Abstract.
This is the first of four papers that study algebraic and
analytic structures associated to the Lerch zeta function.
This paper studies “zeta integrals” associated to the Lerch zeta function
using test functions, and obtains functional equations for them. Special cases include
a pair of symmetrized four-term functional equations
for combinations of Lerch zeta functions,
found by A. Weil, for real parameters 
(a,c) with 
0<a,c<1.
It extends these functions to real 
a and 
c,
and studies limiting cases of these functions where at least one of 
a and 
c
take the values 0 or 1. A main feature is that
as a function of three variables 
(s,a,c), in which 
a and 
c are real,
the Lerch zeta function has discontinuities at integer values of
a and 
c. For fixed 
s,
the function 
(s,a,c) is discontinuous on part of the boundary of the
closed unit square in the 
(a,c)-variables, and the
location and nature of these discontinuities depend on
the real part 
(s) of 
s. Analysis of this behavior is used to determine
membership of these functions in 
L p ([0,1] 2 ,dadc)
for 
1p<, as a function of 
(s). The paper also defines
generalized Lerch zeta functions associated to the oscillator representation,
and gives analogous four-term functional equations for them.
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