  We introduce a new type of multiple zeta functions, which we call bilateral
zeta functions, analogous to the Barnes zeta functions. The bilateral zeta
function is a periodic function and shares certain basic properties of Barnes
zeta function. Especially, we prove that the bilateral zeta function has a nice
Fourier series expansion and the Barnes zeta function can be expressed as a
finite sum of bilateral zeta functions. By these properties of the bilateral
zeta functions, We obtain simple proofs of some formulas, for example the
reflection formula for the multiple gamma function, the inversion formula of
the Dedekind eta function, Ramanujan's formula, Fourier expansion of the Barnes
zeta function and multiple Iseki's formula.
