Abstract
This chapter discusses periodic Bernoulli numbers, summation formulas and applications. Periodic Bernoulli numbers appear in periodic analogues of the Euler–Maclaurin summation formula. The periodic Euler–Maclaurin formula can be derived from the periodic Poisson summation formula. The chapter describes a few of the most important properties of periodic Bernoulli numbers and periodic Bernoulli polynomials. The periodic Poisson and Euler-Maclaurin summation formulas have proven to be enormously useful, especially in applications to number theory. It also discusses trigonometric series of Hardy and Littlewood.
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