Introduction:
In this 1992 paper, J.B. Keiper (an amazing person, who tragically died way too young), derives several power series expansions of the Riemann $\xi$-function that involve infinite sums of powers of the non-trivial zeros $\rho$ (taken in pairs).
$$\sigma_k= \sum_{\rho} \frac{1}{\rho^k}$$
Analogy:
We know that the non-trivial zeros $\rho$ of $\zeta(s)$ drive the oscillating (error) term in the prime counting function(s). Analogously, the zeros $\mu = 0 \pm 2\pi n i$ induce the oscillating (sawtooth) behaviour in the integer counting function. The entire function associated with the $\mu$'s is $\xi_i(s)=\frac{2}{s}\sinh\left(\frac{s}{2}\right)$. Now also define ($\mu$'s taken in pairs):
$$\hat{\sigma}_k= \sum_{\mu} \frac{1}{\mu^k} = \frac{\zeta(1-k)}{\Gamma(k)}$$
Comparison:
I found that all the expansions for $\xi(s)$ from Keiper's paper can be replicated for $\xi_i(s)$ and the results/relations are fully equivalent except for these three:
\begin{align} \sum_{k=1}^\infty \frac{\sigma_k}{k} &=0 \qquad \tag{Kei 16}\\ \sigma_1 &= -\sum_{k=1}^\infty \sigma_k \qquad \tag{Kei 17}\\ \sigma_{j+1} &= (-1)^{j+1} \sum_{k=1}^\infty \binom{k-1}{j}\sigma_k \qquad \tag{Kei 18} \end{align} where in the latter series a $\sigma$ could be derived recursively. This recursion disappears in the equivalent series with the $\hat{\sigma}$'s: \begin{align} \sum_{k=1}^\infty \frac{\hat{\sigma}_k}{k} &=\ln\left(\frac{1}{{\rm e}-1}\right) \\ \frac{1}{{\rm e}-1} &= -\sum_{k=1}^\infty \hat{\sigma}_k \\ -1 + \frac{1}{\Gamma(j+1)} \sum_{k=0}^{j} \Gamma(j+1-k){j+1 \brace j+1-k} ({\rm e}-1)^{k-j-1} &= (-1)^{j+1} \sum_{k=1}^\infty \binom{k-1}{j}\hat{\sigma}_k \\ \end{align} where ${a \brace b}$ is a Stirling number of the second kind. Question:
Where could the differences in recursion between these series of $\sigma$ and $\hat{\sigma}$ originate from? I could conceive of two possible explanations:
- the functional equation $\xi(s)=\xi(1-s)$ versus $\xi_i(s) = \xi_i(-s)$ or
- some connection in the recursion of $\sigma$ to the fundamental theorem of arithmetic?
Or maybe there is a simpler rule at work here? Appreciate any ideas.
