Equivalent formulations of the Riemann hypothesis based on lines of constant phase

W. P. Schleich, I. Bezděková, M. B. Kim, P. C. Abbott, H. Maier, H. L. Montgomery, J. W. Neuberger

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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 languageEnglish
Article number065201
Number of pages11
JournalPhysica Scripta
Issue number6
Publication statusPublished - Jun 2018

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