By Jürgen Fischer

The Notes provide a right away method of the Selberg zeta-function for cofinite discrete subgroups of SL (2,#3) performing on the higher half-plane. the elemental proposal is to compute the hint of the iterated resolvent kernel of the hyperbolic Laplacian on the way to arrive on the logarithmic spinoff of the Selberg zeta-function. prior wisdom of the Selberg hint formulation isn't really assumed. the idea is built for arbitrary actual weights and for arbitrary multiplier structures allowing an method of recognized effects on classical automorphic kinds with no the Riemann-Roch theorem. The author's dialogue of the Selberg hint formulation stresses the analogy with the Riemann zeta-function. for instance, the canonical factorization theorem comprises an analogue of the Euler consistent. ultimately the overall Selberg hint formulation is deduced simply from the houses of the Selberg zeta-function: this can be just like the process in analytic quantity conception the place the categorical formulae are deduced from the homes of the Riemann zeta-function. except the fundamental spectral conception of the Laplacian for cofinite teams the booklet is self-contained and should be precious as a brief method of the Selberg zeta-function and the Selberg hint formulation.

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**Extra resources for An Approach to the Selberg Trace Formula via the Selberg Zeta-Function **

**Sample text**

F% converging of (cf. Ejp [E2], Definition. e. b)). for every case give I ~ Ek , the E i s e n - an a d d i t i o n a l operator. A family an e i g e n 2 a c k e t of A if it Stieltjes I 6 IR, I 6 IR, integral being understood decomposition con- " properties: for all norm Hilbert-space, a symmetric is c a l l e d T • -> I , the for In this f 6 Dk be a s e p a r a b l e , H for all , the on p : I ..... mj) formula vI 6 H three llv -viii= 0 1 = f ~dv. ,T; to the e x p a n s i o n (vl)16iR system sums.

M . jp, (i) being the l-th and . I ¼+r2_~ in- Ejp(Z,½+ir) yields ' Hence, Z one component I ¼+r2-~ • E JP (z,½+ir)dr • ___j___1 • Ejp, (i) (z,½+ir) ¼+r2-~ of the v e c t o r function E . dr , JP Consequently, ] [ {-~k-l) -] G (I) ( z) ] (w) k~ ' T m. J I ~J Z j=1 p=1 Computing ~ the c o l u m n 1 6 {I .... ,d} dxd-matrix made I Z I -i nhO n m. % J I + ~Z Z ~ j I p=1 E(z,s) • Ejp, (i) (z,½+it) • Ejp(W,½+it)dt. 5) for all By tE(z,½+it) / tr(~(½+it) k ) tE(z,½-it) - E(w,½-it) t~(½+it) > t 6 IR. 7), ~(½+it) is unitary, hence the last term equals m, tr E(w,½-it) .

4 follows: = X (S) -I X(P)X(S) jp(SZ) H(Sz,PSz) , S-Ips z 6 IH. With the f u n d a m e n t a l Gklhyp(Z) domain do(z) Fp X = defined tr X (P) {P}r F above, we thus h a v e f jp(Z) Yp H(z,Pz) hs(d(z,Pz)) d~(z)- tr P > 2 Now fix f Yp To p r o v e N(P) P 6 F , tr P > 2. 1. z)) d~(z) A]:p Here, as well as above, the fundamental domain be replaced by IR × [I,N(Po) [. AFp of Z(DN(p)) may Hence, the last integral eauals N(P o) f f <-~P)Zl)x+i(N{P)+1)y/ -" I By the substitution N(P)+I N(P)-] { t+i [ <-~7/ log N(Po) : N(P)+I N(P)-I ~ N(P)+I N(P)-I y x " ks +(N_P)+I)2y 2 4N(P)y ~dy y2 " • t, this term is equal to {(Nm)+1) 2 hs < 4N(P) (I+t2)) dt 4N(P) )s F(s+k)F(s-k) -(N(P)+1) ~ • log N(P o) • 4~F(2s) (1-it)2k (1+t2)s+ k ( F 4N(P) s+k,s-k;2s; - If the hypergeometric 2 (N(P)+I) function is represented I ) dt -1+t 2 by the corresponding power series, the integrand may be integrated term by term.