A Hecke Correspondence For Automorphic Integrals With Innitely Many Poles

In 1936 Hecke proved a correspondence theorem between Dirichlet seriesrnwith functional equations and automorphic forms with certain growthrnconditions. In this correspondence, the Dirichlet series has at most simplernpoles at 0 and k: Since Hecke's original paper, many authors haverngeneralized the correspondence theorem in di_erent directions.rnAmong the generalizations, we shall be interested in is the one by SalomonrnBochner in 1951. In Bochner's version, the correspondence is betweenrnautomorphic integrals with _nite log{polynomial sum and Dirichletrnseries with a functional equation. The most important feature of thisrngeneralization is the presence of the log{polynomial sum. Here a logpolynomialrnsum is a sum of the form q(z) = Pn l=1 z_l Pm(l)rnj=0 _(l; j)(log z)j ;rnthe coe_cients, _(l; j) and _l are complex numbers, n; l; m(l); and j arernnon negative integers. In Bochner's version, the Dirichlet series has arnpole of order m(l) + 1 at _l:rnAustin Daughton in 2012, extended Bochner's result to the Dirichletrnseries with in_nitely many poles for the theta group and for weight k _ 0:rnThis thesis will extend the Bochner correspondence theorem betweenrnautomorphic integrals with in_nite log-polynomial period function andrnof _ weight arbitrary real k on Hecke group which is generated by S_ =rn: The theta group correspond to _ = 2:rnWe will deal with the cases when _ > 2 and _ = 2 cos _=p; p 2 Z andrnp _ 3:

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