In algebraic number theory, the Schwartz–Bruhat functions on the adeles can be used to give an adelic version of the Poisson summation formula from analysis, i.e., for every  one has , where . John Tate developed this formula in his doctoral thesis to prove a more general version of the functional equation for the Riemann zeta function.
Schwartz–Bruhat functions on adeles can be used to provide an adelic form of the Poisson summation formula from analysis, which is equivalent to stating that for every  one has , where . John Tate developed this formula in his doctoral thesis to prove a more general version of the functional equation for the Riemann zeta function.