On quotients of Riemann zeta values at odd and even integer arguments

We show for even positive integers n   that the quotient of the Riemann zeta values ζ(n+1)ζ(n+1) and ζ(n)ζ(n) satisfies the equationζ(n+1)ζ(n)=(1−1n)(1−12n+1−1)L⋆(pn)pn′(0), where pn∈Z[x]pn∈Z[x] is a certain monic polynomial of degree n   and L⋆:C[x]→CL⋆:C[x]→C is a linear functional, which is connected with a special Dirichlet series. There exists the decomposition pn(x)=x(x+1)qn(x)pn(x)=x(x+1)qn(x). If n=p+1n=p+1 where p   is an odd prime, then qnqn is an Eisenstein polynomial and therefore irreducible over Z[x]Z[x].