The dirichlet series for the Vonmangoldt function, $\Lambda(n)$, which is equal to zero when $n$ is not a prime a power, and $\ln(p)$ when it is a prime power say, $n=p^j$, is
$-\frac{\zeta'(s)}{\zeta(s)}=\sum_{k=1}^{\infty}\frac{\Lambda(k)}{k^s}$
Where $\zeta(s)$ is the zeta function, and $\zeta'(s)$ is the derivative of the zeta function with respect to $s$,
This can be re-written as $-\frac{\zeta'(s)}{\zeta(s)}=\sum_{k=1}^{\infty}\frac{\Lambda(k)}{k^s}=\sum_{p}\frac{\ln(p)}{p^s-1},$ with the last sum ranging over all primes p,
Can someone help me brake this sum, up into prime congruence sums similarly $\sum_{k=0}^{\infty}\frac{\Lambda(5k+1)}{(5k+1)^s}$, ie re-write it as prime sums, where the primes are congruent to some b modulo $5$. I have done it before modulo $4$, and $3$ so I know it can be done, I am just having trouble restricting the powers appropietly to account for cases when $p^a\equiv1$ mod $5$, has no solutions.