In 1998, Don Zagier studied the ‘modified Bernoulli numbers’ $B_n^{*}$ whose 6-periodicity for odd $n$ naturally arose from his new proof of the Eichler-Selberg trace formula. These numbers satisfy amusing variants of the properties of the ordinary Bernoulli numbers. Recently, Victor H. Moll, Christophe Vignat and I studied an obvious generalization of the modified Bernoulli numbers, which we call ‘Zagier polynomials’. These polynomials are also rich in structure, and we have shown that a theory parallel to that of the ordinary Bernoulli polynomials exists. Zagier showed that his asymptotic formula for $B_{2n}^{*}$ can be replaced by an exact formula.

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