General infinite series evaluations involving Fibonacci numbers and the Riemann zeta function

Анотація

The purpose of this paper is to present closed forms for various types of infinite series
involving Fibonacci (Lucas) numbers and the Riemann zeta function at integer arguments.
To prove our results, we will apply some conventional arguments and combine the Binet formulas
for these sequences with generating functions involving the Riemann zeta function and some known series evaluations.
Among the results derived in this paper, we will establish that

$\displaystyle
\sum_{k=1}^\infty (\zeta(2k+1)-1) F_{2k} = \frac{1}{2},\quad
\sum_{k=1}^\infty (\zeta(2k+1)-1) \frac{L_{2k+1}}{2k+1} = 1-\gamma,$

where $\gamma$ is the familiar Euler-Mascheroni constant.