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By Dennis A. Hejhal (auth.)
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Extra resources for The Selberg Trace Formula for PSL(2,ℝ) Volume I
He has often stated (in conversations) that his discovery was motivated by Maass[l] and by the classical theory of automorphic forms. The idea of taking the trace seemed quite natural, since it looked like it would be too difficult to get hold of the individual eigenfunctlons. Cf. 177(paragraph 2)]. It is easy to understand why Selberg studied trace formulas so intensively: they bear a very striking resemblance to the so-called explicit formulas of prime number theory. Cf. Well[l]. (**) I Briefly stated, one has: = - + W- where the non-trlvial zeros of the Riemann zeta function --i+ i T 2 The Riemann Hypothesis is equivalent to the fact that the rlght-hand side ('~ E C).
1/2 < h(r) The f u n c t i o n Im(r) - , we must therefore To get around these problems, we use the following trick. 2 of chapter I is To ensure regularity in Re(W) Re(~ ) > ~ I [Im(r)~ = 1/2 + ~ I . ~i ( 'a v~+/~~ = ~. ,- r~ i ~ i. ~p two s e r i e s Putting ~' = s - 112 ~=0 are convergent wlth rJ,(,-~) ~ [using Re(s) ~ i N(P) = m ( P (~'r) dr I ~(P)" ~ o o ( X ) = O(x) ] provided Re('r ) ~'1/2. gXves r~,/~" ) > ] r ~(r} "~- HU,)-'/a ~(r)A l'Pl t~x- + Since i r~+,l~ r%'~ = -- The last ' -~ I ~ Sir) 'a- I~(r) "va Sty)'''~' 1 , we c a n w r i t e i.
35 NOTES FOR CHAPTER ONE Part A (general remarks). 5). Since we are dealing only with PSL(2,~) , there is no need to enter Into a preliminary discussion of Riemannlan symmetric spaces. Readers interested in symmetric spaces are referred to Helgason[l] and Selberg[l,2]. Selberg was first led to the L 2 ( ~ \H) trace formula around 1950-51. He has often stated (in conversations) that his discovery was motivated by Maass[l] and by the classical theory of automorphic forms. The idea of taking the trace seemed quite natural, since it looked like it would be too difficult to get hold of the individual eigenfunctlons.
The Selberg Trace Formula for PSL(2,ℝ) Volume I by Dennis A. Hejhal (auth.)