A finite series expression for infinite sums of powers of non-trivial zeros?

Question

This WolframMathworld-page, mentions:

$$Z(n) = \sum_{k=1}^{\infty} \left( \frac{1}{\rho_k^n} + \frac{1}{(1-\rho_k)^n}\right) \quad n \in \mathbb{N}$$

where $\rho_k$ is the $k$-th non-trivial zero of the Riemann $\zeta$-function.

The page also lists these first 6 finite series expressions for $Z(n)$:

I have been searching for a pattern and found through re-ordering the terms:

$Z(1)= 1 + \frac{\gamma}{2}- \frac{\ln(4\pi)}{2}$

$Z(2)= 1 + \gamma^2- \frac34\zeta(2)+\frac{2}{1}\gamma^0\gamma_1$

$Z(3)= 1 + \gamma^3- \frac78\zeta(3)+\frac{3}{1}\gamma^1\gamma_1+ \frac32\gamma^0\gamma_2 $

$Z(4)= 1 + \gamma^4- \frac{15}{16}\zeta(4)+\frac{4}{1}\gamma^2\gamma_1+\frac{4}{2}\gamma^1\gamma_2+\frac46\gamma^0\gamma_3 \qquad\qquad\qquad\qquad+ 2\gamma^0\gamma_1^2$

$Z(5)= 1 + \gamma^5- \frac{31}{32}\zeta(5)+ \frac{5}{1}\gamma^3\gamma_1+\frac{5}{2}\gamma^2\gamma_2+\frac{5}{6}\gamma^1\gamma_3+\frac{5}{24}\gamma^0\gamma_4 \qquad\qquad\,\,+ 5\gamma^1\gamma_1^2+\frac{5}{2}\gamma^0\gamma_1\gamma_2$

$Z(6)= 1 + \gamma^6- \frac{63}{64}\zeta(6)+ \frac{6}{1}\gamma^4\gamma_1+\frac{6}{2}\gamma^3\gamma_2+\frac{6}{6}\gamma^2\gamma_3+\frac{6}{24}\gamma^1\gamma_4+\frac{6}{120}\gamma^0\gamma_5 \,\,\,+6\gamma^1\gamma_1\gamma_2+ 9\gamma^2\gamma_1^2+2\gamma^0\gamma_1^3+\gamma_1\gamma_3+\frac34\gamma^0\gamma_2^2$

where the left part could be simplified into: $$1+\gamma^{n}-{\frac { \left( {2}^{n}-1 \right)}{{2}^{n}}\,\zeta(n)}+n\sum _{k=1}^{n-1}{\frac {\gamma^{n-k-1}\gamma \left( k \right) }{\Gamma \left( k+1 \right) }}$$

however, I struggle to find a pattern in the remaining terms in the right part.

Q1: Does anybody know whether a full finite series expression exist in the literature?

Q2: Are there expressions for $Z(>6)$ available somewhere in the literature?

ADDED:

Thanks to the references provided in the answers below (esp. eq. 47 in Keiper's paper), I managed to derive this recurrence relation (note I use the more commonly used $\sigma_k$ instead of $Z(k)$):

Set:

$\sigma_1 = 1 + \frac{\gamma}{2}- \frac{\ln(4\pi)}{2}$

and for $k>1, k \in \mathbb{N}$:

$$\sigma_k=1+\left(\frac{1}{2^k}-1 \right )\zeta(k)+\frac{\gamma\,\gamma_{k-2}}{\Gamma(k-1)}+\frac{k\,\gamma_{k-1}}{\Gamma(k)}-\sum_{j=1}^{k-2}\frac{\gamma_{j-1}}{\Gamma(j)}\,\left( 1+\left(\frac{1}{2^{k-j}}-1\right)\zeta(k-j)-\sigma_{k-j}\right)$$

and this perfectly generates $\sigma_7, \sigma_8, \cdots$ in terms of a finite series of Stieltjes constants :-)

Answer 1

Your question is odd. $Z$ is not the usual Hardy $Z$-function.

Are you aware that $$\frac{\zeta'(s)}{\zeta(s)} = C-\frac1{s-1}+\sum_\rho (\frac1{s-\rho}+\frac1{\rho})$$ where $\rho$ runs over the trivial and non-trivial zeros (repeated with multiplicity) ? Differentiating $n-1$ times and letting $s=1$ you get some expressions in term of finitely many coefficients of the power series at $s=1$ of $\frac{\zeta'(s)}{\zeta(s)}+\frac1{s-1}$ thus of $\zeta(s)$.

In your question you are considering a sum over the non-trivial zeros, thus you get an aditionnal $\zeta(n)$ term corresponding to the sum over the trivial zeros.

Answer 2

The page you cite has references. The references for the table you reproduce are

• Lehmer, D. H. "The Sum of Like Powers of the Zeros of the Riemann Zeta Function." Math. Comput. 50, 265-273, 1988.
• Keiper, J. B. "Power Series Expansions of Riemann's xi Function." Math. Comput. 58, 765-773, 1992.
• Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, p. 168, 2003.
• Bombieri, E. and Lagarias, J. C. "Complements to Li's Criterion for the Riemann Hypothesis." J. Number Th. 77, 274-287, 1999.

Lehmer writes

These two numbers are not connected in any obvious way with any other known constants. Their continued fractions show no radical departure from the norm.

So he did not see the patterns you describe at that time. Finch describes forms similar to the ones you reproduce.

McPhedran ("Sum Rules for Functions of the Riemann ZetaType", arXiv:1801.07415v2) writes the sum of reciprocal powers of roots of a functions in a general class including the zeta function in terms of derivatives of the logarithm of the function, evaluated at the origin (eqn. (6), there).

A reference that packages using a recursion to get the sequence of sums of negative integer powers of the roots of the Riemann zeta function together with the power series expansion of the $\xi$ function to get expressions in terms of the various constants you list is Bagdasaryanab et al. ("Analogues of Newton–Girard power-sum formulas for entire and meromorphic functions with applications to the Riemann zeta function", https://doi.org/10.1016/j.jnt.2014.07.006 ).