For p = 2, we have,
$\begin{align}&\sum_{n=1}^\infty[\zeta(pn)-1] = \frac{3}{4}\end{align}$
It seems there is a general form for odd p. For example, for p = 5, define $z_5 = e^{\pi i/5}$. Then,
$\begin{align} &5 \sum_{n=1}^\infty[\zeta(5n)-1] = 6+\gamma+z_5^{-1}\psi(z_5^{-1})+z_5\psi(z_5)+z_5^{-3}\psi(z_5^{-3})+z_5^{3}\psi(z_5^{3}) = 0.18976\dots \end{align}$
with the Euler-Mascheroni constant $\gamma$ and the digamma function $\psi(z)$.
- Anyone knows how to prove/disprove this?
- Also, how do we split $\psi(e^{\pi i/p})$ into its real and imaginary parts so as to express the above purely in real terms?
More details in my blog.