This appears to be a relationship:
$\sum\limits_{p\;\text{prime}} \frac{1}{p^s} = \log\zeta (s) - \sum\limits_{n=1}^{\infty}\frac{\sqrt{a_{n}b_{n}}}{n^{s}}$
where $a_{n}$ is a sequence of fractions beginning $\frac{0}{1}, \frac{1}{1}, \frac{1}{1}, \frac{1}{2}, \frac{1}{1}, \frac{0}{1}, \frac{1}{1}, \frac{1}{3}, \frac{1}{2}, \frac{0}{1}, \frac{1}{1}, \frac{0}{1}...$ and $b_{n}$ is another sequence of fractions $\frac{0}{1}, \frac{0}{1}, \frac{0}{1}, \frac{1}{2}, \frac{0}{1}, \frac{1}{1}, \frac{0}{1}, \frac{1}{3}, \frac{1}{2}, \frac{1}{1}, \frac{0}{1}, \frac{0}{1}...$
for which the Dirichlet series are defined by:
$\sum\limits_{n=1}^{\infty}\frac{a_{n}}{n^{s}} = +\frac{1}{1}(\zeta (s)-1)^1 -\frac{1}{2}(\zeta (s)-1)^2 +\frac{1}{3}(\zeta (s)-1)^3 - \frac{1}{4}(\zeta (s)-1)^4 +\frac{1}{5}(\zeta (s)-1)^5 -...$
$\sum\limits_{n=1}^{\infty}\frac{b_{n}}{n^{s}} = -\frac{0}{1}(\zeta (s)-1)^1 +\frac{1}{2}(\zeta (s)-1)^2 -\frac{2}{3}(\zeta (s)-1)^3 + \frac{3}{4}(\zeta (s)-1)^4 -\frac{4}{5}(\zeta (s)-1)^5 +...$
The square root of the elementwise multiplication of $a_{n}$ times $b_{n}$,$\;$ $\sqrt{a_{n}b_{n}}$ is then: $\frac{0}{1},\frac{0}{1},\frac{0}{1},\frac{1}{2},\frac{0}{1},\frac{0}{1},\frac{0}{1},\frac{1}{3},\frac{1}{2},\frac{0}{1},\frac{0}{1},\frac{0}{1}...$
Is there any literature describing multiplication of coefficients of Dirichlet series?