Approach to a Proof of the Riemann Hypothesis by the Second Mean-Value Theorem of Calculus - Advances in Pure Mathematics

APM > Vol.6 No.13, December 2016

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

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Download as PDF (Size: 3433KB) PP. 972-1021

DOI: 10.4236/apm.2016.613074 2,118 Downloads 4,347 Views Citations

Author(s)

Alfred Wünsche

Affiliation(s)

Humboldt-Universitüt Berlin, Institut für Physik, Nichtklassische Strahlung (MPG), Berlin, Germany.

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 by the second mean-value theorem. Besides this theorem we apply the Cauchy-Riemann differential equation in an integrated operator form derived in the Appendix B. All this means that we prove a theorem for zeros of

on the imaginary axis z=iy for a whole class of function

which includes in this way the proof of the Riemann hypothesis. This whole class includes, in particular, also the modified Bessel functions

for which it is known that their zeros lie on the imaginary axis and which affirms our conclusions that we intend to publish at another place. In the same way a class of almost-periodic functions to piece-wise constant non-increasing functions

belong also to this case. At the end we give shortly an equivalent way of a more formal description of the obtained results using the Mellin transform of functions with its variable substituted by an operator.

KEYWORDS

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

Share and Cite:

Wünsche, A. (2016) Approach to a Proof of the Riemann Hypothesis by the Second Mean-Value Theorem of Calculus. Advances in Pure Mathematics, 6, 972-1021. doi: 10.4236/apm.2016.613074.

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