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A Sequence of Polynomials Related to the Evaluation of the Riemann Zeta Function

Euler was the first to evaluate the sum of the series for even values of s. Before 1740 he had obtained the values of ζ(2) and ζ(4), and in his Introductio in analysin infinitorum (1748) he explicitly gave the values for even s up to s = 26. Later he discovered the well-known formula valid for all even s in terms of the so-called Bernoulli numbers. (See [2] for an account of Euler’s work.) After Euler’s original work many other methods have been given to compute ζ(2k) for integer k. Several of them have in common the use of moments of trigonometric functions (definite integrals of xk cos nx or xk sin nx). We will modify that approach by defining a sequence of interpolating polynomials adapted in a natural way to the evaluation and express the sum ζ(2k) in terms of the values of the polynomials at a point. There is a recursive formula giving these values without using the polynomials themselves. Using the derivatives of these polynomials we obtain integral formulas for ζ(2k + 1). This is achieved using only elementary calculus: trigonometric identities, integration by parts and the differentiability of some particular functions. Writing the Fourier series expansion of our sequence of polynomials we will recognize them as being essentially the Euler polynomials.