Some Arithmetical Properties of the Generating Power Series for the Sequence {ζ(2k+1)}_k=1 ^∞

Abstract

Let f_odd(z):= ∑^∞ _k=1ζ(2k + 1)z^2k be the power series with the values of the Riemann ζ function at odd integers as coefficients. This function can be analytically continued to a meromorphic function over C. We prove that 1 and the values of f_odd at rational points with relatively prime denominators are linearly independent over ―Q. Some arithmetical properties of the sequence {ζ(2k+1)} ^∞_k=1 are deduced.