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A. Juhl's history of generalised Selberg zeta functions This is an excerpt from section 8.2 ("Historical comments") of A. Juhl's book
"In the case of finite volume surfaces Selberg discovered the trace formula and the
related zeta function now bearing his name in the seminal paper [261]. Gelfand, Graev and
Piateski-Shapiro ([104]) discussed the trace formual for compact quotients of
The two classical volumes of Hejhal ([127], [128]) cover many aspects of the trace formula and the zeta function for finite volume surfaces. The methods used by Hejhal belong to the more classical framework, avoiding representation theory. The first volume of [275] is an introduction to the subject which covers its relations to a great variety of problems ranging from number theory to mathematical physics. For a role of the trace formula and the zeta function in quantum mechanics we refer to [150], [151] and the references there. Although already [261] contains some remarks on the higher-dimensional real hyperbolic case,
the Selberg zeta function Z (but not for
_{S}Z itself) and characterized its zeros and poles. Gangolli and Warner
extended these results in [103] also to the finite volume case.
_{S}In 1985 Wakayama [288] introduced even more general Selberg type zeta functions associated
to a locally homogeneous vector bundle on Also motivated by [261] Smale posed in section II.4 of his seminal paper [266] the problem of studying [a] zeta function for general classes of flows. In the case of the geodesic flow of a surface of constant negative curvature the latter zeta function coincides with Selberg's zeta function (as pointed out to Smale by Sinai and Langlands). Then in his 1976 paper [244] Ruelle published the prototype of a new argument for the existence of meromorphic continuations of the zeta functions for hyperbolic flows...with real analytic Anosov foliations on compact manifolds of arbitrary dimension. The basic ingredients of Ruelle's proof are the Markov decomposition of the underlying phase space, the associated symbolic dynamics, the Bowen-Manning countng lemma and a trace formula for the transfer operators. Later Fried ([93]) extended Ruelle's method to cover all generalized Selberg zeta functions. In particular, Fried gave alternative definitions for the zeta functions introduced by Gangolli and Wakayama in terms of the dynamics of a corresponding flow. These definitions emphasize the dynamical nature of the zeta functions. Now in 1988/89 we wrote a paper ([158]) in which we proved that the Selberg trace formula for
a compact quotient...can be reformulated in the form which now is the explicit form of the dynamical
Lefschetz formula (see Chapter 3). The method of [158] was to start with the explicit form of the
trace formula proved in [80] and to express the identity term in the trace formula in terms of
multiplicities...by using suitable index formulas for Dirac operators on The main motivation for doing this was the paper [112] in which Guillemin gave a formal argument for
regarding the Selberg trace formula for compact Riemann surfaces [Juhl continues for several pages, covering more recent work.]
references[80] J.J. Duistermaat and V. Guillemin, "The spectrum of positive elliptic operators and periodic
bicharacteristics", [93] D. Fried, "The zeta functions of Ruelle and Selberg I", [101] R. Gangolli, "Zeta functions of Selberg's type for compact space forms of symmetric spaces of rank
one", [103] R. Gangolli and G. Warner, "Zeta functions of Selberg's type for some non-compact quotients of symmetric
spaces of rank one", [104] I.M. Gelfand, M.I. Graev and I.I. Piatetskii-Shapiro, [112] V. Guillemin, "Lectures on the spectral theory of elliptic operators", [127] D.A. Hejhal, Lecture Notes in
Mathematics (Springer, 1976)
[128] D.A. Hejhal, Lecture Notes in
Mathematics (Springer, 1983)
[150] N. Hurt, [151] N. Hurt, "Zeta functions and periodic orbit theory: a review", [158] A. Juhl, "Quantized geodesic flow and the Selberg trace formula I", In [244] D. Ruelle, "Zeta function for expanding maps and Anosov Flows", [261] A. Selberg, "Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series", [266] S. Smale, "Differentiable dynamical systems", [275] A. Terras, [288] M. Wakayama, "Zeta functions of Selberg's type for non-compact quotient of Selberg trace formula and zeta functions page
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