### Abstract:

In 1936 Hecke proved a correspondence theorem between Dirichlet series
with functional equations and automorphic forms with certain growth
conditions. In this correspondence, the Dirichlet series has at most simple
poles at 0 and k: Since Hecke's original paper, many authors have
generalized the correspondence theorem in di_erent directions.
Among the generalizations, we shall be interested in is the one by Salomon
Bochner in 1951. In Bochner's version, the correspondence is between
automorphic integrals with _nite log{polynomial sum and Dirichlet
series with a functional equation. The most important feature of this
generalization is the presence of the log{polynomial sum. Here a logpolynomial
sum is a sum of the form q(z) = Pn l=1 z_l Pm(l)
j=0 _(l; j)(log z)j ;
the coe_cients, _(l; j) and _l are complex numbers, n; l; m(l); and j are
non negative integers. In Bochner's version, the Dirichlet series has a
pole of order m(l) + 1 at _l:
Austin Daughton in 2012, extended Bochner's result to the Dirichlet
series with in_nitely many poles for the theta group and for weight k _ 0:
This thesis will extend the Bochner correspondence theorem between
automorphic integrals with in_nite log-polynomial period function and
of _ weight arbitrary real k on Hecke group which is generated by S_ =
: The theta group correspond to _ = 2:
We will deal with the cases when _ > 2 and _ = 2 cos _=p; p 2 Z and
p _ 3: