## Abstract

We prove the equivalence of three formulations of the Riemann hypothesis for functions f defined by the four assumptions: (a 1) f satisfies the functional equation f(1 - s) = f(s) for the complex argument s ≡ σ + iτ, (a2) f is free of any pole, (a3) for large positive values of σ the phase θ of f increases in a monotonic way without a bound as τ increases, and (a4) the zeros of f as well as of the first derivative f ′ of f are simple zeros. The three equivalent formulations are: (R1) All zeros of f are located on the critical line σ = 1/2, (R2) All lines of constant phase of f corresponding to merge with the critical line, and (R3) All points where f ′ vanishes are located on the critical line, and the phases of f at two consecutive zeros of f ′ differ by π. Our proof relies on the topology of the lines of constant phase of f dictated by complex analysis and the assumptions (a1)-(a4). Moreover, we show that (R2) implies (R1) even in the absence of (a4). In this case (a4) is a consequence of (R2).

Original language | English |
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Article number | 065201 |

Number of pages | 11 |

Journal | Physica Scripta |

Volume | 93 |

Issue number | 6 |

DOIs | |

Publication status | Published - Jun 2018 |