when Thus is periodic, with period 1, in each element of . More generally,
with , . This is the quasi-periodicity property of the Riemann theta function. It determines the Riemann theta function up to a constant factor. The set of points form a -dimensional lattice, the period lattice of the Riemann theta function.
Again, with ,
Because of this property, the elements of and are usually restricted to , without loss of generality.
For Riemann theta functions with half-period characteristics,
See also §20.2(iii) for the case and classical theta functions.