Primes as sums of Fibonacci numbers

Abstract: The purpose of this paper is to discuss the relationship between prime numbers and sums of Fibonacci numbers. One of our main results says that for every sufficiently large integer $k$ there exists a prime number that can be represented as the sum of $k$ different and non-consecutive Fibonacci numbers. This property is closely related to, and based on, a prime number theorem for certain morphic sequences. The proof of such a prime number theorem, combined with a corresponding local result, is the central contribution of this paper, from which we derive the result stated in the beginning. Problems of this type have been discussed intensively in the context of the base-$q$ expansion. The Gelfond problems (1968/1969), and the Sarnak conjecture, were the driving forces of this development. Mauduit and Rivat resolved the question on the sum of digits of prime numbers (2010) and the sum of digits of squares (2009), thus leaving open only part of the third Gelfond problem. Later the second author (2017) proved Sarnak's conjecture for all automatic sequences, which are based on the q-ary expansion of integers, and which generalize the sum-of-digits function in base $q$ considerably. In order to obtain corresponding results for Fibonacci numbers, we have to extend Mauduit and Rivat's method considerably. In fact, we are departing significantly from this method, proving the statement that $\exp(2\pi i \vartheta\mathsf z(n))$ has \emph{level of distribution} $1$ (here $\mathsf z(n)$ is the number of Fibonacci numbers needed to write $n$ as their sum). This latter result forms an essential part of our treatment of the occurring sums of type $\textrm I$ and $\textrm{II}$ and uses Gowers norms related to $\mathsf z(n)$ as a central technical tool. The appearance of Gowers norms in our method is intimately tied to the iterated application of a new generalization of van der Corput's inequality.