\( \eta(s) = (1-2^{1-s}) \zeta(s) \)
general form, for \( p \in \mathbb{N} \)
\( \zeta(s) = \dfrac{1}{1^s} + \dfrac{1}{2^s} + \dfrac{1}{3^s} + ... \)
\( \dfrac{p}{p^s} \zeta(s) = \dfrac{p}{p^s} + \dfrac{p}{{(2p)}^s} + \dfrac{p}{{(3p)}^s} + ... \)
for \( p=3 \)
\( \zeta(s) - \dfrac{p}{p^s} \zeta(s) = \dfrac{1}{1^s} + \dfrac{1}{2^s} - \dfrac{2}{3^s} + \dfrac{1}{4^s} + \dfrac{1}{5^s} - \dfrac{2}{6^s}... = \rho(s)\)
\( (1 - p^{1-s}) \zeta(s) = \rho(s) \)
\( \zeta(s) = \dfrac{1}{1-p^{1-s}} \rho(s) \)
\( \rho(s,p) = (1 - p^{1-s}) \zeta(s) \)
\( \rho(s,q) = (1 - q^{1-s}) \zeta(s) \)
\( \rho(s,p) - \rho(s,q) = (q^{1-s} - p^{1-s}) \zeta(s) \)
for \( p=7, q=11 \)
\( \rho(s,7) - \rho(s,11) = - \dfrac{7}{7^s} + \dfrac{11}{11^s} - \dfrac{7}{14^s} - \dfrac{7}{21^s} + \dfrac{11}{22^s} - \dfrac{7}{28^s} + \dfrac{11}{33^s}... + \dfrac{4}{77^s}...\)
\( \rho(z,p) - \rho(z,q) = (q^{1-z} - p^{1-z}) \zeta(z) \)
and for \( s=1, p=3 \)
\( \rho(1,3) = \dfrac{1}{1} + \dfrac{1}{2} - \dfrac{2}{3} + \dfrac{1}{4} + \dfrac{1}{5} - \dfrac{2}{6}... = ln(3) \)
\( \rho(1,p) = ln(p) \)
