I recently gave a talk "at Dartmouth"1 1i.e. over zoom, hosted and organized by Dartmouth . The focus of the talk was the (odd-indexed) Fibonacci zeta function: $$ \sum_{n \geq 1} \frac{1}{F(2n-1)^s},$$ where $F(n)$ is the nth Fibonacci number. The theme is that the Fibonacci zeta function can be recognized as coming from an inner product of automorphic forms, and the continuation of the zeta function can be understood in terms of the spectral expansion of the associated automorphic forms.
This is a talk from ongoing research. I do not yet understand "what's really going on". But within the talk I describe a few different generalizations; firstly, there is a generalization to other zeta functions that can be viewed as traces of units on quadratic number fields, and secondly there is a generalization to quadratic forms recognizing solutions to Pell's equation.
I intend to describe additional ideas from this talk in the coming months, as I figure out how pieces fit together. But for now, here are the slides.
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