [[mu].sup.x] H < [infinity]. We take x, x + h [member of] [R.sub.[greater than or equal to]2] \ H.

[mathematical expression not reproducible]

Case II. If x + h [member of] [[e.sup.n+1], [e.sup.n+2])\ H, we shall express the sum [[sigma].sub.[beta].T] (x + h) in the form

[mathematical expression not reproducible]

The first sum is [mathematical expression not reproducible] because [mathematical expression not reproducible]. The second sum is

[mathematical expression not reproducible]

So, in both cases, the relation (2) becomes

[mathematical expression not reproducible]

The optimal bound is achieved with [h = [x.sup.3/4]/[(log x).sup.x].

Thus,

[mathematical expression not reproducible]

The opposite inequality is derived from

[[not upsilon].sub.[GAMMA]](x) [greater than or equal to] 1/h [[DELTA].sup.- .sub.1] [[not upsilon].sub.1,[GAMMA])(x) by the same procedure. If [epsilon] > 0 is arbitrarily small, then [mathematical expression not reproducible] is obviously dominated by the error term. This completes the proof.

doi: 10.3792/pjaa.94.21

References

[1] M. Avdispahic and Dz. Gusic, On the error term in the prime geodesic theorem, Bull. Korean Math. Soc. 49 (2012), no. 2, 367-372.

[2] M. Avdispahic and L. Smajlovic, An explicit formula and its application to the Selberg trace formula, Monatsh. Math. 147 (2006), no. 3, 183-198.

[3] P. Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, 106, Birkhauser Boston, Inc., Boston, MA, 1992.

[4] P. X. Gallagher, Some consequences of the Riemann hypothesis, Acta Arith. 37 (1980), 339-343.

[5] D. A. Hejhal, The Selberg trace formula for PSL(2, R). Vol. I, Lecture Notes in Mathematics, 548, Springer-Verlag, Berlin, 1976.

[6] S. Koyama, Refinement of prime geodesic theorem, Proc. Japan Acad. Ser. A Math. Sci. 92 (2016), no. 7, 77-81.

[7] B. Randol, On the asymptotic distribution of closed geodesics on compact Riemann surfaces, Trans. Amer. Math. Soc. 233 (1977), 241 247.

Department of Mathematics, University of Sarajevo, Zmaja od Bosne 35, BA-71000 Sarajevo, Bosnia and Herzegovina

(Communicated by Kenji FUKAYA, M.J.A., Feb. 13, 2018)
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Author:Avdispahic, Muharem
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Date:Mar 1, 2018
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