Some unusual identities for special values of the Riemann zeta function (Q5952313)

The argument of this paper is based on the arithmetical functions \(\{ a_j \mid j\in \mathbb N \}\) defined by \[ a_{j}(n)= \sum_{m (\bmod n)} \cos\left({2\pi m(m,n)^{j-1}\over n}\right). \] It is shown that for each \(j\in \mathbb N\), \(a_{j}(n)\) is multiplicative, \(a_{j}(p^{\alpha}) = p^{b(j-1)+r}\) if \(\alpha - 1 = bj+r\) with \(0<r<j\) and \(a_{j}(p^{\alpha}) = 0\) if \(\alpha \equiv 1 (\bmod j)\). From these properties of the \(a_j(n)\)'s there quickly follow identities such as \[ \sum_{n\in \mathbb N}{a_{2}(n)^{x}\over n^{s}} = \zeta(2s-x), \quad \sum_{n\in \mathbb N}{a_{3}(n^{2})^{x}\over n^{s}} = {\zeta(s-x-1) \zeta(3s-4x-3)\over \zeta(2s-2x-2)} , \] \[ \sum_{n\in\mathbb N}{a_{2}(n)^{x} a_{3}(n)^{y} a_{6}(n)^{z}\over n^s } = {\zeta(2s-x-y-z)\zeta(6s-3x-4y-5z)\over \zeta(4s-2x-2y-2z)}, \] where \(x, y, z \in \mathbb R^{+}\) and \(s\in \mathbb C\) sufficiently large so that the Dirichlet series for the zeta-function are absolutely convergent at the arguments that occur. These identities in turn imply various representations (involving the Bernoulli numbers and \(a_{j}(n)\)'s) of \(\zeta(k)\) for every integer \(k \geq 2\), such as \[ \zeta(k) = 2^{k-{1\over 2}}\pi^{k}\sqrt{{B_{2k}\over (2k)!}\sum_{n\in \mathbb N} {a_{3}(n)^{k}\over n^{k}}}. \] By this method it is possible to obtain many identities in addition to those included in the paper.