Riemann Zeta Function

\( \zeta (s)  = \dfrac{1}{1-2^{-s}} . \dfrac{1}{1-3^{-s}} . \dfrac{1}{1-5^{-s}} ... \dfrac{1}{1-p^{-s}} ...\)

\( \eta (s)  = (1-2^{1-s}) \zeta (s) , 0 < s < 1\)

\( \eta (s)  = \dfrac{1-2^{1-s}}{1-2^{-s}} . \dfrac{1}{1-3^{-s}} . \dfrac{1}{1-5^{-s}} ... \dfrac{1}{1-p^{-s}} ...\)

\( \rho (s,p)  = (1-p^{1-s}) \zeta (s)\)

\( \rho (s,q)  = (1-q^{1-s}) \zeta (s)\)

\( \rho (s,p) - \rho (s,q)  = (q^{1-s}-p^{1-s}) \zeta (s)\)

\( p=7, q=11 \)

\( \rho(s,7) - \rho(s,11) =- \dfrac{7}{7^s} + \dfrac{11}{11^s} -\dfrac{7}{14^s} - \dfrac{7}{21^s} + \dfrac{11}{22^s} - \dfrac{7}{28^s} + \dfrac{11}{33^s}... + \dfrac{4}{77^s}...\) calculate to \( m \) terms, \( m=p*q*k \) \( \rho (s,p) - \rho (s,q)  = q\sum\limits_{n=1}^{pk}{\dfrac{1}{(nq)^s}} - p\sum\limits_{n=1}^{qk}{\dfrac{1}{(np)^s}}\)