Analytic Continuation of the Fibonacci Dirichlet Series

Functions defined by Dirichlet series J^=l a/f are Interesting because they often code and link properties of an algebraic nature in analytic terms. This is most often the case when the coefficients an are multiplicative arithmetic functions, such as the number or sum of the divisors of w, or group characters. Such series were the first to be studied, and are fundamental in many aspects of number theory. The most famous example of these is undoubtedly g(s) = S*=i n~ (Re(s) > 1), the Riemann zeta function. Initially studied by Euler, who wanted to know the values at the positive integers, it achieved prominence with Riemann, who clarified its intimate connection with the distribution of primes, and gave it lasting notoriety with his hypothesis about the location of its zeros. Another class of Dirichlet series arises in problems of Diophantine approximation, taking an to be the fractional part ofnO, where 0 is an irrational number. Their properties depend on how well one can approximate 9 by rational numbers, and how these fractional parts are distributed modulo 1. The latter is also a dynamical question about the Iterative behavior of the rotation by angle 0 of the unit circle. Such functions were defined and studied by Hardy and Littlewood in [3], and also by Hecke [5], Ostrowski and others. A Dirichlet series typically converges In a half-plane Re(s) > a0. The first step in retrieving the information contained In It Is to study Its possible analytic continuation. Even its existence is not usually something that can be deduced Immediately from the form of the coefficients, however simple their algebraic or analytic nature may be. For Instance, as Is well known, g(s) extends meromorphically to the whole complex plane, with only a simple pole at s = 1. In addition, it has an important symmetry around Re(s) = 1/2, in the form of a functional equation, a hallmark of many arithmetical Dirichlet series. It has "trivial" zeros at 2 , 4 , 6 , . . . , and its values at the negative odd Integers are rational, essentially given by the Bernoulli numbers. The Diophantine series described above also extend to meromorphic functions on C, but there Is no reason to expect a symmetric functional equation. Indeed their poles form the half of a lattice in the left half-plane. Other series, more fancifully defined, are likely not to extend at all. For instance, it is known that Y>p~~\ where p runs over the primes, cannot extend beyond any point on the imaginary axis, even though It Is formed from terms of T^=\n~ (Chandrasekharan's book [1] Is a nice Introduction to these arithmetical connections, whereas Hardy and RIeszs book [4] Is a good source for the more analytical aspects of the general theory of Dirichlet series). The function <p(s) we study In this paper, defined by the Dirichlet series HF~, where F„ is the w* Fibonacci number, shares properties with both types mentioned above. We will show that It extends to a meromorphic function on all of C and that It has, like the Riemann zeta function, "trivial zeros at -2, 6, -10,.. . . However, it has trivial simple poles at 0, 4, 8,.... Again like C(s), we show that at the odd negative Integers Its values are rational numbers, in this case