By Jürgen Fischer

The Notes provide an immediate method of the Selberg zeta-function for cofinite discrete subgroups of SL (2,#3) performing on the higher half-plane. the elemental inspiration is to compute the hint of the iterated resolvent kernel of the hyperbolic Laplacian so as to arrive on the logarithmic by-product of the Selberg zeta-function. earlier wisdom of the Selberg hint formulation isn't really assumed. the speculation is built for arbitrary genuine weights and for arbitrary multiplier platforms 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 includes an analogue of the Euler consistent. eventually the final Selberg hint formulation is deduced simply from the houses of the Selberg zeta-function: this can be just like the method in analytic quantity thought the place the categorical formulae are deduced from the homes of the Riemann zeta-function. except the elemental spectral thought of the Laplacian for cofinite teams the e-book 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**

I,m+I ..... 2 v - I } many 2k elliptic IH As R : Rj - the p r i m i t i v e in to a r o t a t i o n {R} F the 62 0 cos sin | @ cos@ bitrary ing \ ! 8, tr X(R) also . Whenever denotes element depends R half R° on the is an ar- the o r d e r (depend- corresponding F-conjugacy to class only. Proof. As are not elliptic ~lell(Z) x(M) j M = x ( - M ) j _ M = fixed Z points Z (M £ F), we of have for all z 6 IH which F : tr X(S-IRS) JS_IRs (z) H(z, S-1RSz) ks(~(z, S-IRSz)) • {R} r S6Z (Rl\r 0<8<. The inner sum runs through right cosets ment R 6 {R} F .

I series (p. , ~T E(z,s;Vjp,Aj,k,x) will always be . _P_roposition. ) we adopt The zeroth Fourier ~V the following result. coefficient of the expansion of with respect Ejp(Z,s) to has the form I pjp,l(s)y 1-s + 6jl VlpY s , if mI > O if mI = O . Ujp,l(Y,S O Here Pjp,1 is a meromorphic least at the points {s E ~: Re s : ½} Moreover, nate , function of holomorphy (6jl: Kronecker the following Dirichlet of on ~ Ejp(Z, which is holomorphic ), especially at on the line symbol). ,Y ; p:1 ..... mj ; q=1 .....

The g r o u p by any Z(DN(p)) IR × [I,N(P o) [ is is a f u n d a m e n - Therefore f[ks( {z,Pz)) d {z) : f Fp ( iN

] , such class {P}F log N(P o) _< C3 • N(p)-Re C2,C 3 co f -oo s ( log N(P o) depending 1 iRe s 4N(P) (N(P)+t) 2 solely - i+t2/ , on s,k .