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

By the second mean-value theorem of calculus (Gauss-Bonnet theorem) we prove that the class of functionswith 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 functionwith 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.