TITLE:
Approach to a Proof of the Riemann Hypothesis by the Second Mean-Value Theorem of Calculus
AUTHORS:
Alfred Wünsche
KEYWORDS:
Riemann Hypothesis, Riemann Zeta Function, Xi Function, Gauss-Bonnet Theorem, Mellin Transformation
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.6 No.13,
December
20,
2016
ABSTRACT: By the second mean-value theorem of calculus (Gauss-Bonnet theorem) we prove that the class of functionswith an integral representation of the formwith a real-valued functionwhich is non-increasing and decreases in infinity more rapidly than any exponential functions,possesses zeros only on the imaginary axis. The Riemann zeta functionas it is known can be related to an entire functionwith the samenon-trivial zeros as. Then after a trivial argument displacementwe relate it to a functionwith a representation of the formwhereis rapidly decreasing in infinity and satisfies all requirements necessary for the given proof of the position of its zeros on the imaginary axis z=iyby 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 ofon the imaginary axisz=iy for a whole class of functionwhich includes in this way the proof of the Riemann hypothesis. This whole class includes, in particular, also the modified Bessel functionsfor 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 functionsbelong 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.