Fibonacci zeta function

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

Properties

Fibonacci zeta at 1 is irrational
Fibonacci zeta is transcendental at positive even integers
Fibonacci zeta in terms of a sum of binomial coefficients

References

The Fibonacci zeta function by M. Ram Murty
[1]