Difference between revisions of "Fibonacci zeta function"
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$$F(s)=\displaystyle\sum_{k=1}^{\infty} f_n^{-s},$$ | $$F(s)=\displaystyle\sum_{k=1}^{\infty} f_n^{-s},$$ | ||
where $f_n$ denotes the $n$th term in the [[Fibonacci sequence]]. | where $f_n$ denotes the $n$th term in the [[Fibonacci sequence]]. | ||
| + | |||
| + | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The number $F(1)$ is an [[irrational number]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The number $F(2k)$ is a [[transcendental number]] for all $k=1,2,3,\ldots$. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
=References= | =References= | ||
[http://www.mast.queensu.ca/~murty/fibon-tifr.pdf] | [http://www.mast.queensu.ca/~murty/fibon-tifr.pdf] | ||
Revision as of 18:20, 25 July 2015
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: █
