Prime Geodesic Theorem For The Congruence Subgroup (?) (p)

The prime geodesies on the surface F\H are of arithmetic interest. The problem of finding a good error term for the prime geodesic theorem has been intensively studied by many authors. Letπг(χ) count the number of primitive conjugacy classes with norm not exceedingχ. Then the prime geodesic theorem states where E(x) is the error term. It is convenient to consider (?)г(x)=∑A(P), where the sum is over all hyperbolic conjugacy classes{P}.It is conjectured that an error term like O((?)+ε) should hold.For the full modular group case, many authors have studied the error term. A. Selberg [18] introduced a zeta-function Z(s) which in many ways mimics the L-functions of algebraic number fields. Note that Riemann hypothesis is true for Z(s), one should expect that E(x)《(?). But this is out of reach since Z(s) has many more zeros than (?)(s). An error term like O((?)+ε) known for sometime as a consequence of the Selberg trace formula.Iwanicc [6] was the first to break the 3/4 bound. He showed that E(x)《χ35/48+ε. His proof used essentially the Selberg trace formula, Kuznetsov trace formula, Burgess's estimate for character sum, as well as the mean-value esti-mate for Fourier coefficients of Maass cusp forms.Luo and Sarnak [14] captured a considerable cancelation in the sums over the eigcnvalucs.that is As a consequence,they proved the error term O((?)+ε).Cai[2]improved the bound to O((?)+ε)by modifying the argument of lwanicc. Combining the knowledge of class number formula and Diriehlet L-function,Soundararajan and Young[20]proved that.E(x)《(?)For being(:ongruence subgroup case,there are little results.A remainder term of the form.O((?)+ε)was known(see Sarnak[15]).In 1994,Luo, Rudnick and Sarnak[13]made significant advanee in the selberg cigenvaluc eonjccture. Their result yields for any congruence subgroupГ(?) sL(2,Z).In this thesis,inspired by Bykovskii[1]and Soundararajan and Young [20],we consider the gcodesic problem for congruence subgroupГ(p)and give an asymptotic formula for the prime geodesic theorem,here p≥3 being a prime.As a consequence, we obtain the bounnd O((?)-ε).When p=1,we get the result of Soundararajan and Young.Our main result is the followingTheorem 0.1 Letκ(u)be a smooth function with compact support in(O,Y), satisfying∫∞+∞κ(u)du=1 and∫-∞+∞|k(j)(u)|du《Y-j.Let(?)≤Y≤(?) be a parameter.Then forГ-Г(p),we have Here,although p is a priime,we can see Theorem 0.1 holds true for p=1. Then wc find that by using Luo-Sarnak's bound E(x；κ)《x7/8+εY-1/4.This is just Soundararajan and Young's result for the modular group.Let Y=x3/4,,we obtain the followingCorollary0.1 ForГ=Г(p),we have...