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Advances in Pure Mathematics 2016

Approach to a Proof of the Riemann Hypothesis by the Second Mean-Value Theorem of Calculus

DOI: 10.4236/apm.2016.613074, PP. 972-1021

Alfred Wünsche

Keywords: Riemann Hypothesis, Riemann Zeta Function, Xi Function, Gauss-Bonnet Theorem, Mellin Transformation

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Abstract:

By the second mean-value theorem of calculus (Gauss-Bonnet theorem) we prove that the class of functions $\"\"$ with an integral representation of the form $\"\"$ with a real-valued function $\"\"$ which is non-increasing and decreases in infinity more rapidly than any exponential functions $\"\"$ , $\"\"$ possesses zeros only on the imaginary axis. The Riemann zeta function $\"\"$ as it is known can be related to an entire function $\"\"$ with the same non-trivial zeros as . Then after a trivial argument displacement $\"\"$ we relate it to a function $\"\"$ with a representation of the form $\"\"$ where $\"\"$ is rapidly decreasing in infinity and satisfies all requirements necessary for the given proof of the position of its zeros on the imaginary axis z=iy

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