  A high-order accurate quadrature rule for the discretization of boundary
integral equations (BIEs) on closed smooth contours in the plane is introduced.
This quadrature can be viewed as a hybrid of the spectral quadrature of Kress
(1991) and the locally corrected trapezoidal quadrature of Kapur and Rokhlin
(1997). The new technique combines the strengths of both methods, and attains
high-order convergence, numerical stability, ease of implementation, and
compatibility with the "fast" algorithms (such as the Fast Multipole Method or
Fast Direct Solvers). Important connections between the punctured trapezoidal
rule and the Riemann zeta function are introduced, which enable a complete
convergence analysis and lead to remarkably simple procedures for constructing
the quadrature corrections. The paper reports a detailed comparison between the
new method and the methods of Kress, of Kapur and Rokhlin, and of Alpert
(1999).
