2-transitive abstract ovals of odd order by Korchmaros G.

By Korchmaros G.

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54) n-{h) = - h{iK)d\ogZx{^-^K). 52). {K)dlogZx{^ + f^)- Jo On a deja remarque que Q(h) est la somme des quatre integrales (2 —2^)J+(/i),... ,7i~{h). 55). D. 9. Ajoutons quelques remarques concemant demonstration. le theoreme 1 et sa LA FORMULE DES TRACES DE SELBERG 49 a) L'hypothese que ^ n'est pas valeur prop re de —Ax n'intervient que pour assurer que h{—iK)dlog D{K) est holomorphe pour K = 0. Dans le cas ou ^ serait valeur propre, le resultat est encore valable a condition de supposer qu'on a h{0) = 0.

Soit P un operateur pseudo-difFerentiel d'ordre 1, ayant meme symbole principal que la racine carree (positive) de — A M ; c'est un operateur elliptique, et Ton suppose qu'il n'a qu'un nombre fini de valeurs propres non reelles. 1) ^(A)-CA^ pour A-^+oo. 2) n ( r ) = rr(cos rP) = Y^ cos rpn, n ou pn parcourt la suite des valeurs propres de P , multiplicites comprises. En terme de fonctionnelle, on a + 00 / ft(r) cos rpn dr • « ^ pour toute fonction-test ^ ( r ) . 5) Tl{h) = ^ ( / i ( p „ ) + hi-p„))/2.

Soit L un operateur ayant un ensemble discret S de valeurs propres cr, avec une multiplicite finie m{a) pour chaque a. 40) OL{t)= ]^m((7)e-^' aeS converge absolument. 41) OL^I)^ Y. i-irdrnt"^ pour t-^0. m > —mo Ces deux dernieres hypotheses ne sont pas independantes. On pent alors definir le determinant D{X) = det(L+A) pour tout nombre complexe A. De plus, i^(A) est une fonction entiere de la variable complexe A; ses zeros sont les nombres —or, pour a parcourant 5, avec multiplicite egale a m(cr).

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