Fibonacci zeta function

The Fibonacci zeta function is defined by $$F(s)=\displaystyle\sum_{k=1}^{\infty} f_n^{-s},$$ where $f_n$ denotes the $n$th term in the Fibonacci sequence.

Properties

Theorem: The number $F(1)$ is an irrational number.

Proof: █

Theorem: The number $F(2k)$ is a transcendental number for all $k=1,2,3,\ldots$.

Proof: █

References

The Fibonacci zeta function by M. Ram Murty
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