Local Extrema of the Ξ(t) Function and The Riemann Hypothesis
نویسنده
چکیده
In the present paper we obtain a necessary and sufficient condition to prove the Riemann hypothesis in terms of certain properties of local extrema of the function Ξ(t) = ξ( 1 2 + it). First, we prove that positivity of all local maxima and negativity of all local minima of Ξ(t) form a necessary condition for the Riemann hypothesis to be true. After showing that any extremum point of Ξ(t) is a saddle point of the function R{ξ(s)}, we prove that the above properties of local extrema of Ξ(t) are also a sufficient condition for the Riemann hypothesis to hold at t ≫ 1. We present a numerical example to illustrate our approach towards a possible proof of the Riemann hypothesis. Thus, the task of proving the Riemann hypothesis is reduced to the one of showing the above properties of local extrema of Ξ(t).
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