Control Theory and the Riemann Hypothesis: a Roadmap

All this seems to be started by Georg Friedrich Bernhard Riemann’s original paper whose English translation was included at least in [12], without, however forgetting the enormous influence of Leonhard Euler and his studies on the function now known as the Riemann zeta-function. After this, a huge number of papers is available in the internet on the Riemann hypothesis and the zeta-function. Furthermore, several books, classical, like [21] & [12], and new ones, like [14] & [7], have been published on the Riemann zeta-function. The official problem statement on the Riemann hypothesis, which claims that real parts of the complex zeros of the Riemann zeta-function all are 1 2 , is described in [3] & [19]. A disputed proposal for the proof has been presented [8] among many erroneous ones. But the only control-theoretic paper, which we have found, where the Riemann hypothesis and its relation to stability of a given dynamic control system is considered, is [18]. On the other hand several authors have considered dynamic systems from the spectral viewpoint relating them to the location of non-trivial, i.e. complex, zeros of the Riemann zeta-function, see e.g. [1] & [9]. There are several statements, which have been proved equivalent to the Riemann hypothesis, see a comprehensive list in [7]. Among them we mention the condition Λ ≤ 0 of the de Bruijin-Newman constant improved by Odlyzko Λ [17], Lagarias’ statement including harmonic sums [15], and Li’s condition on the positivity of a certain λn-sequence [7]. The condition, which has a direct connection to our studies concerns the Chebyschev function ψ(x) = ∑