The generalized Bernoulli numbers and its relation with the Riemann zeta function at odd-integer arguments

By using the generalized Bernoulli numbers, we deduce new integral representations for the Riemann zeta function at positive odd-integer arguments. The explicit expressions enable us to obtain criteria for the dimension of the vector space spanned over the rational by the $\zeta (2n+1)\mathrm{/}{\pi }^{2n}$, $n\ge 1$.